一类二阶常微分方程组边值问题解的存在性与唯一性

一类二阶常微分方程组边值问题解的存在性与唯一性 Existence and Uniqueness of Solutions for Boundary Value Problems of a Class of Second-Order Ordinary Differential Equation System first-author 王 丹，女，硕士生，研究方向为非线性泛函分析。 E-mail： 1593274650@qq.com 王 丹，女，硕士生，研究方向为非线性泛函分析。 E-mail： 1593274650@qq.com 1593274650@qq.com corresp E-mail： liyx@nwnu.edu.cn E-mail： liyx@nwnu.edu.cn liyx@nwnu.edu.cn 西北师范大学 数学与统计学院， 甘肃 兰州 730070 西北师范大学 数学与统计学院， 甘肃 兰州 730070 College of Mathematics and Statistics， Northwest Normal University， Lanzhou 730070， Gansu， China College of Mathematics and Statistics， Northwest Normal University， Lanzhou 730070， Gansu， China 本文讨论二阶常微分方程组边值问题 <math id="M1"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><mi>u</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930557&type= 57.15000153 15.91733265 解的存在性与唯一性，其中 <math id="M2"><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>→</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930570&type= 37.25333405 3.47133350 连续。在非线性项 <math id="M3"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi mathvariant="normal">与</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930603&type= 33.44333267 4.14866638 关联的不等式条件，以及 <math id="M4"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi mathvariant="normal">与</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930603&type= 33.44333267 4.14866638 关于 <math id="M5"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930620&type= 1.60866666 3.13266683 满足Nagumo型增长条件下，运用Leray-Schauder不动点定理，获得了该问题解的存在性及唯一性。 This paper discusses the existence and uniqueness of solutions to the system of second-order ordinary differential equations boundary value problems <math id="M6"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><mi>u</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930637&type= 55.88000107 15.91733265 where <math id="M7"><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>→</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930647&type= 37.16867065 3.47133350 continuous. Applying the Leray-Schauder fixed point theorem, the existence and uniqueness of solutions is obtained under the condition that the nonlinear term <math id="M8"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930666&type= 14.73200035 4.14866638 and <math id="M9"><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930685&type= 16.25600052 4.14866638 satisfy the associated inequality condition, and <math id="M10"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930699&type= 14.73200035 4.14866638 and <math id="M11"><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930720&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930713&type= 14.98600006 4.14866638 satisfy the Nagumo-type growth condition respectively on <math id="M12"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930728&type= 1.60866666 3.13266683 . 二阶常微分方程组 边值问题 存在性与唯一性 Nagumo型增长条件 Leray-Schauder不动点定理 second⁃order ordinary differential equations system boundary value problem existence and uniqueness Nagumo-type growth condition Leray-Schauder fixed point theorem 1　 引言与主要结果 1 本文讨论二阶常微分方程组边值问题 <math display="block" id="M13"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><mi>u</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930739&type= 65.27799988 18.62666702 解的存在性与唯一性，其中， <math id="M14"><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>→</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930749&type= 43.51866913 4.06400013 连续。该问题在流体动力学及物理学等领域有重要的应用背景，对该问题不含导数项的一些特殊情形，已有许多学者进行了研究 ［ 1 1 ~ 11 11 1~11 ］ 。文献［ 1 1 ， 3 3 ， 5 5 ， 8 8 ~ 11 11 8~11 ］运用了锥上的不动点指数理论，文献［ 2 2 ， 6 6 ］运用了锥上不动点定理，以及文献［ 4 4 ， 7 7 ］运用单调迭代求解的方法等。 1993年Fink和Gatica ［ 1 1 ］ 运用锥上的不动点指数理论，在 <math id="M15"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930937&type= 1.10066664 3.64066648 与 <math id="M16"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932088&type= 1.94733346 3.64066648 严格单调条件下，获得边值问题 <math display="block" id="M17"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>x</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>λ</mi><mi>a</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi>f</mi><mfenced separators="|"><mrow><mi>x</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>y</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>y</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>λ</mi><mi>b</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi>g</mi><mfenced separators="|"><mrow><mi>x</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>y</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>x</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>x</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>y</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>y</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930792&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930785&type= 66.37866211 18.88066673 非负解的存在性。后来，Ma ［ 2 2 ］ 将上述条件弱化为单调条件，通过举例得到了边值问题（2）新的非负解的存在性结论。文献［ 6 6 ］研究了下列边值问题 <math display="block" id="M18"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><mi>u</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930802&type= 56.81133270 18.62666702 该问题的非线性项都是非负的，且在超线性或次线性的条件下获得了正解的存在性。 文献［ 10 10 ］研究了边值问题（1）不含导数项的特殊情形，在 <math id="M19"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932084&type= 1.10066664 3.64066648 与 <math id="M20"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932257&type= 1.94733346 3.64066648 非负的情形下，建立乘积锥上不动点指数的乘积公式，获得边值问题 <math display="block" id="M21"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mi>u</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930857&type= 56.81133270 18.62666702 正解的存在性。 以上文献都是在非线性项 <math id="M22"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930937&type= 1.10066664 3.64066648 与 <math id="M23"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932088&type= 1.94733346 3.64066648 各自独立的条件下，讨论不含导数项 <math id="M24"><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930913&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930908&type= 2.37066650 3.04800010 与 <math id="M25"><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930922&type= 2.37066650 3.04800010 的方程组非负解或正解的存在性，对于边值问题（1）含导数项的情形未见研究。本文对 <math id="M26"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930937&type= 1.10066664 3.64066648 与 <math id="M27"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932088&type= 1.94733346 3.64066648 是否非负不限制，在 <math id="M28"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932216&type= 17.27199936 4.82600021 与 <math id="M29"><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930991&type= 17.61066628 4.82600021 关联的不等式条件，以及 <math id="M30"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi mathvariant="normal">与</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931197&type= 39.11600113 4.82600021 关于 <math id="M31"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932223&type= 1.86266661 3.64066648 满足Nagumo型增长条件下，运用Leray-Schauder不动点定理，获得边值问题（1）解的存在性结果，讨论了解的唯一性。本文的主要结果如下： 定理1 设 <math id="M32"><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>→</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931035&type= 43.51866913 4.06400013 连续，若 <math id="M33"><mi>f</mi><mo>,</mo><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931218&type= 4.82600021 3.64066648 满足下列条件 （F1） 对 <math id="M34"><mo>∀</mo><mi>M</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931070&type= 12.61533356 2.96333337 ， 存在递增连续函数 <math id="M35"><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>:</mo><mfenced open="[" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mfenced><mo>→</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931085&type= 40.04733276 4.14866638 ，满足 <math display="block" id="M36"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mfrac><mrow><mi>ρ</mi><mi mathvariant="normal">d</mi><mi>ρ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mi>ρ</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931099&type= 29.80266571 9.56733322 使得 <math display="block" id="M37"><mfenced open="|" close="|" separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>p</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>p</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931116&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931111&type= 77.04666901 5.58799982 其中， <math id="M38"><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>≤</mo><mi>M</mi><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi></mrow></mfenced><mo>≤</mo><mi>M</mi><mo>,</mo><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931123&type= 53.34000015 4.82600021 。 （F2） 存在常数 <math id="M39"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931227&type= 21.50533485 3.47133350 且 <math id="M40"><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931144&type= 29.71800041 7.87400007 及 <math id="M41"><mi>e</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931154&type= 8.21266651 2.79399991 ，使得 <math display="block" id="M42"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced><mi>y</mi><mo>≤</mo><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931163&type= 90.33933258 4.82600021 其中， <math id="M43"><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931172&type= 54.27133560 4.82600021 则边值问题（1）至少存在一个解。 注1 在定理1中，条件（F2）允许 <math id="M44"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi mathvariant="normal">与</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931197&type= 39.11600113 4.82600021 关于 <math id="M45"><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931187&type= 9.31333351 3.72533321 超线性增长。而Nagumo型增长条件（F1）限制 <math id="M46"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi mathvariant="normal">与</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931197&type= 39.11600113 4.82600021 关于 <math id="M47"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932223&type= 1.86266661 3.64066648 至多二次增长。 加强定理1的条件（F2），则有如下存在唯一性结果： 定理2 设 <math id="M48"><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>→</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931211&type= 43.51866913 4.06400013 连续，若 <math id="M49"><mi>f</mi><mo>,</mo><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931218&type= 4.82600021 3.64066648 满足条件（F1）及下列条件 （F3） 存在常数 <math id="M50"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931227&type= 21.50533485 3.47133350 且 <math id="M51"><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931234&type= 29.71800041 7.87400007 ，使得 <math display="block" id="M52"><mfenced separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931240&type= 135.29733276 5.58799982 <math display="block" id="M53"><mi>a</mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931245&type= 84.58200073 5.58799982 其中， <math id="M54"><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931253&type= 77.47000122 4.91066647 则边值问题（1）存在唯一解。 2　 预备知识 1 记 <math id="M55"><mi>I</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mo>,</mo><mi>C</mi><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931261&type= 24.46866608 3.97933316 表示定义在 <math id="M56"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931931&type= 1.26999998 2.79399991 上的全体连续函数按范数 <math id="M57"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mi>C</mi></mrow></msub><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>t</mi><mo>∈</mo><mi>I</mi></mrow></munder><mfenced open="|" close="|" separators="|"><mrow><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931276&type= 28.95599937 5.41866684 构成的Banach空间。 <math id="M58"><mi mathvariant="normal">对</mi><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931286&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931281&type= 22.86000061 3.97933316 表示定义在 <math id="M59"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931931&type= 1.26999998 2.79399991 上的全体 <math id="M60"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931297&type= 1.86266661 2.79399991 阶连续可微函数按范数 <math id="M61"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>t</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy="false">{</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mi>C</mi></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>C</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>C</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931304&type= 58.67399979 5.50333309 构成的Banach空间。 <math id="M62"><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931311&type= 7.95866632 3.97933316 表示 <math id="M63"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931931&type= 1.26999998 2.79399991 上的全体Lebesgue平方可积函数按内积 <math id="M64"><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931324&type= 34.29000092 6.94266701 构成的Hilbert空间，其内积范数为 <math id="M65"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931334&type= 33.86666870 7.19666624 。记 <math id="M66"><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>=</mo><mo stretchy="false">{</mo><mi>u</mi><mo>∈</mo><mi>C</mi><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo stretchy="false">|</mo><mtext> </mtext><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931340&type= 32.34266663 3.97933316 在 <math id="M67"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931931&type= 1.26999998 2.79399991 上绝对连续，且 <math id="M68"><mi>u</mi><mi>'</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931355&type= 16.59466553 4.06400013 。 设 <math id="M69"><mi>X</mi><mo>,</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931362&type= 7.28133297 3.21733332 分别为范数是 <math id="M70"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mo>⋅</mo></mrow></mfenced></mrow><mrow><mi>X</mi></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mo>⋅</mo></mrow></mfenced></mrow><mrow><mi>Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931369&type= 15.07066536 5.07999992 的Banach空间。 <math id="M71"><mi>X</mi><mo>×</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931377&type= 10.58333302 2.79399991 表示 <math id="M72"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931384&type= 2.62466669 2.79399991 与 <math id="M73"><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931393&type= 2.87866688 2.79399991 的乘积空间按范数 <math id="M74"><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">{</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>X</mi></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>y</mi></mrow></mfenced></mrow><mrow><mi>Y</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931403&type= 44.87333298 6.26533365 构成的Banach空间。 引理1 设 <math id="M75"><mi>u</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931410&type= 14.56266594 3.97933316 满足边界条件 <math id="M76"><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931418&type= 28.53266525 3.97933316 则 <math id="M77"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931426&type= 23.70666695 7.53533268 。 证 正弦函数系 <math id="M78"><mo stretchy="false">{</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>k</mi><mi mathvariant="normal">π</mi><mi>t</mi><mo stretchy="false">|</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931437&type= 31.41133499 3.55599999 为 <math id="M79"><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931494&type= 7.95866632 3.97933316 中的完备直交系，故 <math id="M80"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931454&type= 1.86266661 2.79399991 可展为正弦级数 <math display="block" id="M81"><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>k</mi><mi mathvariant="normal">π</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931460&type= 30.31066895 8.38199997 其中， <math id="M82"><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>k</mi><mi mathvariant="normal">π</mi><mi>t</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>,</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931468&type= 53.34000015 6.94266701 ，且有Parseval等式 <math display="block" id="M83"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931477&type= 23.79133224 8.38199997 另一方面，余弦函数系 <math id="M84"><mo stretchy="false">{</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>k</mi><mi mathvariant="normal">π</mi><mi>t</mi><mo stretchy="false">|</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931485&type= 32.00400162 3.55599999 也为 <math id="M85"><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931494&type= 7.95866632 3.97933316 中的完备直交系，故 <math id="M86"><mi>u</mi><mi>'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931503&type= 2.70933342 2.79399991 可展为余弦级数 <math display="block" id="M87"><mi>u</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>k</mi><mi mathvariant="normal">π</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931515&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931512&type= 40.97866440 8.46666718 其中， <math id="M88"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi>u</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>=</mo><mn mathvariant="normal">0</mn><mo>;</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi>u</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>k</mi><mi mathvariant="normal">π</mi><mi>t</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>=</mo><mi>k</mi><mi mathvariant="normal">π</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931525&type= 100.24533844 6.94266701 ，又有Parseval等式 <math display="block" id="M89"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi><mi>'</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≥</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931532&type= 93.72599792 9.48266602 因此，引理1的结论成立。 引理2 设 <math id="M90"><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>I</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>→</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931546&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931539&type= 36.99933624 3.64066648 连续，满足条件（F1）。对 <math id="M91"><mo>∀</mo><mi>M</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>∃</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>M</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>M</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931551&type= 67.14067078 4.06400013 <math id="M92"><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931558&type= 8.04333401 3.21733332 使得边值问题（1）的解 <math id="M93"><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931562&type= 8.12800026 3.97933316 <math id="M94"><mi mathvariant="normal">满</mi><mi mathvariant="normal">足</mi><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><mi>M</mi><mi mathvariant="normal">时</mi><mo>,</mo><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>≤</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931569&type= 72.98266602 4.82600021 。 证 设 <math id="M95"><mi>M</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931574&type= 10.41399956 2.87866688 ，由（5）式存在 <math id="M96"><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931580&type= 17.94933319 4.14866638 ，使得 <math display="block" id="M97"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi>ρ</mi><mi mathvariant="normal">d</mi><mi>ρ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>></mo><mn mathvariant="normal">2</mn><mi>M</mi><mo>,</mo></mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi>η</mi><mi mathvariant="normal">d</mi><mi>η</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>></mo><mn mathvariant="normal">2</mn><mi>M</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931583&type= 56.38800049 8.55133343 设 <math id="M98"><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931588&type= 34.12067032 3.97933316 为边值问题（1）的解，且满足 <math id="M99"><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931593&type= 20.65866661 4.74133301 ，下证 <math id="M100"><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>≤</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931600&type= 39.11600113 4.82600021 。不妨设 <math id="M101"><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931606&type= 18.45733452 4.74133301 不恒等于0，则存在 <math id="M102"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced><mi mathvariant="normal">及</mi><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>∈</mo><mi>I</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931612&type= 38.35400009 4.06400013 ，使得 <math id="M103"><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,0</mn></mrow></mfenced><mo>,</mo><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931619&type= 125.89932251 6.85799980 。有以下8种情形： <math display="block" id="M104"><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mtext> </mtext><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931625&type= 47.49800110 4.82600021 ； <math display="block" id="M105"><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mtext> </mtext><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931631&type= 47.66733551 4.82600021 ； <math display="block" id="M106"><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo><mtext> </mtext><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931640&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931637&type= 47.92133331 4.82600021 ； <math display="block" id="M107"><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo><mtext> </mtext><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931643&type= 48.00600052 4.82600021 ； <math display="block" id="M108"><mn mathvariant="normal">5</mn><mo stretchy="false">)</mo><mtext> </mtext><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931647&type= 47.83666611 4.82600021 ； <math display="block" id="M109"><mn mathvariant="normal">6</mn><mo stretchy="false">)</mo><mtext> </mtext><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931653&type= 47.83666611 4.82600021 ； <math display="block" id="M110"><mn mathvariant="normal">7</mn><mo stretchy="false">)</mo><mtext> </mtext><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931658&type= 47.92133331 4.82600021 ； <math display="block" id="M111"><mn mathvariant="normal">8</mn><mo stretchy="false">)</mo><mtext> </mtext><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931665&type= 47.92133331 4.82600021 我们仅证明情形1）与情形5），其他情形类似可证。首先证明情形1），取 <math display="block" id="M112"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>∈</mo><mo stretchy="false">[</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931674&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931671&type= 64.17733002 5.50333309 则 <math id="M113"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">且</mi><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931676&type= 51.22333527 5.58799982 。当 <math id="M114"><mi>t</mi><mo>∈</mo><mfenced close="]" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">时</mi><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931685&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931682&type= 50.71533585 4.82600021 。因此 <math display="block" id="M115"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi>u</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mi>v</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931688&type= 85.85199738 12.61533356 所以，对 <math id="M116"><mo>∀</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931694&type= 17.18733406 4.82600021 ，有 <math display="block" id="M117"><mfrac><mrow><mi>u</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>≤</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mfrac><mrow><mi>v</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>≤</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931700&type= 62.39933777 9.90600014 对上式从 <math id="M118"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">到</mi><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931705&type= 8.55133343 4.40266657 积分，得 <math display="block" id="M119"><msubsup><mo>∫</mo><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi>u</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>≤</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>,</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi>v</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>≤</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931710&type= 93.81066895 9.90600014 令 <math id="M120"><mi>ρ</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>η</mi><mo>=</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931716&type= 28.95599937 4.06400013 ，有 <math display="block" id="M121"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mrow><mfrac><mrow><mi>ρ</mi><mi mathvariant="normal">d</mi><mi>ρ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mi>ρ</mi></mrow></mfenced></mrow></mfrac></mrow><mo>≤</mo><mi>u</mi><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>-</mo><mi>u</mi><mfenced separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">2</mn><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><mn mathvariant="normal">2</mn><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931722&type= 68.58000183 9.56733322 <math display="block" id="M122"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mrow><mfrac><mrow><mi>η</mi><mi mathvariant="normal">d</mi><mi>η</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mi>η</mi></mrow></mfenced></mrow></mfrac></mrow><mo>≤</mo><mi>v</mi><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>-</mo><mi>v</mi><mfenced separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">2</mn><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><mn mathvariant="normal">2</mn><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931727&type= 68.49533844 9.56733322 由（9）式知， <math id="M123"><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931733&type= 38.01533508 4.82600021 。因此， <math id="M124"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931818&type= 62.31467056 5.50333309 。所以 <math display="block" id="M125"><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced><mo>≤</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931824&type= 69.42666626 6.85799980 下面证明情形5），取 <math display="block" id="M126"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>∈</mo><mo stretchy="false">[</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931748&type= 63.16133118 5.50333309 则 <math id="M127"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">且</mi><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931754&type= 51.13866425 5.58799982 。当 <math id="M128"><mi>t</mi><mo>∈</mo><mfenced close="]" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">时</mi><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931762&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931759&type= 50.71533585 4.82600021 。因此 <math display="block" id="M129"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mi>u</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931765&type= 86.02133942 12.61533356 所以，对 <math id="M130"><mo>∀</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931770&type= 17.18733406 4.82600021 ，有 <math display="block" id="M131"><mo>-</mo><mfrac><mrow><mi>u</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>≤</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mo>-</mo><mfrac><mrow><mi>v</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>≤</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931779&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931776&type= 70.95066071 9.90600014 对上式从 <math id="M132"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">到</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931782&type= 8.55133343 4.40266657 积分，得 <math display="block" id="M133"><mo>-</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi>u</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>≤</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>,</mo><mo>-</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi>v</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>≤</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931788&type= 102.36200714 9.90600014 令 <math id="M134"><mi>ρ</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>η</mi><mo>=</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931793&type= 28.95599937 4.06400013 ，有 <math display="block" id="M135"><mo>-</mo><msubsup><mo>∫</mo><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mrow><mfrac><mrow><mi>ρ</mi><mi mathvariant="normal">d</mi><mi>ρ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mi>ρ</mi></mrow></mfenced></mrow></mfrac></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mrow><mfrac><mrow><mi>ρ</mi><mi mathvariant="normal">d</mi><mi>ρ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mi>ρ</mi></mrow></mfenced></mrow></mfrac></mrow><mo>≤</mo><mi>u</mi><mfenced separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>-</mo><mi>u</mi><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">2</mn><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><mn mathvariant="normal">2</mn><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931799&type= 96.35066223 9.56733322 <math display="block" id="M136"><mo>-</mo><msubsup><mo>∫</mo><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mrow><mfrac><mrow><mi>η</mi><mi mathvariant="normal">d</mi><mi>η</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mi>η</mi></mrow></mfenced></mrow></mfrac></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mrow><mfrac><mrow><mi>η</mi><mi mathvariant="normal">d</mi><mi>η</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mi>η</mi></mrow></mfenced></mrow></mfrac></mrow><mo>≤</mo><mi>v</mi><mfenced separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>-</mo><mi>v</mi><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">2</mn><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><mn mathvariant="normal">2</mn><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931805&type= 96.26599884 9.56733322 由（9）式知， <math id="M137"><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931812&type= 38.01533508 4.82600021 。因此， <math id="M138"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931818&type= 62.31467056 5.50333309 。所以 <math display="block" id="M139"><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced><mo>≤</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931824&type= 69.42666626 6.85799980 因此，引理2的结论成立。 3　 主要结果的证明 1 定理1的证明 对 <math id="M140"><mo>∀</mo><mi>h</mi><mo>∈</mo><mi>C</mi><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931830&type= 15.07066536 3.97933316 ，线性二阶边值问题 <math display="block" id="M141"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><mi>u</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>h</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931836&type= 32.51200104 11.00666618 存在唯一解 <math id="M142"><mi>u</mi><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi mathvariant="normal">G</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></mfenced><mi>h</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>s</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931841&type= 31.75000000 6.94266701 。其中 <math id="M143"><mi mathvariant="normal">G</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi>t</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>s</mi></mrow></mfenced><mo>,</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>s</mi><mo>≤</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mi>s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>t</mi></mrow></mfenced><mo>,</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931847&type= 52.06999969 10.92199993 为相应的Green函数。作积分算子 <math id="M144"><mi mathvariant="normal">A</mi><mo>:</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931853&type= 53.42466354 3.97933316 如下 <math display="block" id="M145"><mi mathvariant="normal">A</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi mathvariant="normal">G</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></mfenced><mi>f</mi><mfenced separators="|"><mrow><mi>s</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>,</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi mathvariant="normal">G</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></mfenced><mi>g</mi><mfenced separators="|"><mrow><mi>s</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">d</mi><mi>s</mi></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931859&type= 122.25867462 8.04333401 则 <math id="M146"><mi mathvariant="normal">A</mi><mo>:</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931864&type= 53.42466354 3.97933316 为全连续算子，边值问题（1）的解 <math id="M147"><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931873&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931870&type= 8.12800026 3.97933316 等价于算子 <math id="M148"><mi mathvariant="normal">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931887&type= 2.87866688 2.79399991 的不动点。我们对 <math id="M149"><mi mathvariant="normal">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931887&type= 2.87866688 2.79399991 应用Leray-Schauder不动点定理，证明 <math id="M150"><mi mathvariant="normal">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931887&type= 2.87866688 2.79399991 有不动点。为此，考查方程簇 <math display="block" id="M151"><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>=</mo><mi>λ</mi><mi mathvariant="normal">A</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>,</mo><mn mathvariant="normal">0</mn><mo><</mo><mi>λ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931891&type= 43.43399811 3.97933316 设 <math id="M152"><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931895&type= 34.12067032 3.97933316 为方程簇（11）中某个 <math id="M153"><mi>λ</mi><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931900&type= 14.13933277 3.97933316 对应的方程的解，则 <math display="block" id="M154"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi>u</mi><mo>=</mo><mi>λ</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi mathvariant="normal">G</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></mfenced><mi>f</mi><mfenced separators="|"><mrow><mi>s</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">d</mi><mi>s</mi></mrow></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mi>λ</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi mathvariant="normal">G</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></mfenced><mi>g</mi><mfenced separators="|"><mrow><mi>s</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">d</mi><mi>s</mi></mrow></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931903&type= 60.87533569 16.76399994 按线性方程解的Green函数表示， <math id="M155"><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931909&type= 34.12067032 3.97933316 满足方程 <math display="block" id="M156"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mi>u</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>λ</mi><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>λ</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931913&type= 59.18199921 18.62666702 将方程（13）第一式和第二式的两边分别同乘以 <math id="M157"><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">和</mi><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931919&type= 16.67933464 3.97933316 ，两式相加，由条件（F2），得 <math display="block" id="M158"><mo>-</mo><mi>u</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><mi>v</mi><mi>'</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>λ</mi><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>λ</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931923&type= 132.92666626 4.74133301 <math display="block" id="M159"><mi>λ</mi><mfenced separators="|"><mrow><mi>a</mi><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>b</mi><msup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>c</mi><msup><mrow><mi>u</mi><mi>'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>d</mi><msup><mrow><mi>v</mi><mi>'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>e</mi></mrow></mfenced><mo>≤</mo><mi>a</mi><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>b</mi><msup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>c</mi><msup><mrow><mi>u</mi><mi>'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>d</mi><msup><mrow><mi>v</mi><mi>'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>e</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931927&type= 140.46200562 4.74133301 对上式在 <math id="M160"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931931&type= 1.26999998 2.79399991 上积分，左边利用分部积分和方程（13）的边界条件，得 <math display="block" id="M161"><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≤</mo><mi>a</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>b</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>c</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>d</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>e</mi><mo>≤</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mfenced separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mfenced separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><mo>+</mo><mi>e</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931935&type= 89.57733154 13.20800018 由引理1， <math id="M162"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi><mi>'</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi><mi>'</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931939&type= 49.36066818 7.53533268 ，有 <math display="block" id="M163"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≤</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow></mfrac><mo>∶</mo><mo>=</mo><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931943&type= 68.07199860 13.71599960 所以 <math display="block" id="M164"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≤</mo><mroot><mrow><mfrac><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow></mfrac></mrow><mrow/></mroot><mo>=</mo><mroot><mrow><mi>M</mi></mrow><mrow/></mroot><mo>,</mo><mtext>  </mtext><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≤</mo><mroot><mrow><mfrac><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow></mfrac></mrow><mrow/></mroot><mo>=</mo><mroot><mrow><mi>M</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931949&type= 126.40732574 15.07066536 因此，对 <math id="M165"><mo>∀</mo><mi>t</mi><mo>∈</mo><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931953&type= 8.55133343 3.13266683 ，有 <math display="block" id="M166"><mfenced open="|" close="|" separators="|"><mrow><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mi>u</mi><mi>'</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>s</mi></mrow></mrow></mfenced><mo>≤</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>u</mi><mi>'</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>≤</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi><mi>'</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mroot><mrow><mi>M</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931957&type= 79.75599670 8.04333401 <math display="block" id="M167"><mo stretchy="false">|</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo stretchy="false">|</mo><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mi>v</mi><mi>'</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>s</mi></mrow></mrow></mfenced><mo>≤</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>v</mi><mi>'</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>≤</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi><mi>'</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mroot><mrow><mi>M</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931962&type= 78.40132904 8.04333401 所以 <math id="M168"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><mroot><mrow><mi>M</mi></mrow><mrow/></mroot><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><mroot><mrow><mi>M</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931966&type= 39.62400055 5.58799982 ,故 <math id="M169"><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced><mo>≤</mo><mroot><mrow><mi>M</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931970&type= 54.27133560 6.85799980 。 对边值问题（13），相应的非线性项满足 <math display="block" id="M170"><mtable columnalign="left"><mtr><mtd><mfenced open="|" close="|" separators="|"><mrow><mi>λ</mi><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><mfenced open="|" close="|" separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mo stretchy="false">|</mo><mi>p</mi><mo stretchy="false">|</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="|" close="|" separators="|"><mrow><mi>λ</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><mfenced open="|" close="|" separators="|"><mrow><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub><mfenced separators="|"><mrow><mo stretchy="false">|</mo><mi>p</mi><mo stretchy="false">|</mo></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931974&type= 63.92333221 13.03866673 所以 <math id="M171"><mi>λ</mi><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi mathvariant="normal">与</mi><mi>λ</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi mathvariant="normal">满</mi><mi mathvariant="normal">足</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931978&type= 50.54600143 4.82600021 （F1）。由引理2，存在 <math id="M172"><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>M</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>M</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931982&type= 57.99666977 4.06400013 ，使得 <math display="block" id="M173"><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>≤</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>≔</mo><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931986&type= 46.22800064 4.82600021 因此 <math display="block" id="M174"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup></mrow></msub><mo>,</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup></mrow></msub></mrow></mfenced><mo>≤</mo><mroot><mrow><mi>M</mi></mrow><mrow/></mroot><mo>+</mo><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931990&type= 65.36266327 6.85799980 即方程簇（11）的解集在 <math id="M175"><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931997&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931994&type= 21.59000015 3.97933316 中有界。由Leray-Schauder不动点定理可知， <math id="M176"><mi mathvariant="normal">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931999&type= 2.87866688 2.79399991 在 <math id="M177"><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932004&type= 21.59000015 3.97933316 中有不动点，该不动点为边值问题（1）的解。 定理2的证明 先证（F3） <math id="M178"><mo>⇒</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932009&type= 3.72533321 2.79399991 （F2）。对 <math id="M179"><mo>∀</mo><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>∈</mo><mi>I</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932015&type= 54.10200119 4.82600021 ，在（F3）中，取 <math id="M180"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>x</mi><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>p</mi><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>q</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932020&type= 82.80400085 4.57200003 令 <math id="M181"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>t</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy="false">{</mo><mo stretchy="false">|</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo stretchy="false">|</mo><mo>,</mo><mo stretchy="false">|</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo stretchy="false">|</mo><mo stretchy="false">}</mo><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932026&type= 67.73333740 4.99533367 ，则由条件（F3），有 <math display="block" id="M182"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced><mi>y</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932033&type= 49.44533539 4.82600021 <math display="block" id="M183"><mfenced separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow></mfenced><mi>x</mi><mo>+</mo><mfenced separators="|"><mrow><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow></mfenced><mi>y</mi><mo>+</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mi>x</mi><mo>+</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mi>y</mi><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932038&type= 149.86000061 5.58799982 <math display="block" id="M184"><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">|</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mi>x</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mi>y</mi><mo stretchy="false">|</mo><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932043&type= 90.33933258 4.06400013 <math display="block" id="M185"><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932047&type= 61.55266953 4.91066647 <math display="block" id="M186"><mtable columnalign="left"><mtr><mtd><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfrac><mrow><mroot><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>⋅</mo><mfrac><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mroot><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow/></mroot></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">2</mn><mo>⋅</mo><mfrac><mrow><mroot><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi></mrow></mfenced><mo>⋅</mo><mfrac><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mroot><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow/></mroot></mrow></mfrac><mo>≤</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932052&type= 143.08666992 25.48466682 <math display="block" id="M187"><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow></mfrac><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932058&type= 132.92666626 20.23533440 <math display="block" id="M188"><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow></mfenced><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfenced separators="|"><mrow><mi>b</mi><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow></mfenced><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932064&type= 135.72065735 20.57399940 令 <math display="block" id="M189"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>a</mi><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>b</mi><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>c</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>d</mi><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932069&type= 114.46933746 18.79599953 则由上式，可得 <math display="block" id="M190"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced><mi>y</mi><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>∈</mo><mi>I</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>×</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932075&type= 144.77999878 4.82600021 因为 <math id="M191"><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932080&type= 71.03533173 11.93799973 ，所以 <math id="M192"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932084&type= 1.10066664 3.64066648 与 <math id="M193"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932088&type= 1.94733346 3.64066648 满足条件（F2）。由定理1可知，边值问题（1）至少存在一个解。 再证唯一性。设 <math id="M194"><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>,</mo><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>×</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>I</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932092&type= 49.10666656 4.82600021 为边值问题（1）的两个解，则有方程 <math display="block" id="M195"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932097&type= 61.55266953 19.64266586 <math display="block" id="M196"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932102&type= 61.55266953 19.64266586 将方程（15）第一式与方程（14）第一式相减，方程（15）第二式与方程（14）第二式相减，得 <math display="block" id="M197"><mo>-</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932108&type= 113.28399658 6.34999990 <math display="block" id="M198"><mo>-</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932112&type= 113.62266541 6.34999990 将方程（16）两边同时乘以 <math id="M199"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932116&type= 20.06599998 4.06400013 ，方程（17）两边同时乘以 <math id="M200"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932120&type= 19.98133469 4.06400013 ，两式相加，由条件（F3），得 <math display="block" id="M201"><mo>-</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>-</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal"></mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932126&type= 106.34132385 6.34999990 <math display="block" id="M202"><mfenced separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932131&type= 208.87266541 5.58799982 <math display="block" id="M203"><mi>a</mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo><mi>t</mi><mo>∈</mo><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932135&type= 128.94734192 5.58799982 对上式在 <math id="M204"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932139&type= 1.26999998 2.79399991 上积分，左边利用分部积分和方程（14）、（15）的边界条件，得 <math display="block" id="M205"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≤</mo><mi>a</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>b</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>c</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>d</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932144&type= 135.29733276 6.09600019 <math display="block" id="M206"><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mfenced separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mfenced separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932150&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932148&type= 113.62266541 7.02733374 <math display="block" id="M207"><mfenced separators="|"><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mfenced separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932153&type= 65.95532990 8.97466660 即 <math display="block" id="M208"><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfenced separators="|"><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932159&type= 82.97332764 8.97466660 又因为 <math id="M209"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932165&type= 44.11133575 5.24933338 ，故 <math display="block" id="M210"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932170&type= 44.02666473 5.24933338 所以， <math id="M211"><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932176&type= 39.11600113 4.40266657 即 <math id="M212"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932182&type= 43.01066971 4.14866638 其中 <math id="M213"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">为</mi><mi mathvariant="normal">常</mi><mi mathvariant="normal">数</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932187&type= 20.74333191 4.48733330 。由边界条件得， <math id="M214"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932193&type= 19.38866615 3.72533321 。因此 <math id="M215"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932199&type= 24.04533195 4.14866638 ，所以边值问题（1）存在唯一解。 例1 考虑如下二阶微分方程组 <math display="block" id="M216"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><mi>u</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">6</mn><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><msup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><msup><mrow><mi>u</mi><mi>'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mo>″</mo><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mo>-</mo><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><msup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>-</mo><mi>v</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><msup><mrow><mi>v</mi><mi>'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>v</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932204&type= 104.81732941 17.10266685 方程（18）相应的非线性项为 <math display="block" id="M217"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">6</mn><mi>x</mi><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mi>x</mi><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced><mo>=</mo><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>y</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>y</mi><mo>-</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mi>y</mi><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932211&type= 120.22666168 4.82600021 易见 <math id="M218"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932216&type= 17.27199936 4.82600021 关于 <math id="M219"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932223&type= 1.86266661 3.64066648 二次增长， <math id="M220"><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932231&type= 17.61066628 4.82600021 关于 <math id="M221"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932239&type= 1.86266661 3.64066648 二次增长，满足条件（F1）。 因为 <math id="M222"><mi>f</mi><mi mathvariant="normal">与</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932247&type= 4.82600021 3.97933316 <math id="M223"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932257&type= 1.94733346 3.64066648 不是非负的，文献［ 1 1 ~ 11 11 1~11 ］的结果对边值问题（18）不适用。下面验证 <math id="M224"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932264&type= 17.27199936 4.82600021 与 <math id="M225"><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932272&type= 17.61066628 4.82600021 满足条件（F2）。取 <math id="M226"><mi>a</mi><mo>=</mo><mn mathvariant="normal">6</mn><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><mo>,</mo><mi>b</mi><mo>=</mo><mn mathvariant="normal">2</mn><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><mo>,</mo><mi>c</mi><mo>=</mo><mi>d</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>e</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn><mtext> </mtext></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932280&type= 87.46067047 7.95866632 ，则 <math id="M227"><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">8</mn></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932287&type= 46.22800064 7.95866632 。由（19）式可得 <math display="block" id="M228"><mtable columnalign="left"><mtr><mtd><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced><mi>y</mi><mo>=</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>x</mi><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>-</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>t</mi><mi>y</mi><mo>≤</mo><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi></mrow></mfenced><mo>=</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932296&type= 134.11199951 11.51466751 <math display="block" id="M229"><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfrac><mrow><mroot><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>⋅</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow/></mroot></mrow></mfrac><mo>+</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfrac><mrow><mroot><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi></mrow></mfenced><mo>⋅</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow/></mroot></mrow></mfrac><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932305&type= 111.08267212 9.73666668 <math display="block" id="M230"><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow></mfrac><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932313&type= 90.76266479 7.95866632 <math display="block" id="M231"><mfenced separators="|"><mrow><mn mathvariant="normal">6</mn><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow></mfenced><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow></mfenced><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn></mrow></mfrac><mo>=</mo><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932323&type= 95.33467102 9.22866726 因此， <math id="M232"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi></mrow></mfenced><mi mathvariant="normal">与</mi><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>q</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932330&type= 39.11600113 4.82600021 满足条件（F2）。由定理1可知，方程（18）有解。 参 考 文 献 FINK A M ， GATICA J A . Positive solutions of second order systems of boundary value problems ［J］. Journal of Mathematical Analysis and Applications ， 1993 ， 180 （ 1 ）： 93 - 108 . DOI： 10.1006/jmaa.1993.1385 http://dx.doi.org/10.1006/jmaa.1993.1385 . Positive solutions of second order systems of boundary value problems MA R Y . Multiple nonnegative solutions of second-order systems of boundary value problems ［J］. Nonlinear Analysis： Theory ， Methods & Applications， 2000 ， 42 （ 6 ）： 1003 - 1010 . DOI： 10.1016/s0362-546x（99）00152-2 http://dx.doi.org/10.1016/s0362-546x（99）00152-2 . Multiple nonnegative solutions of second-order systems of boundary value problems ZHOU Y M ， XU Y . Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations ［J］. Journal of Mathematical Analysis and Applications ， 2006 ， 320 （ 2 ）： 578 - 590 . DOI： 10.1016/j.jmaa.2005.07.014 http://dx.doi.org/10.1016/j.jmaa.2005.07.014 . Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations WU L B ， SUN T ， HE X Q . The existence and uniqueness of solutions to systems of second-order ordinary differential equations boundary value problem in absract space ［J］. Chinese Quarterly Journal of Mathematics ， 2011 ， 26 （ 4 ）： 573 - 577 . DOI: 10.1360/012010-187 http://dx.doi.org/10.1360/012010-187 . The existence and uniqueness of solutions to systems of second-order ordinary differential equations boundary value problem in absract space YANG Z L ， WANG X M ， LI H Y . Positive solutions for a system of second-order quasilinear boundary value problems ［J］. Nonlinear Analysis ， 2020 ， 195 ： 111749 . DOI： 10.1016/j.na.2020.111749 http://dx.doi.org/10.1016/j.na.2020.111749 . Positive solutions for a system of second-order quasilinear boundary value problems 杨志林 ， 孙经先 . 非线性二阶常微分方程组边值问题的正解 ［J］. 数学学报 ， 2004 ， 47 （ 1 ）： 111 - 118 . DOI： 10.3321/j.issn： 0583-1431.2004.01.016 http://dx.doi.org/10.3321/j.issn：0583-1431.2004.01.016 . 非线性二阶常微分方程组边值问题的正解 YANG Z L ， SUN J X . Positive solutions of boundary value problems for systems of nonlinear second order ordinary differential equations ［J］. Acta Mathematica Sinica ， 2004 ， 47 （ 1 ）： 111 - 118 . DOI： 10.3321/j.issn： 0583-1431.2004.01.016（Ch http://dx.doi.org/10.3321/j.issn：0583-1431.2004.01.016（Ch ）. Positive solutions of boundary value problems for systems of nonlinear second order ordinary differential equations 刘炳妹 ， 刘立山 . 二阶方程组解的存在唯一性 ［J］. 工程数学学报 ， 2007 ， 24 （ 4 ）： 757 - 760 . DOI： 10.3969/j.issn.1005-3085.2007.04.028 http://dx.doi.org/10.3969/j.issn.1005-3085.2007.04.028 . 二阶方程组解的存在唯一性 LIU B M ， LIU L S . The unique solution for systems of second-order equations ［J］. Chinese Journal of Engineering Mathematics ， 2007 ， 24 （ 4 ）： 757 - 760 . DOI： 10.3969/j.issn.1005-3085.2007.04.028（Ch http://dx.doi.org/10.3969/j.issn.1005-3085.2007.04.028（Ch ）. The unique solution for systems of second-order equations 程锡友 . “超-次线性”二阶椭圆型方程组的正解 ［J］. 兰州大学学报（自然科学版） ， 2008 ， 44 （ 3 ）： 113 - 117 . DOI: 10.3901/JME.2008.06.095 http://dx.doi.org/10.3901/JME.2008.06.095 . “超-次线性”二阶椭圆型方程组的正解 CHENG X Y . Positive solutions for super-sublinear elliptic system of second order ［J］. Journal of Lanzhou University （Natural Sciences） ， 2008 ， 44 （ 3 ）： 113 - 117 （Ch） . DOI: 10.3901/JME.2008.06.095 http://dx.doi.org/10.3901/JME.2008.06.095 . Positive solutions for super-sublinear elliptic system of second order CHENG X Y ， ZHONG C K . Existence of positive solutions for a second-order ordinary differential system ［J］. Journal of Mathematical Analysis and Applications ， 2005 ， 312 （ 1 ）： 14 - 23 . DOI： 10.1016/j.jmaa.2005.03.016 http://dx.doi.org/10.1016/j.jmaa.2005.03.016 . Existence of positive solutions for a second-order ordinary differential system CHENG X Y . Existence of positive solutions for a class of second-order ordinary differential systems ［J］. Nonlinear Analysis： Theory ， Methods & Applications， 2008 ， 69 （ 9 ）： 3042 - 3049 . DOI： 10.1016/j.na.2007.08.074 http://dx.doi.org/10.1016/j.na.2007.08.074 . Existence of positive solutions for a class of second-order ordinary differential systems CHENG X Y ， ZHANG Z T . Positive solutions for a multi-parameter system of second-order ordinary differential equations ［J］. Science China Mathematics ， 2011 ， 54 （ 5 ）： 959 - 972 . DOI： 10.1007/s11425-011-4213-x http://dx.doi.org/10.1007/s11425-011-4213-x . Positive solutions for a multi-parameter system of second-order ordinary differential equations 10.14188/j.1671-8836.2021.0035 O175.15 A 1671-8836（2021）05-0433-08 WANG Dan，LI Yongxiang.Existence and Uniqueness of Solutions for Boundary Value Problems of a Class of Second-Order Ordinary Differential Equation System ［J］. J Wuhan Univ （ Nat Sci Ed ），2021，67（5）：433-440. DOI： 10.14188/j.1671-8836.2021.0035（Ch http://dx.doi.org/10.14188/j.1671-8836.2021.0035(Ch ）. WANG Dan，LI Yongxiang.Existence and Uniqueness of Solutions for Boundary Value Problems of a Class of Second-Order Ordinary Differential Equation System ［J］.J Wuhan Univ（Nat Sci Ed），2021，67（5）：433-440. DOI：10.14188/j.1671-8836.2021.0035（Ch）. 王丹,李永祥.一类二阶常微分方程组边值问题解的存在性与唯一性[J].武汉大学学报（理学版）,2021,67(5):433-440. DOI:10.14188/j.1671-8836.2021.0035 王丹,李永祥.一类二阶常微分方程组边值问题解的存在性与唯一性[J].武汉大学学报（理学版）,2021,67(5):433-440. DOI:10.14188/j.1671-8836.2021.0035 国家自然科学基金（11661071；12061062） 2021-02-20 2021-10-24 2021-10-12 武汉大学学报（理学版） 5 67