奇异半正微分方程与积分方程正解的存在性
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摘要
非线性奇异微分方程边值问题与奇异积分方程问题是方程理论中的重要课题,是科学研究和解决技术问题的主要工具,具有广泛的应用,它的丰富理论和先进方法为解决当今科技领域中层出不穷的非线性问题提供了富有成效的理论工具,处理实际问题时发挥着不可替代的作用,对于这类方程的求解也因此成为了研究的热点和难点之一.本人在前人研究的基础上,受到Pedro J.Torres论文的启示,在储继峰已经对二阶半线性奇异方程周期解存在性作了一定工作的情况下,进行进一步的讨论和研究,利用不动点定理证明出了弱奇性条件下奇异微分方程周期正解的存在性,以及奇异积分方程的正解的存在性.这里我们强调的是,弱奇性有助于周期解的存在.
全文共分三章.
第一章,简要综述微分方程,积分方程的发展历史,并且给出了本文要用到的相关知识内容,介绍本文的主要工作.
第二章,首先利用Schauder不动点定理证明了半正情形下奇异微分方程周期解的存在性以及二阶非自治奇异耦合系统周期解的存在性；其次,利用Leray-Schauder二择一原则和锥不动点定理证明带有非线性扰动Hill方程周期正解的存在性和多重性,给出具体例子,对于储继峰教授之前所研究的弱排斥Hill方程多重正周期解存在性给出更具普遍性的例子进行解释说明.
第三章,首先在弱奇性条件下利用Schauder不动点定理对奇异积分方程和耦合方程组正解的存在性进行讨论；其次我们利用锥不动点定理和Leray-Schauder非线性二择一定理,讨论了在正的情形和半正情形下奇异积分方程多重正解的存在性；最后举例加以说明.
The singular boundary problems of nonlinear diferential equations and inte-gral equations are important subjects in the theory of equations. With a wide rangeof applications, it is the main tool for scientific research and solving the engineeringtechnique problems. Its rich theory and advanced methods to solve the nonlinearproblem of the modern science and technology field provide an efective theoreti-cal tool.Also it plays an irreplaceable role in dealing with practical problems. Tosolving this kind of equations becomes one of hot and difcult researches. On thebasis of previous studies and Pedro J.Torres’ paper, I try my best to discuss andresearch deeply on Chu Jifeng’s work of periodic problem for second-order semilin-ear singular equation. We will prove the existence of periodic solutions for singulardiferential equations and positive solution to integral equations in some suitableweak singularities by using the fixed point theorem.
The whole contents is divided into three chapters.
In chapter1, as the beginning of this paper, we give a brief overview to thedevelopment and background of diferential equations, integral equations. Then,we ofered some relative knowledge. Moreover, we introduce the main results ofthis paper.
In chapter2,as the second part of this thesis, we study the existence of posi-tive solutions to second order singular periodic diferential equations. Second, weestablish the existence of periodic solutions for the second order non-autonomoussingular coupled systems. The proof relies on Schauder’s fixed point theorem. Atlast, by using the Leray-Schauder nonlinear alternative theorem and Krasnoselskiifixed point theorem in cones, we establish the existence of multiple positive solu-tions to the Hill’s equations with singular perturbations. Here, we present morerepresentative examples. Diferent from former literatures, in some suitable weaksingularities, it may help the existence of periodic solutions.
In chapter3, as one of the main parts of this thesis, we state the applications ofSchauder’s fixed point theorem to semipositone singular integral equations. Then,by using the Leray-Schauder nonlinear alternative theorem and Krasnoselskii fixedpoint theorem in cones, we discussed the multiple positive solutions to singular posi-tone and semipositone problems of integral equation. Next, we will use Schauder’sfixed point theorem to obtain the existence of positive solutions of singular integralcoupled systems. At last, give some examples to illustrate theorems.
全文共分三章.
第一章,简要综述微分方程,积分方程的发展历史,并且给出了本文要用到的相关知识内容,介绍本文的主要工作.
第二章,首先利用Schauder不动点定理证明了半正情形下奇异微分方程周期解的存在性以及二阶非自治奇异耦合系统周期解的存在性；其次,利用Leray-Schauder二择一原则和锥不动点定理证明带有非线性扰动Hill方程周期正解的存在性和多重性,给出具体例子,对于储继峰教授之前所研究的弱排斥Hill方程多重正周期解存在性给出更具普遍性的例子进行解释说明.
第三章,首先在弱奇性条件下利用Schauder不动点定理对奇异积分方程和耦合方程组正解的存在性进行讨论；其次我们利用锥不动点定理和Leray-Schauder非线性二择一定理,讨论了在正的情形和半正情形下奇异积分方程多重正解的存在性；最后举例加以说明.
The singular boundary problems of nonlinear diferential equations and inte-gral equations are important subjects in the theory of equations. With a wide rangeof applications, it is the main tool for scientific research and solving the engineeringtechnique problems. Its rich theory and advanced methods to solve the nonlinearproblem of the modern science and technology field provide an efective theoreti-cal tool.Also it plays an irreplaceable role in dealing with practical problems. Tosolving this kind of equations becomes one of hot and difcult researches. On thebasis of previous studies and Pedro J.Torres’ paper, I try my best to discuss andresearch deeply on Chu Jifeng’s work of periodic problem for second-order semilin-ear singular equation. We will prove the existence of periodic solutions for singulardiferential equations and positive solution to integral equations in some suitableweak singularities by using the fixed point theorem.
The whole contents is divided into three chapters.
In chapter1, as the beginning of this paper, we give a brief overview to thedevelopment and background of diferential equations, integral equations. Then,we ofered some relative knowledge. Moreover, we introduce the main results ofthis paper.
In chapter2,as the second part of this thesis, we study the existence of posi-tive solutions to second order singular periodic diferential equations. Second, weestablish the existence of periodic solutions for the second order non-autonomoussingular coupled systems. The proof relies on Schauder’s fixed point theorem. Atlast, by using the Leray-Schauder nonlinear alternative theorem and Krasnoselskiifixed point theorem in cones, we establish the existence of multiple positive solu-tions to the Hill’s equations with singular perturbations. Here, we present morerepresentative examples. Diferent from former literatures, in some suitable weaksingularities, it may help the existence of periodic solutions.
In chapter3, as one of the main parts of this thesis, we state the applications ofSchauder’s fixed point theorem to semipositone singular integral equations. Then,by using the Leray-Schauder nonlinear alternative theorem and Krasnoselskii fixedpoint theorem in cones, we discussed the multiple positive solutions to singular posi-tone and semipositone problems of integral equation. Next, we will use Schauder’sfixed point theorem to obtain the existence of positive solutions of singular integralcoupled systems. At last, give some examples to illustrate theorems.
引文
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[17] X.Ding, J.Jiang. Multiple periodic solutions in delayed Gause-type ratiodependentpredator-Prey systems with nonmonotonic numerical responses[J]. Mathematicaland Computer Modelling,2008,47:1323-1331.
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[20] D.Jiang, J.Chu,M.Zhang. Multiplicity of positive periodic solutions to Superlinearrepulsive singular equations[J]. Diferential Equations,2005,211:282-302.
[21] Z.Jing. Periodic solutions of seeond order diferential equations[J]. Kexue Tongbao,1981,26(16):964-967.
[22] Z.Jing. On the existence of periodic solutions of the second-order diferntial equa-tions[J], Acta Math. Appl. Sinica,1982,5(1):15-18.
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