Existence Of Solutions For Semipositone Singular Differential And Integral Equations

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.