Study On The Fixed Points Theorems Of Nonlinear Operator And The Existence Of Positive Solutions Of Ordinary Diferential Equations

The aim of this article is, on basis of the partial order theory, to study the fixedpoints theorems of nonlinear operator and the existence of positive solution of ordinarydiferential equations in Banach space by using nonlinear functional analysis method.By deep study, we obtained many new results.The paper is divided into four chapters according to the contents:Chapter1Preference, we present a short background and history on boundaryvalue problems for the nonlinear functional analysis and the main work of the paper.Chapter2In this chapter, we mainly consider three class decreasing operators.In the section one, by using the famous Schauder fixed points theorem, we obtains afixed point theorem for decreasing operators. The result is applied to singular bound-ary value problems which has the decreasing nonlinearity on u, and the nonlinearitymay be singular u=0. The section two, we study a class of random fixed pointtheorems for decreasing operator without assuming any continuity and compactness.As applications, two examples of random integral equations are given. In the sectionthree, we present some existence and uniqueness theorems for ordered contractive mapin Banach lattices. Moreover, we present some applications to first-order ordinarydiferential equations with initial value conditions, proving the existence of a uniquesolution without the existence of a lower solution or upper solution.Chapter3In this chapter, we mainly consider three class mixed monotone oper-ators. In the section one, by introducing τ-ψ-mixed monotone operators in orderedBanach spaces, the existence and uniqueness theorems for the operators are obtainedand the new results are used to study the existence of positive solutions to second orderequations with Neumann boundary conditions. The section two, we discuss a class ofoperator equations. Without any continuity or compactness of the operators, the ex-istence and uniqueness theorems of solutions for the operator equations are obtained.In the end, we apply the new results presented in this paper to Initial problems ofOrdinary diferential equations. In the section three, without any continuity or com-pactness of the operators, the existence and uniqueness theorems of solutions for the operator equations are obtained. In the end, we apply the new results presented inthis paper to Hammerstein integral equations.Chapter4On basis of the partial order theory, this chapter studies the existenceof positive solution of ordinary diferential equations in Banach space by using nonlinearfunctional analysis method. In the section one, we deal with a class of singular secondorder Sturm Liouville boundary value problems by using fixed point theorem. A newresult on the existence of C_p~1[0,+∞) positive solution for the question is obtained. Thesection two, in case of not requiring f(t, u) to be nonnegative, we consider the positivesolutions for the second order boundary value problem by transforming the boundaryvalue problem into the integral equation systems and using fixed point theory in cones.New results on the existence of positive solution to the boundary value problem areguaranteed. In the section three, we consider some m-point boundary value problems ofsecond-order diferential equations. We give uniqueness and its exact expressions of thesolutions for the linear diferential equations with some m-point boundary conditions.As applications, we study the solution by iteration method for some nonlinear singularboundary value problems.