The Boundary Value Problems For Ordinary Differential Equations And The Fixed Point Theorems

The thesis consists of six chapters. The subjects of the paper are the boundary value problems for four-class nonlinear ordinary differential equations and the fixed point theorems of a class of mixed monotone operators and applications.In Chapter 1, we present a short background and history on boundary value problems for nonlinear ordinary differential equations and the main work of the paper.In Chapter 2, we consider the existence of solutions for a class of second-order differential equations with nonlinear boundary conditions. By a new definition of upper and lower solutions coupled with monotone iterative technique, we obtain the existence of extremal solutions for three-point boundary value problems. We improve some relevant results. As an application, an example is given to illustrate the results.Chapter 3 deals with the existence of symmetric positive solutions for a class of four-orderφ-Laplacian operator with integral boundary conditions. We obtain sufficient conditions for the existence, multiplicity, and nonexistence of symmetric positive solutions to the problem by employing the fixed point theorem of cone expansion and compression of norm type. The contributions of this paper are twofold. The first is that the function f depends on the first-order derivative of the unknown functions, and the second one is that the equation covers the two important cases whenφ(u)= u and p-Laplacian operatorφ>(u)=|u|p-2u, p> 1. Our paper improves and generalizes the related results to some degree. Moreover, we give an example to illustrate our main results.In Chapter 4 we focus on the existence of symmetric positive solutions for a class of higher-order differential equation with boundary value conditions. By applying the monotone iterative technique, we obtain a sufficient and necessary condition for the existence of at least one symmetric positive solution for the prob-lem, we also discuss the uniqueness, a iterative sequence and an error estimation for the symmetric positive solution to the problem. Moreover, an example will be presented to illustrate the applicability of our results.Chapter 5 concerns the existence of pseudo-symmetric positive solutions to a class of second-order boundary value problems. By applying the monotone itera-tive technique, we obtain a sufficient and necessary condition for the existence of at least one pseduo-symmetric positive solution for the problem, we also discuss the uniqueness, a iterative sequence and an error estimation for the pseduo-symmetric positive solution to the problem. An example will be presented to illustrate the applicability of our results.In Chapter 6 we study a class of mixed monotone operators defined in a partially ordered Banach space. We obtain the uniqueness and existence of fixed points of the operators, without assuming operators to be continuous or compact. Our conclusions extend and improve previous results. As an application, we prove the existence and uniqueness of a positive solution for a class of boundary value problems and a class of integral equations.