Types Of Nonlinear Operator Of The Son, And Equation

Nonlinear functional analysis is organic combined of nonlinear problem and functional analysis, it is an important branch of mathmatics, and it can explain several kinds of natural phenomena, so it is highly recognised as one of the most active fields in analyse mathematics at present. Among them, the fixed point of operaters and the solutions of differential equations come from a lot of branches of applied mathematics and physics, and it is very meaningful in both practical application and theoretical study aspects. In the present paper, we employed the cone theory, fixed point theory, topological degree theory, the property of connected sets and upper-lower solutions method and so on to investigate the existence of solutions to some kinds of nonlinear operators and differential equations. The results obtained are either new or essentially generalize and improved previous relevant ones under weaker conditions.The thesis is divided into three chapters according to contents as following:In Chapter 1, by using the Banach contraction mapping principle and the theory of cone, under more general conditions, the existence and uniqueness theorems of solutions for a class of systems of nonlinear non-monotone abstract two binary operator equationsin Banach spaces are studied. The results presented here have improved and unified many known results. As an application, by using our theorems, the existence and uniqueness of solutions for systems of nonlinear non-monotone binary operator integral equations both of which were mixed type in Banach spaces are given.In Chapter 2, on the base of Chapter 1, by using the Banach contraction mapping principle and the theory of cone, the existence and uniqueness theorems of quasi-fixed point for a class of abstract two binary operator in Banach spaces are obtained without the conditions of continuous or compact. The results presented in this charter are different from the ones in Charter 1 and have improved and unified many known results.In Chapter 3, by using the method of properties of the connectivity and the method of upper and lower solutions with opposite order, the existence of solutions for the following multi-point boundary value problem at resonancewas investigated, where f :[0,1]×R×Râ†'R be a continuous function,a:[0,1]â†'R be a continuous function.α∈[0,+∞),β∈(1,+∞),ξ∈[0,1),η∈(0,1),andβ(η+α)=1+α,1-2ξ>βη(η-2ξ).