Solvability Of Nonlinear Operator Equations (Systems) And Its Applications

Many nonlinear equations arising in theory and applications, including nonlinear differential equations, nonlinear intergral equations,nonlinear difference equations and functional equations, may be of different types in form, but they can be transformed into the same operator equantion in essnce.Hence,when the solvability of one operator equation is proved,the existence results of many equations are obtained .There arise many techniques in the study of existence of solution for operator equations,such as topological transversality method,upper and lower solutions method,monotone iterative method, and variation method,and bountiful results are obtained. These methods and results have been successfully applied to various of nonlinear equations.The aim of this paper is to study the solvability of some operator equations by using partial order method.Firstly, the solvability of an operator equation is considered The existence results are obtained and then applied to some boundary value problems of differential equations and intergral-differential equations.Further ,we extend this research to the case of operator equations systems;Due to the wide applications of setvalued operators in modern mathematics,the solvability of the setvalued operator equations are studied. At last,the existence results of solutions for some third order ordinary differential equations are obtained , by means of transforming them into corresponding operator equations in proper spaces.The main results are listed in the following: The concept of order continuous operator is introduced,then partial order method is employed to establish some existence results for an operator equation in metric space and Banach space respectively. Iterative sequences are also constructed,which converge to the solutions of the operator equation. When the operators are not continuous,existence results for the operator equation are obtained by the use of partial order method and Zorn's Lemma. The solvability of operator equations systems is studied.When the operators sat-isfy some monotone conditions, the existence results for the operator equations systems are verified, and,iterative sequences are constructed,which converge to the solutions of the operator equations systems. Some set-valued increasing operators are defined and fixed point theorems for these operators are proved in metric space and Banach space respectively. The study of existence of fixed point for contractive operators is extended to the set-valued case, and some fixed point theorems for set-valued contractive operators are proved. Caristi's fixed point theorem is generalized to the setvalued case,some Caristi's fixed point theorems for set-valued operators are presented. The existence of three positive solutions for boundary value problem of third order differential equation is studied. Some new maximum principals are proved and then employed to prove the existence of solution and positive solution for two boundary value problems of third order differential equations.