Research For Some Problems On Coupled Fixed Points Of Nonlinear Operators

This paper mainly study some coupled fixed point and ?-coupled fixed point theorems of mixed monotone operators in the generalized metric spaces.Moreover,the obtained results are applied to the solution of varied equations.This parper is composed of five chapters.In the first chapter,the origination?development?research issues and research significance of the mixed monotone operators and the generalized metric spaces are introduced,respectively.In addition,some related results and notions are given.In the second chapter,some common fixed point theorems of two operators are given.These results are obtained by exploring the corresponding initial value problems for a class of mixed monotone operators.As an application,the existence and uniqueness of solution to integro-differential equations is discussed.In the third chapter,a new ?-coupled fixed point theorem for a class of mixed monotone operators in metric spaces is given.To this end,the notions of ?-coupled fixed point and modified F-control function are introduced.These obtained results are applied to derive some ?-coupled fixed point theorems in partial metric spaces and quasi-metric spaces,respectively.Some applications to matrix equations and integral equations are also presented.In the fourth chapter,by means of a partial order ? and the mixed monotonicity of a operator ?,a ?-coupled fixed point theorem and its corollaries are obtained in b-metric spaces.And the existence of solution to integral equations is discussed.In the fifth chapter,by the characteristic of a partial order ?q in partial quasi-metric spaces,the notion of mixed monotone operators respecting to ?q is proposed.Furthermore,some coupled fixed point theorems are obtained and are applied to study the existence of solution to quadratic integral equations.