Multivariate Fixed Point Theory And Its Application

Recently,the fixed point theory of multivariate operators has attracted the atten-tion of many authors,this paper is mainly based on the multivariate fixed point,theorems for contractive mapping in Banach space which proposed by them.First,the concept of multivariate best approximation point is introduced in this paper,and the proof of its existence and uniqueness is given.It is a generalization of multivariate fixed point theorems.second,we also study a kind of system of N fixed point operator equations in metric spaces.The existence and uniqueness and the iterative algorithm of solution are studied,and some interest results are obtained.This system of fixed point equations is a generalized form of fixed point equation x = Tx.Furthermore,some applications of the system of fixed point equations are also given,such as the system of first order differential equations,the system of nonlinear equations,the system of linear equations and the system of integral equations.In addition,The concept of the homeomorphism metric space is firstly introduced,and the multiplicative metric space is a special form of the homeomorphism metric space.In the end,we also study the quasi-?-nonexpanse mapping and the generalized mixed equilibrium problem.A more generalized hy-brid shrinking projection algorithm for finding a common solution for a system of generalized mixed equilibrium problems was introduced in this paper.This paper includes five parts.In the first part,we introduce the significance of fixed point theory and its application in nonlinear functional analysis,recall the the history and present situation of fixed point theory of contraction mappings.In the second part,the concept of multivariate best proximity point is introduced,then we prove an existence and uniqueness theorems of the multivariate best proximity point in the complete metric spaces and give its iterative algorithm.The third part is to introduce the concept of the homeomorphism metric space.and to prove the fixed point theorems and the best proximity point theorems for generalized contractions in such spaces.The fourth part is to introduce and investigate a more generalized hybrid shrinking projection algorithm for finding a common solution for a system of generalized mixed equilibrium problem.A accelerated strong convergence theo-rem of common solutions is established in the framework of a non-uniformly convex Banach space.In the last part,we summarize the main content of this paper,and then analyze several aspects of future research.