Existence And Construction Of The Fixed Points In B-metric Space

The fixed point theory has already constituted one of the core contents of functional analysis,and has been widely used in algebraic equation,differential equation,integral equation,mathematical economics and other fields.This paper mainly studies the fixed point problems in three kinds of metric spaces.The first is topological space,in which the basic properties of the topological degree of set-valued maps are studied and the fixed points of monotone set-valued maps are considered.Second,in b-metric space,the fixed point theorem of Reich type compression image is obtained in complete b-metric space,and the result substantially weakens the compression condition of an existing fixed point theorem.The number and construction theory of a class of fixed points are obtained.The third is the rectangular b-metric space,which mainly solves an open problem proposed by S.Czerwik.The main results of this paper are as follows:Frist,succeeded in turning the Reich in metric space type contractive mapping fixed point theorem is generalized to the metric space and the rectangular b-metric space respectively,got the Reich type contractive mapping the existence and uniqueness of fixed point theorem.The result not only weakens the compression condition of the fixed point theorem in metric space,but also solves an open problem proposed by s.zerwik in one of his papers.Finally,an example is given to illustrate the rationality of the theorem.Second,the equivalence of Ekeland variational principle in metric space and Caristi fixed point theorem is studied.Based on Caristi fixed point theorem,the number and construction of fixed points are discussed.Third,by the properties of topological degree of set-valued mapping are,and then the fixed point theorem of monotone single-valued mapping is extended to monotone set-valued mappings.