Common Fixed Point Results Of Generalized Contractive Mappings

Fixed point theory has become an important part of nonlinear functional analysis and a rapidly developing subject.The research results of this problem have been successfully applied in the fields of partial differential equations,cybernetics,economic balance theory and game theory,and are closely related to many branches of modern mathematics.Banach contraction mapping principle is a rewarding result in fixed point theory.Its importance is that it develops the idea of iteration,which can approach the fixed point to any degree,and provides a theoretical support for the existence,uniqueness and iterative algorithm of many equation solutions.In this paper,we mainly study the fixed point theorems of various types of generalized contractive mappings in b-metric s^paces and b-metric-like s^paces,and we provide some examples to support the validity of our conclusions.Some applications to prove the existence and uniqueness of solutions of a class of integral equations and differential equations are also presented.In chapter 1,we introduce some related concepts,properties and lemmas of b-metric s^paces and b-metric-like s^paces and also give some concepts of weak compatibility and coincidence point.In chapter 2,combined with the concepts ofg-α_s^p-admissible mappings and orbital admissible,we propose the definitions of generalized(g-a_s^p,ψ,φ)contractive mappings type I、generalized(g-α_s^p,ψ,φ)contractive mappings type II and generalizeda_s-y-Geraghty contractive mappings in b-metric s^paces and introduce a number of common fixed point theorems of the new types of contractive mappings by using the properties of the given mappings.Firstly,we construct a sequence{y _n}and prove that{y _n}is a Cauchy sequence in X by means of reduction to absurdity.Secondly,according to the completeness of X and our new contractive mappings,we conclude that f and g have a unique common fixed point in X.Finally,we also provide some examples and applications in proving the existence and uniqueness of solutions of integral equations and differential equations for our results.In chapter 3,combined with the concept of weak compatibility and the continuity of functions,we prove some common fixed point theorems of generalized（ψ,φ）-weakly compatible contractive mappings in b-metric-like s^paces.Firstly,we construct a sequence{y _n}in X,and prove that lim from （n→+∞） d(y_n,y_n+1)=0 and{y _n}is a Cauchy sequence in X by means of reduction to absurdity.Secondly,according to the completeness of X and weak compatibility,we prove that f and g have a unique common fixed point in X.Finally,an example is given to illustrate the effectiveness of one of the obtained results.Meanwhile,a corollary is obtained according to the common fixed point theorem,and the existence and unique solution of a class of integral equations is proved by using this corollary.