Research On Fixed Point Problems Of Several Classes Of Contractive Mappings On Generalized Metric Spaces

The research of fixed point problem has always attracted the attention of scholars.It also plays an important role in mathematics,physics and other fields,and has a wide range of applications in many fields.A well-known result of fixed point theory is the Banach contraction principle.With the continuous in-depth research and exploration of various researchers,it has been extended to different generalized metric spaces.These conclusions should be used to prove the existence and uniqueness of the solutions of nonlinear operator equations,and several examples were obtained through exploration.This paper mainly studies the fixed point theory of various types of generalized contraction mappings in b-metric spaces,and generalizes the b-metric to rectangular b-metric for research.At the same time,some examples are given to illustrate the validity of the results,and the application of the results in proving the existence and uniqueness of solutions of integral and differential equations is also given.The first chapter mainly introduces the concepts,properties and lemma of bmetric spaces and rectangular b-metric spaces,and gives concepts such as weak compatibility,common fixed points,coincident points,and multivariate fixed points.The second chapter examines the common fixed points and coincident points of generalized α-φE-Geraghty contraction mappings on b-metric space,and proposes the definition of αi,j-φEM,N-Geraghty contraction mapping andαi,j-φEN-Geraghty contraction mapping,as well as the definition of triangularαi,j-orbital admissible and some related lemmas.Using the properties of the given mapping,the existence uniqueness of the common fixed point of the αi,j-φEM,NGeraghty contraction mapping and the existence of the coincident point of theαi,j-φEN-Geraghty contraction mapping are proposed.Two examples are also given to verify our results.Chapter 3 gives the multivariate fixed point and multiple common fixed point theorems for(Ψ,φ)weakly contractive mappings in b-metric spaces and their applications.The concepts of multivariate*-metric function,multivariate coincident point,multivariate common fixed point and multivariate commutative are introduced,and the multivariate fixed point theorem for single mappings and the multivariate common fixed point theorem for double mappings are proposed.Concurrently,we give the application of our results in solving the problem of the existence of solutions to integral equations.Chapter 4 expands the results of b-metric spaces,and studies the fixed point theorem for injection and the solvability of nonlinear integral equations in rectangular b-metric spaces.The concept of triangular αsp orbital admissible mappings is introduced and corresponding lemmas are given.Two fixed point results for special types of contractive mappings in rectangular b-metric spaces are obtained.Finally,we give the application of the existence and uniqueness of the integral equation solution.