Coupled Common Fixed Point Theorems And Stability Of Some Hybrid Contractive Mapping Pairs

Fixed point theory is one of the most active topics in the nonlinear analysis and research,it is widely used in many domains.Inspired by the coupled fixed points of Bhaskar and Lakshmikantham,this paper follows the following steps:Firstly,this paper introduce the research status and preliminary knowledge.Mainly introduce the results related to this paper that other scholars have done before and came to some important conclusions,including coupled fixed point and coupled coincidence point theorems of single-valued contractive mappings in the partially order metric spaces and coupled fixed point theorems of set-valued contractive mappings in metric spaces.Some basic knowledge,symbols,definitions and lemmas are also explained.Secondly,eleven theorems are demonstrated and part of their proof processes are provided,prove some coupled common fixed point theorems for hybrid contractive mapping pairs and single-valued contractive mapping pairs,coupled fixed point theorems for set-valued contractive mappings and single-valued contractive mappings in metric spaces,the contractive types are as follows:(?),some proofs are omitted for theorems that have the same method of proof.Thirdly,five theorems are demonstrated and part of their proof processes are provided,prove the existence of coupled common fixed points for hybrid contractive mapping pairs and single-valued contractive mapping pairs,coupled fixed points for set-valued contractive mappings and single-valued contractive mappings in partially ordered metric spaces,the contractive types are as follows:(?).Fourthly,also to discuss the stability of coupled common fixed point sets for uniformly convergent sequences of hybrid contractive mapping pairs and coupled fixed point sets for uniformly convergent sequences of set-valued contractive mappings in metric spaces.Fifthly,four illustrative examples are constructed in this paper,by means of the given space,mappings and functions,to explain some of the theorems presented and illustrate the differences between the results of the coupled fixed points and the results of Sintunavarat et al and Abbas et al.Finally,some results obtained in this paper is used to solve the system of nonlinear integral equations,the application of the coupled fixed point theorem to integral equations for a single-valued contractive mapping in a complete metric space is investigated.