Common Fixed Point Theorems via Inverse    C   k   −  Class Functions in Metric Spaces

Mi Zhou,Jia Deng,Arslan Hojat Ansari,Mukesh Kumar Jain,Xiao-Lan Liu,Naeem Saleem

doi:10.1155/2021/6648993

Abstract

In this paper, we firstly introduce a new notion of inverse C k − class functions which extends the notion of inverse C − class functions introduced by Saleem et al., 2018. Secondly, some common fixed point theorems are stated under some compatible conditions such as weak semicompatible of type A , weak semicompatibility, and conditional semicompatibility in metric spaces. Moreover, we introduce a new kind of compatibility called S τ − compatibility which is weaker than E . A . property and also present a common fixed point theorem in metric spaces via inverse C k − class functions. Some examples are provided to support our results.

Highlights

Introduction and PreliminariesAs a follow-up work of A.H
Some common fixed point theorems are stated under some compatible conditions such as weak semicompatible of type (A), weak semicompatibility, and conditional semicompatibility in metric spaces
We introduce a new kind of compatibility called Sτ− compatibility which is weaker than (E.A.) property and present a common fixed point theorem in metric spaces via inverse Ck− class functions

Summary

Introduction

Introduction and PreliminariesAs a follow-up work of A.H. Ansari’s research on fixed point (or common fixed point) theory via auxiliary C-class functions, very recently, Saleem et al [1] introduced the new concept of inverse C-class functions and obtained some corresponding fixed point theorems under certain weak compatibility assumption via inverse C-class functions. A pair (f, g) of self-maps of a metric space (X, d) is said to be semicompatible, if limn⟶+∞fgxn gt holds whenever 􏼈xn􏼉 is a sequence in X such that limn⟶+∞fxn limn⟶+∞gxn t for some t ∈ X. A pair (f, g) of self-maps of a metric space (X, d) is said to be semicompatible of type (A), if limn⟶+∞fgxn gt and limn⟶+∞gfxn ft hold whenever 􏼈xn􏼉 is a sequence in X such that limn⟶+∞fxn limn⟶+∞gxn t for some t ∈ X.

Results

Conclusion