A fixed point theorem for (\phi, \shi)-convex contraction in metric spaces

Deepak Khantwal,U.C Gai̇rola,Surbhi Aneja

doi:10.31197/atnaa.735372

Abstract

In the present paper, we introduce the notion of (\phi, \shi)-convex contraction mapping of order m and establish a fixed point theorem for such mappings in complete metric spaces. The present result extends and generalizes the well known result of Dutta and Choudhary (Fixed Point Theory Appl. 2008 (2008), Art. ID 406368), Rhoades (Nonlinear Anal., 47(2001), 2683-2693), Istratescu (Ann. Mat. Pura Appl., 130(1982), 89-104) and besides many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results.

Highlights

A mapping f : X → X, where (X, d) is a metric space, is said to be a contraction mapping if for all x, y ∈ X, d(f x, f y) ≤ kd(x, y), where 0 ≤ k < 1. (1)the Banach contraction principle, which states that every contraction mapping on a complete metric space (X, d) has a unique xed point, is one of the pivotal result in xed point theory
The Banach contraction principle, which states that every contraction mapping on a complete metric space (X, d) has a unique xed point, is one of the pivotal result in xed point theory
For m = 1, one can see that the convex contraction (4) reduces to Banach contraction (1) Istr μescu's xed point theorem is an eective generalization of Banach contraction principle and has been extended by several authors

Summary

Introduction

The Banach contraction principle, which states that every contraction mapping on a complete metric space (X, d) has a unique xed point, is one of the pivotal result in xed point theory. If f is a weakly contractive mapping on a complete metric space (X, d) f has a unique xed point. On the other hand the study of convex contraction, which does not imply the contraction condition (1) but ensure the existence and uniqueness of the xed point, was initiated by Istr μescu (see [12, 13, 14]).

Results

Conclusion