F-Convex Contraction via Admissible Mapping and Related Fixed Point Theorems with an Application

Y Singh,Shin Kang,Mohammad Khan

doi:10.3390/math6060105

Abstract

In this paper, we introduce F-convex contraction via admissible mapping in the sense of Wardowski [Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl., 94 (2012), 6 pages] which extends convex contraction mapping of type-2 of Istrǎţescu [Some fixed point theorems for convex contraction mappings and convex non-expansive mappings (I), Libertas Mathematica, 1(1981), 151–163] and establish a fixed point theorem in the setting of metric space. Our result extends and generalizes some other similar results in the literature. As an application of our main result, we establish an existence theorem for the non-linear Fredholm integral equation and give a numerical example to validate the application of our obtained result.

Highlights

Introduction and PreliminariesThe Banach’s contraction principle [1] first appeared in explicit form in 1922, where it was used to establish the existence of a solution for an integral equation
In 1982, Istrǎţescu [4] introduced the class of convex contractions in metric space, where he considered seven values d( x, y), d( Tx, Ty), d( x, Tx ), d( Tx, T 2 x ), d(y, Ty), d( Ty, T 2 y) and d( T 2 x, T 2 y) for all x, y ∈ X
He showed with example that T is in the class of convex contraction but it is not a contraction

Summary

Introduction

Introduction and PreliminariesThe Banach’s contraction principle [1] first appeared in explicit form in 1922, where it was used to establish the existence of a solution for an integral equation. Note that T is not an α∗ -admissible, we extend the notion of convex contraction [4] to an α-F-convex contraction and prove a fixed point theorem in the setting of metric space.

Results

Conclusion