Fixed point theory in the setting of (alpha,beta,psi,phi)-interpolative contractions

Erdal Karapınar,Andreea Fulga,Antonio Francisco Roldán López De Hierro

doi:10.1186/s13662-021-03491-w

Abstract

In this manuscript we introduce the notion of (alpha,beta,psi,phi)-interpolative contraction that unifies and generalizes significant concepts: Proinov type contractions, interpolative contractions, and ample spectrum contraction. We investigate the necessary and sufficient conditions to guarantee existence and uniqueness of the fixed point of such mappings.

Highlights

For many years researchers in the field of fixed point theory have known and employed contractivity conditions of the type:ψ d (Tu, Tv) ≤ φ d (u, v) for all u, v ∈ X , where (X , d ) is a metric space, T : X → X is a self-mapping, and ψ, φ : [0, +∞) → [0, +∞) are two auxiliary functions satisfying certain general conditions whose main aim is to help in the task of proving existence and, in most of cases, uniqueness of fixed points of the mapping T (Boyd and Wong [1], Geraghty [2], Amini-Harandi and Petruşel [3], Jleli and Samet [4], Wardowski [5], etc.) This is only a simple, but interesting, way to generalize the original Banach’s contractivity condition that firstly appeared in [6]
After a series of successive research papers deepening this area, this last author and Shahzad presented the notion of ample spectrum contraction in [15] with a double aim: to analyze the essential properties, from an abstract point of view, that any contraction that may arise in the future must satisfy and generalize as many results as possible in the field of fixed point theory
Before that, we point out that this class properly extends the family of Proinov contractions since, we proved that all Proinov contractions are particular cases of the Roldán López de Hierro and Shahzad ample spectrum contractions

Summary

Introduction

For many years researchers in the field of fixed point theory have known and employed contractivity conditions of the type:ψ d (Tu, Tv) ≤ φ d (u, v) for all u, v ∈ X , where (X , d ) is a metric space, T : X → X is a self-mapping, and ψ, φ : [0, +∞) → [0, +∞) are two auxiliary functions satisfying certain general conditions whose main aim is to help in the task of proving existence and, in most of cases, uniqueness of fixed points of the mapping T (Boyd and Wong [1], Geraghty [2], Amini-Harandi and Petruşel [3], Jleli and Samet [4], Wardowski [5], etc.) This is only a simple, but interesting, way to generalize the original Banach’s contractivity condition that firstly appeared in [6]. Theorem 3 guarantees that each Proinov contraction from a complete metric space into itself has a unique fixed point.1 Inspired by this class of contractions and by ω-interpolative Ćirić–Reich–Rus-type contractions [22], in this paper we are going to study the existence and uniqueness of fixed points of self-mapping satisfying a more general contractivity condition.

Results

Conclusion