Common fixed points for (psi -varphi )-weak contractions type in b-metric spaces

José R Morales,Edixon M Rojas

doi:10.1007/s40065-021-00347-9

Abstract

The aim of this paper is to prove the existence and uniqueness of common fixed points for a pair of (psi -varphi )-weak contractive self-maps in the setting of b-metric spaces satisfying the minimal requirement of weakly compatibility, and other weak commuting properties as compatibility, R-weakly commuting and R-weakly commuting of types (A_T), (A_S) and (A_P). Also, we will analyze the convergence and stability of the Jungck-Noor iterative scheme for this class of pairs of mappings on b-metric spaces endowed with a convex structure.

Highlights

Motivated from these facts, in this paper we are going to prove the existence and uniqueness of common fixed points for a pair of (ψ − φ)-weak contractive type self-maps in the setting of b-metric spaces satisfying the minimal requirement of weakly compatibility, and other weak commuting properties as compatibility, Rweakly commuting and R-weakly commuting of types ( AT ), ( AS) and ( AP )
We will analyze the convergence and stability of the Jungck-Noor iterative scheme for this class of pairs of mappings on b-metric spaces endowed with a convex structure
We recall some definitions, results and properties of b-metric spaces that will be useful in this paper

Summary

Motivation

The beginning of the metric fixed point theory could be set in 1922 with the Banach’s seminal paper [9], in which he introduced the so-called Banach contraction mappings and analyzed the existence and uniqueness of its fixed points in the setting of metric spaces. Rhoades in his paper [41] extended the BCP to φ-weak contraction mappings without using the condition limt→+∞ φ(t) = +∞, as follows: Theorem 1.1 Let (M, d) be a complete metric space and let S : M −→ M be a φ-weak contraction, S has a unique fixed point when φ is a nondecreasing continuous function with φ(t) > 0 for all t > 0 and φ(0) = 0. With sequences (αn), (βn), (γn) ⊂ [0, 1] Motivated from these facts, in this paper we are going to prove the existence and uniqueness of common fixed points for a pair of (ψ − φ)-weak contractive type self-maps in the setting of b-metric spaces satisfying the minimal requirement of weakly compatibility, and other weak commuting properties as compatibility, Rweakly commuting and R-weakly commuting of types ( AT ), ( AS) and ( AP ). We will analyze the convergence and stability of the Jungck-Noor iterative scheme for this class of pairs of mappings on b-metric spaces endowed with a convex structure

Some fundamental results in b-metric spaces

Some consequences

Convergence results

Stability result