A New Common Fixed Point Theorem for Three Commuting Mappings

Meryeme El Harrak,Ahmed Hajji

doi:10.3390/axioms9030105

Abstract

In the present paper, we propose a common fixed point theorem for three commuting mappings via a new contractive condition which generalizes fixed point theorems of Darbo, Hajji and Aghajani et al. An application is also given to illustrate our main result. Moreover, several consequences are derived, which are generalizations of Darbo’s fixed point theorem and a Hajji’s result.

Highlights

Introduction and PreliminariesSchauder’s fixed point theorem [1] plays a crucial role in nonlinear analysis.Namely, Schauder [1] has proved that if a self-mapping T is continuous on compact and convex subset of Banach spaces, T has at least one fixed point
In 2013, Hajji [6] established a common fixed point theorems for commuting mappings verifying α(ST ( A)) ≤ kα( A), α(ST ( A)) < α( A), which generalize Darbo’s and Sadovskii’s fixed point theorems
As examples and applications, he studied the existence of common solutions of equations in Banach spaces using the Axioms 2020, 9, 105; doi:10.3390/axioms9030105

Summary

Introduction and Preliminaries

Schauder’s fixed point theorem [1] plays a crucial role in nonlinear analysis. Namely, Schauder [1] has proved that if a self-mapping T is continuous on compact and convex subset of Banach spaces, T has at least one fixed point. Schauder’s fixed point theorem for α-set contraction that is, such that α( T ( A)) ≤ kα( A), with k ∈ [0, 1), on a closed, bounded and convex subsets of Banach spaces. In Reference [7], we made use of some axioms of measure of noncompactness to establish the following contractive condition σ ( H ( A)) ≤ φ(S( A)) − φ(S(conv( T ( A)))), giving rise to common fixed point theorem for three commuting and continuous mappings. Motivated by contractive conditions investigated in b-metric spaces [9,10,11] and using a measure of noncompactness, we derive from our main theorem some consequences, which are generalizations of Darbo’s fixed point theorem [2] and a Hajji’s result [6].

Main Result

Application

Consequences