Common Fixed Point Results for Generalized Wardowski Type Contractive Multi-Valued Mappings

Hüseyin Işık,Vahid Parvaneh,Ishak Altun,Babak Mohammadi

doi:10.3390/math7111130

Abstract

In this paper, we introduce generalized Wardowski type quasi-contractions called α - ( φ , Ω ) -contractions for a pair of multi-valued mappings and prove the existence of the common fixed point for such mappings. An illustrative example and an application are given to show the usability of our results.

Highlights

For a metric space (Λ, d), let CB(Λ) be the class of all nonempty closed and bounded subsets ofΛ and K(Λ) be the class of all nonempty compact subsets of Λ (it is well known that K(Λ) ⊆ CB(Λ)).The mapping H : CB(Λ) × CB(Λ) → R+ ∪ {0} defined byH( P, Q) = max{sup d( p, Q), sup d(q, P)}, f or any P, Q ∈ CB(Λ)p∈ P q∈ Q is called the Pompeiu–Hausdorff metric induced by d, where d( p, Q) = inf{d( p, q) : q ∈ Q}is the distance from p to Q ⊆ Λ
We introduce the concept of α-( φ, Ω)-contraction for a pair of multi-valued mappings and prove the existence of common fixed
Ω(d(η2, η3 )) ≤ φ2 (Ω(d(η0, η1 ))). Continuing this process, either we find a common fixed point of Υ and Γ or we can construct a sequence {ηn } in Λ such that η2n+1 ∈ Υη2n, η2n+2 ∈ Γη2n+1, d(ηn, ηn+1 ) > 0, α(ηn, ηn+1 ) ≥ 1 for all n ∈ N ∪ {0} and

Summary

Introduction

For a metric space (Λ, d), let CB(Λ) be the class of all nonempty closed and bounded subsets of. Let (Λ, d) be a complete metric space and Υ : Λ → CB(Λ) be a multi-valued mapping such that. Wardowski [2] gave a new generalization of Banach contraction to show the existence of the fixed point for such contraction by a more simple method of proof than the Banach’s one. Aydi et al [9] studied a common fixed point for generalized multi-valued contractions. We introduce the concept of α-( φ, Ω)-contraction for a pair of multi-valued mappings and prove the existence of common fixed. An illustrative example and an application to the system of Volterra-type integral inclusions are given

Preliminaries

Main Results

An Application to Volterra-Type Integral Inclusions

Conclusions