Common Fixed Point Theorems of Generalized Multivalued (ψ,ϕ)-Contractions in Complete Metric Spaces with Application

Eskandar Ameer,Muhammad Arshad,Sungsik Yun,Dong Shin

doi:10.3390/math7020194

Abstract

The purpose of this paper is to introduce the notion of generalized multivalued ψ , ϕ -type contractions and generalized multivalued ψ , ϕ -type Suzuki contractions and establish some new common fixed point theorems for such multivalued mappings in complete metric spaces. Our results are extension and improvement of the Suzuki and Nadler contraction theorems, Jleli and Samet, Piri and Kumam, Mizoguchi and Takahashi, and Liu et al. fixed point theorems. We provide an example for supporting our new results. Moreover, an application of our main result to the existence of solution of system of functional equations is also presented.

Highlights

Introduction and PreliminariesIn the fixed point theory of continuous mappings, a well-known theorem of Banach [1] states that if ( X, d) is a complete metric space and if S is a self-mapping on X which satisfies the inequality d(Sx, Sy) ≤ kd( x, y) for some k ∈ [0, 1) and all x, y ∈ X, S has a unique fixed point x ∗ and the sequence of successive approximations {Sxn } converges to x ∗ for all x ∈ X, the Banach’s theorem [1]has been extensively studied and generalized on many settings.Suzuki [16] proved the following fixed point theorem.Theorem 1 ([16])
Wardowski [17] introduced the notion of F-contractions and proved fixed point theorems concerning F-contractions as follows
A mapping T : X −→ X is said to be a θ-contraction if there exist a constant k ∈ (0, 1) and θ ∈ Θ such that x, y ∈ X, d ( Tx, Ty) 6= 0 =⇒ θ (d ( Tx, Ty)) ≤ [θ (d ( x, y))]k, where Θ is the set of functions θ : (0, ∞) −→ (1, ∞) satisfying the following conditions: (Θ1) θ is nondecreasing, (Θ2) for each sequence {tn } ⊂ (0, ∞), lim θ = 1 if and only if lim tn = 0, n→∞

Summary

Introduction and Preliminaries

In the fixed point theory of continuous mappings, a well-known theorem of Banach [1] states that if ( X, d) is a complete metric space and if S is a self-mapping on X which satisfies the inequality d(Sx, Sy) ≤ kd( x, y) for some k ∈ [0, 1) and all x, y ∈ X, S has a unique fixed point x ∗ and the sequence of successive approximations {Sxn } converges to x ∗ for all x ∈ X, the Banach’s theorem [1]. A mapping T : X −→ X is said to be a θ-contraction if there exist a constant k ∈ (0, 1) and θ ∈ Θ such that x, y ∈ X, d ( Tx, Ty) 6= 0 =⇒ θ (d ( Tx, Ty)) ≤ [θ (d ( x, y))]k , where Θ is the set of functions θ : (0, ∞) −→ (1, ∞) satisfying the following conditions:. Liu et al [21] proved new fixed point theorems for (ψ, φ)-type Suzuki contractions in complete metric spaces as follows. The pair ( T, S) is said to be a generalized multivalued (ψ, φ)-type Suzuki contraction if there exist a comparison function ψ and φ ∈ Φ such that for all x, y ∈ X with Sx 6= Ty, min { D ( x, Sx ) , D (y, Ty)} < d ( x, y) =⇒ φ ( H (Sx, Ty)) ≤ ψ (φ ( M ( x, y)))

Main Results

Some Consequences

Application