On ( α ∗ , ψ ) -contractive multi-valued mappings

Muhammad Usman Ali,Tayyab Kamran

doi:10.1186/1687-1812-2013-137

Abstract

In this paper, we generalize the contractive condition for multi-valued mappings given by Asl, Rezapour and Shahzad in 2012. We establish some fixed point theorems for multi-valued mappings from a complete metric space to the space of closed or bounded subsets of the metric space satisfying generalized -contractive condition. MSC:47H10, 54H25.

Highlights

Samet et al [ ] introduced the notion of α-ψ-contractive self-mappings of a metric space
Contractive mappings and prove some fixed point theorems for such mappings
Where ψn is the nth iterate of ψ

Summary

Introduction

Samet et al [ ] introduced the notion of α-ψ-contractive self-mappings of a metric space. We generalize the notion of α∗-ψ - contractive mappings and prove some fixed point theorems for such mappings. Such a map H is called generalized Hausdorff metric induced by d. A mapping G : X → CL(X) is called α∗-ψ contractive if there exist two functions α : X × X → [ , ∞) and ψ ∈ such that α∗(Gx, Gy)H(Gx, Gy) ≤ ψ d(x, y). [ ] Let (X, d) be a complete metric space, let α : X × X → [ , ∞) be a function, let ψ ∈ be a strictly increasing map and T be a closed-valued, α∗-admissible and α∗-ψ -contractive multi-function on X. Let (X, d) be a complete metric space and let G : X → CL(X) be an α∗-admissible strictly generalized (α∗, ψ)-contractive mapping. Since ψ is strictly increasing, by ( . ), we have ψ d(x , x ) < ψ qψ d(x , x )

Then by

For each