Coincidence Points of a Sequence of Multivalued Mappings in Metric Space with a Graph

Muhammad Khan,Akbar Azam,Nayyar Mehmood

doi:10.3390/math5020030

Abstract

In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we obtain an existence theorem for a special type of fractional integral equation. Moreover, we establish a result on the convergence of successive approximation of a system of Bernstein operators on a Banach space.

Highlights

Introduction and PreliminariesFor the metric space ( X, d), using the notions of Nadler [1] and Hu [2], denote CB( X ), C ( X )and 2X by the collection of nonempty closed and bounded, compact and all nonempty subsets of X respectively
Well known idea of Hausdorff–Pompeiu distance H on CB( X ) induced by d is used to define a metric on CB ( X ) as follows: H ( A, B) = inf{e > 0 : A ⊆ N (e, B), B ⊆ N (e, A)}, where: N (e, A) = { x ∈ X : d( x, a) < e, for some a ∈ A}
In 1969, Nadler [1] proved fixed point results for multivalued mappings in complete metric spaces, using the Hausdorff distance H, which was the generalization of Banach contraction principle in the settings of set-valued mappings

Summary

Introduction

Introduction and PreliminariesFor the metric space ( X, d), using the notions of Nadler [1] and Hu [2], denote CB( X ), C ( X )and 2X by the collection of nonempty closed and bounded, compact and all nonempty subsets of X respectively. In 1969, Nadler [1] proved fixed point results for multivalued mappings in complete metric spaces, using the Hausdorff distance H, which was the generalization of Banach contraction principle in the settings of set-valued mappings. Covitz and Nadler [3] extended the idea of multivalued mappings in the generalized metric spaces.

Results

Conclusion