A note on the equivalence of some metric and H-cone metric fixed point theorems for multivalued contractions

Huaping Huang,Dragan Ðorić,Stojan Radenović

doi:10.1186/s13663-015-0289-2

Huaping Huang, Dragan Ðorić + Show 1 more

Open Access

https://doi.org/10.1186/s13663-015-0289-2

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Journal: Fixed Point Theory and Applications	Publication Date: Mar 26, 2015
Citations: 3	License type: CC BY 4.0

Abstract

AbstractIn this paper, by using Minkowski functional introduced by Kadelburg et al. (Appl. Math. Lett. 24:370-374, 2011) or nonlinear scalarization function introduced by Du (Nonlinear Anal. 72:2259-2261, 2010), we prove some equivalences between vectorial versions of fixed point theorems for H-cone metrics in the sense of Arshad and Ahmad and scalar versions of fixed point theorems for (general) Hausdorff-Pompeiu metrics (in usual sense).

Highlights

1 Introduction Recently, the investigation of possible equivalence between fixed point results in cone metric spaces and metric spaces has become a hot topic in many mathematical activities
By using the properties either of the Minkowski functional qe or the nonlinear scalarization function ξe, some scholars have made a conclusion that many fixed point results in the setting of cone metric spaces or tvs-cone metric spaces can be directly obtained as a consequence of the corresponding results in metric spaces
Definition . ([ ]) Let (X, d) be a cone metric space and let A be a collection of nonempty subsets of X

Summary

Introduction

The investigation of possible equivalence between fixed point results in cone metric spaces (or tvs-cone metric spaces) and metric spaces has become a hot topic in many mathematical activities. ([ ]) Let (X, d) be a cone metric space and A a collection of nonempty subsets of X. A map H : A × A → E is called an H-cone metric in the sense of Arshad and Ahmad if the following conditions hold:. (Nadler [ ]) Let (X, d) be a complete metric space and A be a collection of nonempty, closed, and bounded subsets of X. (Arshad and Ahmad [ ]) Let (X, d) be a complete cone metric space. Let A be a collection of nonempty closed subsets of X, and let H : A × A → E be an H-cone metric in the sense of Arshad and Ahmad. Let A be a family of nonempty, closed, and bounded subsets of X and let there exist an H-cone metric H : A ×. Suppose that T, S : X → A are two multivalued mappings and suppose that there is λ ∈ [ , ) such that, for all x, y ∈ X, at least one of the following conditions holds:

Sy and each fixed u

Ty and