Fixed Point Theorems for Set-Valued Mappings on TVS-Cone Metric Spaces

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Abstract In the context of tvs-cone metric spaces, we prove a Bishop-Phelps and a Caristi’s type theorem. These results allow us to prove a fixed point theorem for \((\delta,L)\)-weak contraction according to a pseudo Hausdorff metric defined by means of a cone metric.

Keywords and phrases tvs-cone metric space; Bishop-Phelps type theorem, Caristi type theorem, Berinde weak contraction; set-valued mapping.

2010 MSC Primary 47H10; Secondary 47H04.

Introduction

Huang and Zhang in (Huang 1468), introduced the concept of cone metric space as a generalization of metric space. The most relevant of their work is that these authors gave an example of a contraction on a cone metric space, which is not contraction in a standard metric space. This fact makes it clear that the theory of metric spaces are not flexible enough for the fixed point theory, which it has prompted several authors to publish numerous works on fixed point theory for operators defined on cone metric spaces. Most of these are based in cone metrics taking values in a Banach space, and even, some of them suppose this space is normal, in the sense that this space has a base of neighborhood of zero consisting of order-convex subsets. The main aim of this paper is to provide results for set-valued mappings defined on a cone metric space, whose metric takes values in a quite general topological vector space, since it is only assumed this space is \(\sigma\)-order complete. In (Agarwal 2011) (see also, (I. Altun 1145)), Agarwal and Khamsi proved a version of Caristi’s theorem based in a Bishop-Phelps type result for a cone metric taking values in a Banach space. In this paper, we extend this result, which enables us to prove a more general version of Caristi’s theorem for cone metric spaces. Natural consequences are deduced from this fact and, as an application, we prove existence of fixed point for an analogous weak contraction of set-valued mapping defined by Berinde and Berinde in (M. Berinde 2007), which, in our case, is defined according to a pseudo Hausdorff cone metric.

The paper is organized as follows. In Section 2 some preliminary definitions and facts are given, while in Section 3, Bishop-Phelps and Caristi’s theorems are proved. Finally, Section 4 is devoted to an application to set-valued weak contractions defined by means of a cone metric.