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

TL;DR

This paper extends fixed point theorems to tvs-cone metric spaces, establishing Bishop-Phelps and Caristi's theorems, and introduces a fixed point result for weak contractions using a cone-based pseudo Hausdorff metric.

Contribution

It provides new fixed point theorems in tvs-cone metric spaces, generalizing classical results to a broader setting with cone-valued metrics.

Findings

Proved Bishop-Phelps and Caristi's theorems in tvs-cone metric spaces

Established fixed point theorem for $( ext{}\delta, L)$-weak contractions

Introduced a pseudo Hausdorff metric based on cone metrics

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.