The role of regularity to reach the vector valued version of Caristi's fixed point theorem

TL;DR

This paper investigates the importance of cone regularity in establishing a vector valued Caristi's fixed point theorem, demonstrating that regularity is crucial for the theorem's validity in vector valued metric spaces.

Contribution

It introduces a vector valued Caristi's theorem with weakened hypotheses, emphasizing the essential role of cone regularity, and generalizes previous results.

Findings

Regularity of the cone is essential for the theorem.

Absence of regularity invalidates some previous results.

The main theorem generalizes prior Caristi-type theorems.

Abstract

In this paper we discuss on the vector valued version of Caristi's theorem. We show that the regularity of the cone is an essential condition to reach the vector valued version of Caristi's theorem in vector valued metric spaces. It is shown that how the absence of the regularity causes some previous Caristi type theorems and results fail to be correct. Our main result give a vector valued version of Caristi's theorem for correspondences with weakened hypotheses in comparison with previous results and generalize them.