On Caristi fixed point theorem for set-valued mappings

TL;DR

This paper addresses a generalization of Caristi's fixed point theorem for set-valued mappings in ordered metric spaces, providing negative results on Penot's problem and introducing new fixed point theorems.

Contribution

It resolves Penot's problem negatively and presents novel fixed point theorems for set-valued mappings in ordered metric spaces.

Findings

Penot's problem on generalization is settled negatively

New fixed point theorems are established for set-valued mappings

Results extend fixed point theory in ordered metric spaces

Abstract

The aim of this paper is to discuss Penot's problem on a generalization of Caristi's fixed point theorem. We settle this problem in the negative and we present some new theorems on the existence of fixed points of set-valued mappings in ordered metric spaces.