Functional type Caristi-Kirk theorem on two metric spaces and applications

Karim Chaira,Elmiloudi Marhrani

doi:10.1186/s13663-016-0586-4

Abstract

In this paper, we give some generalizations of the functional type Caristi-Kirk theorem (see Functional Type Caristi-Kirk Theorems, 2005) for two mappings on metric spaces. We investigate the existence of some fixed points for two simultaneous projections to find the optimal solutions of the proximity two functions via Caristi-Kirk fixed point theorem.

Highlights

1 Introduction Recall that a real-valued function φ defined on a metric space X is said to be lower semi-continuous if for any sequencen of X which converges to x ∈ X, we have φ(x) ≤ lim infn φ(xn) (φ(x) ≥ lim supn φ(xn))
In, Caristi obtained the following fixed point theorem on complete metric spaces, known as the Caristi fixed point theorem
We will say that x ∈ M is a maximal element of M if and only if (x ≤ y, y ∈ M ⇒ x = y)

Summary

Introduction

Recall that a real-valued function φ defined on a metric space X is said to be lower (upper) semi-continuous if for any sequence (xn)n of X which converges to x ∈ X, we have φ(x) ≤ lim infn φ(xn) (φ(x) ≥ lim supn φ(xn)).In , Caristi (see [ ]) obtained the following fixed point theorem on complete metric spaces, known as the Caristi fixed point theorem.Theorem. Ekeland [ ]) Let (X, d) be a complete metric space and φ : X → R+ be a lower semi-continuous function.

Results

Conclusion