Extension of Kirk-Saliga Fixed Point Theorem in a Metric Space with a Reflexive Digraph

Karim Chaira,Abderrahim Eladraoui,Mustapha Kabil,Samih Lazaiz

doi:10.1155/2018/1471256

Karim Chaira, Abderrahim Eladraoui + Show 2 more

Open Access

https://doi.org/10.1155/2018/1471256

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Abstract

We extend the result of Kirk-Saliga and we generalize Alfuraidan and Khamsi theorem for reflexive graphs. As a consequence, we obtain the ordered version of Caristi’s fixed point theorem. Some concrete examples are given to support the obtained results.

Highlights

Fixed point theory is one of the most useful tools in mathematics; it is used to solve many existence problems such as differential equations, control theory, optimization, and several other branches
The most well-known fixed point result is Banach contraction principle [2]; it is famous for its applications, proving the existence of solution of integral equations by converting the problem to fixed point problem
It is worth mentioning that Caristi’s fixed point theorem is equivalent to the Ekeland variational principle [8]. It characterizes the completeness of the metric space as showed by Kirk in [9]

Summary

Introduction

Fixed point theory is one of the most useful tools in mathematics; it is used to solve many existence problems such as differential equations, control theory, optimization, and several other branches (for the literature see [1]). There is KirkSaliga fixed point theorem (see [10]) which states that any map T : X → X has a fixed point provided that X is complete metric space and there exist an integer p ∈ N and a lower semicontinuous function φ : X → [0, ∞) such that d (x, Tx) ≤ φ (x) − φ (Tpx) and φ(Tx) ≤ φ(x) for any x ∈ X. [16] Alfuraidan and Khamsi gave an analogue version of Caristi’s fixed point theorem in the setting of partially ordered metric space where the inequality holds only for comparable elements.

Results

Conclusion