A New Fixed Point Theorem in Modular Metric Spaces

Ali Mutlu ,Kübra Özkan ,Utku Gürdal

doi:10.28924/2291-8639-16-2018-472

Abstract

In this article, we first give a new fixed point theorem which is main theorem of our study in modular metric spaces. After that, by using this theorem, we express some interesting results. Moreover, we characterize completeness in modular metric spaces via this theorem. Finally, we use our main result to show the existence of solution for a specific problem in dynamic programming.

Highlights

The fixed point theory is used in many different fields of mathematics such as topology, analysis, nonlinear analysis and operator theory
Let ω : (0, ∞) × X × X → [0, ∞] be a metric modular on X, Xω be a modular metric space, C ⊆ Xω and ψ : C → R+ be a function on C. ψ is called lower semi-continuous (l.s.c.) on C if lim ωλ(xn, x) = 0 ⇒ ψ(x) ≤ lim inf(ψ(xn))
Let ω be a metric modular on X, Xω be a complete modular metric space, ψ : Xω → R+ be a lower semi-continuous function on Xω and T : Xω → Xω be a mapping satisfying the inequality ωλ(x, T x) ≤ ψ(x) − ψ(T x) for all x ∈ Xω and λ > 0

Summary

Introduction

The fixed point theory is used in many different fields of mathematics such as topology, analysis, nonlinear analysis and operator theory. We first give a new fixed point theorem which is main theorem of our study in modular metric spaces. Modular metric space; fixed point theorem; complete modular metric.

Results

Conclusion