On one-local retract in modular metrics.

Abstract

We continue the study of the concept of one-local retract in the settings of modular metrics. This concept has been studied in metric spaces and quasi-metric spaces by different authors with different motivations. In this article, we extend the well-known results on one-local retract in metric point of view to the framework of modular metrics. In particular, we show that any self-map ψ : Xw −! Xw satisfying the property w(λ, ψ(x), ψ(y)) ≤w(λ, x, y) for all x, y ∈ X and λ > 0, has at least one fixed point whenever the collection of all qw-admissible subsets of Xw is both compact and normal.