Best Proximity Point Results in Non-Archimedean Modular Metric Space

Mohadeshe Paknazar,Manuel Sen

doi:10.3390/math5020023

Mohadeshe Paknazar, Manuel Sen

Open Access

https://doi.org/10.3390/math5020023

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Abstract

In this paper, we introduce the new notion of Suzuki-type ( α , β , θ , γ ) -contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak P λ -property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature.

Highlights

Introduction and PreliminariesModular metric spaces are a natural and interesting generalization of classical modulars over linear spaces, like Lebesgue, Orlicz, Musielak–Orlicz, Lorentz, Orlicz–Lorentz, Calderon–Lozanovskii spaces and others
Let A and B be two nonempty subsets of a non-Archimedean modular metric space Xω with ω regular, such that A is ω −complete and A0λ is nonempty for all λ > 0
Let ( A, B) be a pair of nonempty subsets of a non-Archimedean modular metric space Xω with ω regular, such that A is complete and A0λ is nonempty for all λ > 0

Summary

Introduction and Preliminaries

Modular metric spaces are a natural and interesting generalization of classical modulars over linear spaces, like Lebesgue, Orlicz, Musielak–Orlicz, Lorentz, Orlicz–Lorentz, Calderon–Lozanovskii spaces and others. We investigate the Suzuki-type result of Zhang et al [16] in the setting of non-Archimedean modular metric space as follows: Corollary 4. Let ( A, B) be a pair of nonempty and closed subsets of a complete non-Archimedean modular metric space Xω with ω regular, such that A0λ is nonempty for all λ > 0. (Suzuki-type result of Suzuki [21]) Let ( A, B) be a pair of nonempty and closed subsets of a complete non-Archimedean modular metric space Xω with ω regular, such that A0λ is nonempty for all λ > 0. Let ( A, B) be a pair of nonempty subsets of a non-Archimedean modular metric space Xω with ω regular, such that A is complete and A0λ is nonempty for all λ > 0.

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