Common Fixed Points Results on Non-Archimedean Metric Modular Spaces

Wissam Kassab

doi:10.3390/sym11111355

Abstract

This paper introduces two new contractive conditions in the setting of non-Archimedean modular spaces, via a C-class function, an altering distance function, and a control function. A non-Archimedean metric modular is shaped as a parameterized family of classical metrics; therefore, for each value of the parameter, the positivity, the symmetry, the triangle inequality, or the continuity is ensured. The main outcomes provide sufficient conditions for the existence of common fixed points for four mappings. Examples are provided in order to prove the usability of the theoretical approach. Moreover, these examples use a non-Archimedean metric modular, which is not convex, making the study of nonconvex modulars more appealing.

Highlights

Various modular structures, viewed as alternatives to classical normed or metric spaces, have been intensely studied in connection with the fixed point theory
The new modular proves to be a parameterized family of classical metrics; for each value of the parameter, the triangle inequality or the continuity is ensured
If f Xω is not closed and one of the sets in Condition (7) is closed, we follow the similar arguments as above to prove the common fixed point of the four mappings f, g, T, S. This papers defines the notions of the almost nonlinear (S, T, L, F, ψ, φ)-convex contractive condition of type I and type II on a non-Archimedean modular space, as two distinct extensions for a similar contractive condition defined on metric spaces

Summary

Introduction

Various modular structures, viewed as alternatives to classical normed or metric spaces, have been intensely studied in connection with the fixed point theory. The new modular proves to be a parameterized family of classical metrics; for each value of the parameter, the triangle inequality or the continuity is ensured This makes the newly defined object a very good instrument for analyzing various contractive conditions or for using non-standard iterative procedures. This paper uses the setting of a non-Archimedean metric modular space and defines and studies new nonlinear contractive conditions. The source for this approach is the work of Shatanawi et al [14], who developed a similar theory, but in the framework of a complete metric space. By properly including concepts as weakly compatible mappings (see Jungck [18]) or dominating and weak annihilators (see Abbas et al [19]), several new outcomes regarding the existence of common fixed points are obtained

Preliminaries

First Extension to Partially Ordered Non-Archimedean Metric Modular Spaces

Second Extension to Partially Ordered Non-Archimedean Modular Spaces

Conclusions