Nadler’s Theorem on Incomplete Modular Space

Fatemeh Lael,Monica Felicia Bota,Liliana Guran,Naeem Saleem

doi:10.3390/math9161927

Abstract

This manuscript is focused on the role of convexity of the modular, and some fixed point results for contractive correspondence and single-valued mappings are presented. Further, we prove Nadler’s Theorem and some fixed point results on orthogonal modular spaces. We present an application to a particular form of integral inclusion to support our proposed version of Nadler’s theorem.

Highlights

The fixed point results in modular spaces have several applications in various branches of sciences
Nguyen Van Dung, in [17], talked about the importance of the results proved on orthogonal set and showed that many existence results on fixed points in orthogonal complete metric spaces can be proved by using the corresponding existence results in complete metric spaces
The aim of this paper is to prove some generalizations of fixed point results in an orthogonal modular space by relaxing some strong assumptions for the modular spaces case such as convexity, continuity and Fatou property

Summary

Introduction

The fixed point results in modular spaces have several applications in various branches of sciences. These results strongly depend on some assumptions which are more theoretic and have no applications in normed vector spaces. Some recent research trends are dedicated to the study of the well-known fixed point theorems by relaxing the assumptions and considering the case of modular spaces. Fixed point theory in modular spaces had been considered a starting point in the research field, after being identified as a generalization of normed spaces. The fixed point theorems in modular spaces presented in the literature deal with rigorous statements and proofs of many interesting problems that give some applications in a wide variety of fields, including machine learning, programming, quantum mechanics, etc

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