Abstract

Very recently, by considering a self-mapping T on a complete metric space satisfying a general contractivity condition of the form ψ(d(Tx,Ty))≤φ(d(x,y)), Proinov proved some fixed-point theorems, which extended and unified many existing results in the literature. Accordingly, inspired by Proinov-type contraction conditions, Roldán López de Hierro et al. introduced a novel family of contractions in fuzzy metric spaces (in the sense of George and Veeramani), whose main advantage is the very weak constraints imposed on the auxiliary functions that appear in the contractivity condition. They also proved the existence and uniqueness of fixed points for the discussed family of fuzzy contractions in the setting of non-Archimedean fuzzy metric spaces. In this paper, we introduce a new family of fuzzy contractions based on Proinov-type contractions for which the involved auxiliary functions are not supposed to satisfy any monotonicity assumptions; further, we establish some new results about the existence and uniqueness of fixed points. Furthermore, we show how the main results in the above-mentioned paper can be deduced from our main statements. In this way, our conclusions provide a positive partial solution to one of the open problems posed by such authors for deleting or weakening the hypothesis of the nondecreasingness character of the auxiliary functions.

Highlights

  • Motivated by the contributions of [9,14], in this paper, we introduce a novel family of contractions based on the Proinovtype contractions for which the involved auxiliary functions are supposed to satisfy weaker constraints, and we describe some new results about the existence of unique fixed points in non-Archimedean fuzzy metric spaces

  • In the following lemma, we introduce a new condition on the non-Archimedean fuzzy metric space in order to guarantee that the sequences involved in the proofs of fixed-point theorems satisfy additional properties, which are of great importance

  • Inspired by Proinov contractions, very recently, some authors extended his main results to the setting of fuzzy metric spaces

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Summary

A New Approach to Proinov-Type Fixed-Point Results in

Mi Zhou 1,∗ , Naeem Saleem 2 , Xiaolan Liu 3,4,5,∗ , Andreea Fulga 6 and Antonio Francisco Roldán López de Hierro 7,∗. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in School of Science and Technology, University of Sanya, Sanya 572022, China. College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China. Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationlization and Internet of Things, Zigong 643000, China

Introduction
Preliminaries
The Contractive Condition and a Class of Auxiliary Functions
Fixed-Point Theory in the Setting of Non-Archimedean Fuzzy Metric Spaces
Application
Conclusions
Full Text
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