A New Approach to Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces

Mi Zhou, Naeem Saleem, Xiaolan Liu, Andreea Fulga and Antonio Francisco Roldán López de Hierro
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Mi Zhou: School of Science and Technology, University of Sanya, Sanya 572022, China
Naeem Saleem: Department of Mathematics, University of Management and Technology, Lahore 54770, Punjab, Pakistan
Xiaolan Liu: College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China
Andreea Fulga: Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania
Antonio Francisco Roldán López de Hierro: Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain

Mathematics, 2021, vol. 9, issue 23, 1-21

Abstract: Very recently, by considering a self-mapping T on a complete metric space satisfying a general contractivity condition of the form ψ ( d ( T x , T y ) ) ≤ 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.

Keywords: fuzzy metric space; fixed point; Proinov-type contraction; non-Archimedean fuzzy metric space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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