A New Approach to Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces
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AuthorZhou, Mi; Saleem, Naeem; Liu, Xiaolan; Fulga, Andreea; Roldán López de Hierro, Antonio Francisco
Fuzzy metric spaceFixed pointProinov-type contractionNon-Archimedean fuzzy metric space
Zhou, M.; Saleem, N.; Liu, X.; Fulga, A.; Roldán López de Hierro, A.F. A New Approach to Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces. Mathematics 2021, 9, 3001. https://doi.org/ 10.3390/math9233001
SponsorshipXiao-lan Liu is partially supported by National Natural Science Foundation of China (Grant No.11872043), Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Internationalization and Internet of Things (Grant No.2020WYJ01), Sichuan Science and Technology Program (Grant No. 2019YJ0541) and Scientific Research Project of Sichuan University of Science and Engineering (Grant Nos. 2017RCL54, 2019RC42, and 2019RC08), Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (Grant No. 2019QZJ03), Open Fund Project of Artificial Intelligence Key Laboratory of Sichuan Province (Grant No. 2018RYJ02), Zigong Science and Technology Program (Grant No. 2020YGJC03), 2020 Graduate Innovation Project of Sichuan University of Science and Engineering (Grant No. y2020078). A.F. Roldán López de Hierro is grateful to Project of Ministerio de Ciencia e Innovación (Grant No. PID2020-119478GB-I00) and also to Junta de Andalucía of the Andalusian PAIDI (Grant No. FQM-365). A.F. Roldán López de Hierro is grateful to Project of Ministerio de Ciencia eInnovación (Grant No. PID2020-119478GB-I00) , Junta de Andalucía of the Andalusian PAIDI (Grant No. FQM-365) and Program FEDER Andalucía 2014-2020 (Grant No. Project A-FQM170-UGR20).
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 fixedpoint 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.