Best Proximity Point Results for Generalization of α ̌–η ̌ Proximal Contractive Mapping in Fuzzy Banach Spaces

Abstract

<p>The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems provide an approximate solution to the fixed-point equation Tҳ = ҳ. It is used to solve the problem to determine an approximate solution that is optimum. The main goal of this paper is to present new types of proximal contraction for nonself mappings in a fuzzy Banach space. At first, the notion of the best proximity point is presented. We introduce the notion of α ̌–η ̌-β ̌ proximal contractive. After that, the best proximity point theorem for such type of mappings in a fuzzy Banach space is proved. In addition, the concept of α ̌–η ̌-φ ̌ proximal contractive mapping is presented in a fuzzy Banach space and under specific conditions, the best proximity point theorem for such type of mapping is proved. Additionally, some examples are supplied to show the results' applicability.</p>

Best Proximity Point Results in Fuzzy Normed Spaces

“…A F N space (D, F N ,⊗) is called complete if every Cauchy sequence in D is convergent in D. In a F N space (D,F N ,⊗), Sabri and Ahmed (2022) presented the notion of fuzzy distance. Consider Ũ and Ṽ be subsets of (D,F M ,⊗) which are nonempty and Ũ • (𝜏), Ṽ • (𝜏) denoted by the following sets :…”

“…F N space was subsequently introduced in different methods by a large number of other mathematicians. F N space has been the topic of a considerable number of publications; for instance, see (Miheţ and Zaharia, 2014;Sabri and Ahmed, 2022;Kider and Kadhum, 2019;Sharma and Hazarika, 2020;Sabre, 2012;Konwar and Debnath, 2023;Sabri and Ahmed, 2023;Raghad, 2021).…”

Another Type of Fuzzy Inner Product Space

“…Raghad [7] introduced some types of fuzzy convergence sequences of operators that are defined on a standard fuzzy normed space and investigated some properties and relationships between these concepts. Authors in [8][9][10] proved some results about the ISSN: 0067-2904 best proximity point theorem and fixed point theorem in fuzzy normed spaces. J. R.Kider and M. N. Gheeab [11] introduced the definition of a general fuzzy normed space as a generalization of the notion of fuzzy normed space after that they investigated and proved the basic properties of this space.…”

Some Results of Fixed Point for Single Value Mapping in Fuzzy Normed Space with Applications

“…Many other mathematicians, such as Felbin [16], Cheng and Mordeson [17], and others, afterward presented the concept of fuzzy normed linear spaces in various ways. Besides, fuzzy normed linear spaces have been studied in several works, see [18][19][20][21][22][23]. In this paper, we will utilize Nadaban and Dzitac's notion of the fuzzy norm [24].…”