An approach of Banach algebra in fuzzy metric spaces with an application

Saif Ur Rehman; Arjamand Bano; Hassen Aydi; Choonkil Park; Saif Ur Rehman; Arjamand Bano; Hassen Aydi; Choonkil Park

doi:10.3934/math.2022527

AIMS Mathematics

2022, Volume 7, Issue 5: 9493-9507. doi: 10.3934/math.2022527

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Research article

An approach of Banach algebra in fuzzy metric spaces with an application

1.
Institute of Numerical Sciences, Department of Mathematics, Gomal University, Dera Ismail Khan 29050, Pakistan
2.
Université de Sousse, Institut Supérieur d'Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
3.
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa
5.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

Received: 09 January 2022 Revised: 27 February 2022 Accepted: 01 March 2022 Published: 14 March 2022
MSC : 47H07, 47H10, 54H25

The purpose of this paper is to present a new concept of a Banach algebra in a fuzzy metric space (FM-space). We define an open ball, an open set and prove that every open ball in an FM-space over a Banach algebra $ \mathcal{A} $ is an open set. We present some more topological properties and a Hausdorff metric on FM-spaces over $ \mathcal{A} $. Moreover, we state and prove a fuzzy Banach contraction theorem on FM-spaces over a Banach algebra $ \mathcal{A} $. Furthermore, we present an application of an integral equation and will prove a result dealing with the integral operators in FM-spaces over a Banach algebra.
- fixed point,
- Banach algebra $ \mathcal{A} $,
- fuzzy metric space over $ \mathcal{A} $,
- integral equation
Citation: Saif Ur Rehman, Arjamand Bano, Hassen Aydi, Choonkil Park. An approach of Banach algebra in fuzzy metric spaces with an application[J]. AIMS Mathematics, 2022, 7(5): 9493-9507. doi: 10.3934/math.2022527

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Abstract

The purpose of this paper is to present a new concept of a Banach algebra in a fuzzy metric space (FM-space). We define an open ball, an open set and prove that every open ball in an FM-space over a Banach algebra $ \mathcal{A} $ is an open set. We present some more topological properties and a Hausdorff metric on FM-spaces over $ \mathcal{A} $. Moreover, we state and prove a fuzzy Banach contraction theorem on FM-spaces over a Banach algebra $ \mathcal{A} $. Furthermore, we present an application of an integral equation and will prove a result dealing with the integral operators in FM-spaces over a Banach algebra.

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