On fixed points, their geometry and application to satellite web coupling problem in $ \mathcal{S}- $metric spaces

Meena Joshi; Anita Tomar; Thabet Abdeljawad; Meena Joshi; Anita Tomar; Thabet Abdeljawad

doi:10.3934/math.2023220

AIMS Mathematics

2023, Volume 8, Issue 2: 4407-4441. doi: 10.3934/math.2023220

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On fixed points, their geometry and application to satellite web coupling problem in $ \mathcal{S}- $metric spaces

1.
S. S. J. Campus, Soban Singh Jeena University, Almora 263601, Uttarakhand, India
2.
Pt. L. M. S. Campus, Sridev Suman Uttarakhand University, Rishikesh 249201, Uttarakhand, India
3.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 17 September 2022 Revised: 18 November 2022 Accepted: 23 November 2022 Published: 05 December 2022
MSC : 47H10, 54H25, 55M20, 37E10

We introduce an $ \mathcal{M-} $class function in an $ \mathcal{S-} $metric space which is a viable, productive, and powerful technique for finding the existence of a fixed point and fixed circle. Our conclusions unify, improve, extend, and generalize numerous results to a widespread class of discontinuous maps. Next, we introduce notions of a fixed ellipse (elliptic disc) in an $ \mathcal{S}- $metric space to investigate the geometry of the collection of fixed points and prove fixed ellipse (elliptic disc) theorems. In the sequel, we validate these conclusions with illustrative examples. We explore some conditions which eliminate the possibility of the identity map in the existence of an ellipse (elliptic disc). Some remarks, propositions, and examples to exhibit the feasibility of the results are presented. The paper is concluded with a discussion of activation functions that are discontinuous in nature and, consequently, utilized in a neural network for increasing the storage capacity. Towards the end, we solve the satellite web coupling problem and propose two open problems.
- {continuity,
- fixed ellipse,
- fixed elliptic disc,
- $ \mathcal{M-} $class function,
- $ \mathcal{S}- $Caristi map }
Citation: Meena Joshi, Anita Tomar, Thabet Abdeljawad. On fixed points, their geometry and application to satellite web coupling problem in $ \mathcal{S}- $metric spaces[J]. AIMS Mathematics, 2023, 8(2): 4407-4441. doi: 10.3934/math.2023220

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Abstract

We introduce an $ \mathcal{M-} $class function in an $ \mathcal{S-} $metric space which is a viable, productive, and powerful technique for finding the existence of a fixed point and fixed circle. Our conclusions unify, improve, extend, and generalize numerous results to a widespread class of discontinuous maps. Next, we introduce notions of a fixed ellipse (elliptic disc) in an $ \mathcal{S}- $metric space to investigate the geometry of the collection of fixed points and prove fixed ellipse (elliptic disc) theorems. In the sequel, we validate these conclusions with illustrative examples. We explore some conditions which eliminate the possibility of the identity map in the existence of an ellipse (elliptic disc). Some remarks, propositions, and examples to exhibit the feasibility of the results are presented. The paper is concluded with a discussion of activation functions that are discontinuous in nature and, consequently, utilized in a neural network for increasing the storage capacity. Towards the end, we solve the satellite web coupling problem and propose two open problems.

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