Fixed point to fixed circle and activation function in partial metric space

• Autorzy: Tomar Anita , Joshi Meena , Padaliya S. K.

Identyfikatory

DOI

10.1515/jaa-2021-2057

• Języki publikacji: EN

Abstrakty

EN

We familiarize a notion of a fixed circle in a partial metric space, which is not necessarily the same as a circle in a Euclidean space. Next, we establish novel fixed circle theorems and verify these by illustrative examples with geometric interpretation to demonstrate the authenticity of the postulates. Also, we study the geometric properties of the set of non-unique fixed points of a discontinuous self-map in reference to fixed circle problems and responded to an open problem regarding the existence of a maximum number of points forwhich there exist circles. This paper is concluded by giving an application to activation function to exhibit the feasibility of results, thereby providing a better insight into the analogous explorations.

Słowa kluczowe

EN

Banach contraction; fixed circle; ρ-Caristi map; neural network

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2022
• Tom: Vol. 28, nr 1
• Strony: 57--66
• Opis fizyczny: Bibliogr. 15 poz., wykr.

Twórcy

autor

Tomar Anita

• Department of Mathematics, Government Degree College Thatyur (Tehri Garhwal), Uttarakhand, India

• https://orcid.org/0000-0001-8033-856X

autor

Joshi Meena

• Department of Mathematics, S. G. R. R. (P. G.) College, Dehradun, India

• https://orcid.org/0000-0002-1562-0988

autor

Padaliya S. K.

• Department of Mathematics, S. G. R. R. (P. G.) College, Dehradun, India

Bibliografia

• [1] Ö. Acar, I. Altun and S. Romaguera, Caristi’s type mappings on complete partial metric spaces, Fixed Point Theory 14 (2013), no. 1, 3-9.
• [2] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133-181.
• [3] R. K. Bisht and N. Y. Özgür, Geometric properties of discontinuous fixed point set of (ϵ − δ) contractions and applications to neural networks, Aequationes Math. 94 (2020), no. 5, 847-863.
• [4] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251.
• [5] S. G. Matthews, Partial metric topology, Ann. New York Acad. Sci. 728 (1994), 183-197.
• [6] X. Nie and W. X. Zheng, On multistability of competitive neural networks with discontinuous activation functions, in: 4th Australian Control Conference, IEEE Press, Piscataway (2014), 245-250.
• [7] N. Y. Özgür, Fixed-disc results via simulation functions, Turkish J. Math. 43 (2019), no. 6, 2794-2805.
• [8] N. Y. Özgür and N. Taş, Some fixed-circle theorems and discontinuity at fixed circle, AIP Conf. Proc. 1926 (2018), Article ID 020048.
• [9] N. Y. Özgür and N. Taş, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. 42 (2019), no. 4, 1433-1449.
• [10] R. P. Pant, N. Özgür, N. Taş, A. Pant and M. C. Joshi, New results on discontinuity at fixed point, J. Fixed Point Theory Appl. 22 (2020), no. 2, Paper No. 39.
• [11] R. P. Pant, N. Y. Özgür and N. Taş, Discontinuity at fixed points with applications, Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 4, 571-589.
• [12] R. P. Pant, N. Y. Özgür and N. Taş, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 1, 499-517.
• [13] B. E. Rhoades, Contractive definitions and continuity, in: Fixed Point Theory and Its Applications (Berkeley 1986), Contemp. Math. 72, American Mathematical Society, Providence (1988), 233-245.
• [14] S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl. 2010 (2010), Article ID 493298.
• [15] N. Taş, Bilateral-type solutions to the fixed-circle problem with rectified linear units application, Turkish J. Math. 44 (2020), no. 4, 1330-1344.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).