An introduction to fixed-circle problem on soft metric spaces

Subject Areas : Fixed point theory

N. Taş ¹ (Department of Mathematics, Balikesir University, 10145, Bal\i kesir, Turkey)
O. B. Özbakir ² (Department of Mathematics, Faculty of Sciences, Ege University, 35100, Izmir, Turkey)

Received: 2023-10-24 Accepted : 2023-12-13 Published : 2023-12-14

Keywords: Soft circle, soft fixed circle, soft metric space,

Abstract :

Recently, soft set theory has been extensively studied both theoretically and practically with different approaches. On the other hand, fixed-circle problem has been investigated as a geometric generalization of fixed-point theory and this problem can be applied to some applicable areas. With these two perspectives, in this paper, we obtain some soft fixed-circle results using different auxiliary functions on a soft metric space. To do this, we are inspired various contractive conditions. The obtained results can be considered as an existence or uniqueness theorem. The proved theorems are supported by some illustrative examples. Finally, we give a list of geometric consequences of these results.‎

References:

[1] M. Abbas, G. Murtaza, S. Romaguera, Soft contraction theorem, J. Nonlinear Convex Anal. 16 (2015), 423-435.

[2] S. Banach, Sur les opÌerations dans les ensembles abstraits et leur application auxÌequations intÌegrales, Fund. Math. 3 (1) (1922), 133-181.

[3] J. Caristi, Fixed point theorems for mapping satisfying inwardness conditions, Trans. Am. Math. Soc. 215 (1976), 241-251.

[4] S. K. Chatterjea, Fixed point theorems, C.R. Acad. Bulgare Sci. 25 (1972), 727-730.

[5] S. Das, S. K. Samanta, Soft metric, Ann. Fuzzy Math. Inform. 6 (1) (2013), 77-94.

[6] S. Das, S. K. Samanta, Soft real sets, soft real numbers and their properties, J. Fuzzy Math. 20 (3) (2012), 551-576.

[7] I. Demir, N-soft mappings with application in medical diagnosis, Math. Methods Appl. Sci. 44 (8) (2011), 7343-7358.

[8] I. Demir, O. B. Özbakır, An extension of Lowen’s uniformity to the fuzzy soft sets, Konuralp J. Math. 6 (2) (2018), 321-331.

[9] I. Demir, O. B.Özbakır, Some properties of fuzzy soft proximity spaces, Sci. World J. 10 (2015), 10:752634.

[10] G. Z. Ercinar, Some Geometric Properties of Fixed Points, Ph.D. Thesis, Eskisehir Osmangazi University, 2020.

[11] F. Feng, Y. B. Jun, X. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008), 2621-2628.

[12] F. Feng, C. X. Li, B. Davvaz, M. I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput. 14 (2010), 8999-9911.

[13] A. C. Güler, E. D. Yıldırım, O. B.Ö zbakır, A fixed point theorem on soft G-metric spaces, J. Nonlinear Sci. Appl. 9 (2016), 885-894.

[14] R. Irkin, N. Y.Özgür, N. Tas, Optimization of lactic acid bacteria viability using fuzzy soft set modelling, Int. J. Optim. Control, Theor. Appl. 8 (2) (2018), 266-275.

[15] M. Joshi, A. Tomar, S. K. Padaliya, Fixed point to fixed ellipse in metric spaces and discontinuous activation function, Appl. Math. E-Notes. 21 (2021), 225-237.

[16] A. Kalaichelvi, P.H. Malini, Application of fuzzy soft sets to investment decision making problem, Int. J. Math. Sci. Appl. 1 (3) (2011), 1583-1586.

[17] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.

[18] F. Karaca, N. Tas, Decision making problem for life and non-life insurances, J. BAUN Inst. Sci. Technol. 20 (1) (2018), 572-588.

[19] P. K. Maji, P. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003), 555-562.

[20] N. Mlaiki, N.Özgür, N. Tas, D. Santina, On the fixed circle problem on metric spaces and related results, Axioms. 12 (2023), 12:401.

[21] D. Molodtsov, Soft set theory-first results, Comput. Math. Appl. 37 (1999), 19-31.

[22] N.Özgür, Fixed-disc results via simulation functions, Turkish J. Math. 43 (6) (2019), 2794-2805.

[23] N. Y.Özgür, N. Tas, A note on “application of fuzzy soft sets to investment decision making problem”, J. New Theory. 7 (2015), 1-10.

[24] N.Özgür, N. Tas, Geometric properties of fixed points and simulation functions, Adv. Stud. Euro-Tbil. Math. J. 16 (4) (2023), 91-108.

[25] N. Y.Özgür, N. Tas, Some Fixed-Circle Theorems and Discontinuity at Fixed Circle, AIP Conference Proceedings 1926, 2018.

[26] N. Y.Özgür, N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. 42 (4) (2019), 1433-1449.

[27] M. Riaz, S. T. Tehrim, On bipolar fuzzy soft topology with decision-making, Soft Comput. 24 (24) (2020), 18259-18272.

[28] M. Riaz, N. Cagman, I. Zareef, M. Aslam, N-soft topology and its applications to multi-criteria group decision making, J. Intell. Fuzzy Syst. 36 (6) (2019), 6521-6536.

[29] M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl. 61 (7) (2011), 1786-1799.

[30] N. Tas, Bilateral-type solutions to the fixed-circle problem with rectified linear units application, Turkish J. Math. 44 (4) (2020), 1330-1344.

[31] N. Tas, Suzuki-Berinde type fixed-point and fixed-circle results on S-metric spaces, J. Linear. Topol. Algebra. 7 (3) (2018), 233-244.

[32] N. Tas, N.Özgür, New Fixed-Figure Results on Metric Spaces, Fixed Point Theory and Fractional Calculus. Forum for Interdisciplinary Mathematics, Singapore, 2022.

[33] N. Tas, N. Y.Özgür, On a generalized soft metric space, Acta. Univ. Apul. 57 (2019), 41-62.

[34] N. Tas, N. Y.Özgür, P. Demir, An application of soft set and fuzzy soft set theories to stock management, J. Nat. Appl. Sci. 21 (2) (2017), 791-196.

[35] R. Uma, P. Murugadas, S. Sriram, Generalized inverse of fuzzy neutrosophic soft matrix, J. Linear. Topol. Algebra. 6 (2) (2017), 109-123.

[36] D. Wardowski, On a soft mapping and its fixed points, Fixed Point Theory Appl. (2013), 2013:182.

[37] M. I. Yazar, C. G. Aras, S. Bayramov, Fixed point theorems of soft contractive mappings, Filomat. 30 (2) (2016), 269-279.

[38] I. Zorlutuna, M. Akdag, W. K. Min, S. Atmaca, Remarks on soft topological spaces, Ann. Fuzzy Math. Inform. 3 (2) (2012), 171-185.