Document Type : Research Paper


Department of Mathematics, Siksha-Bhavana, Visva-Bharati, Santiniketan-731235, West Bengal, India.


Our present work is the extension of the line of research in the context of $\phi$-metric spaces. We introduce the notion of fixed circle and obtain suitable conditions for the existence and uniqueness of fixed circles for self mappings. Additionally, we present some figures and examples in support of our  results. 


Main Subjects

1. M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), pp. 1–72.
2. F. Hausdorff, Grundzuge der Mengenlehre (Fundamentals of Set Theory), Leipzig, Von Veit, (1914).
3. S. Banach, Sur les operations dans las ensembles abstraits et leur application aux equations integrales, Find. Math., 3 (1922), 133-181.
4. S. Gahler, 2-metrische raume und ihre topologische struktur, Math. Nachr., 26 (1963), pp. 115-148.
5. S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav., 1 (1993), pp. 5-11.
6. M. Iqbal, A. Batool, O. Ege and M. de la Sen, Fixed point of generalized weak contraction in b-metric spaces, J. Funct. Spaces, (2021), Article ID 2042162, 8 pages.
7. M. Iqbal, A. Batool, O. Ege and M. de la Sen, Fixed point of almost contraction in b-metric spaces, Journal of Mathematics, (2020), Article ID 3218134, pp. 1-6.
8. A.J. Gnanaprakasam, G. Mani, O. Ege, A. Aloqaily and N. Mlaiki, New fixed point results in orthogonal b-metric spaces with related applications, Mathematics, 11 (2023), pp. 1-18.
9. Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), pp. 289–297.
10. O. Ege, C. Park and A.H. Ansari, A different approach to complex valued Gb-metric spaces, Adv. Difference Equ., 152 (2020), pp. 1-13.
11. S. Sedghi, N. Shobe and A. Aliouche, A generalization of fixed point theorems in S-metric space, Math. Vesn., 64 (2012), pp. 258-266.
12. T.L. Shateri, O. Ege and M. de la Sen, Common fixed point on the bv(s)-metric space of function-valued mappings, AIMS Math., 6 (2021), pp. 1065-1074.
13. L.G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 33 (2007), pp. 1468-1476.
14. A. Das and T. Bag, A generalization to parametric metric spaces, Int. J. of Nonlinear Anal. Appl., 14 (2023), pp. 229-244.
15. O. Ege and M. de la Sen, A new perspective on parametric metric spaces, Mathematics, 7 (2019), pages 1008.
16. A. Das, A. Kundu and T. Bag, A new approach to generalize metric functions, Int. J. Nonlinear Anal. Appl., 14 (2023), pp. 279–298.
17. N.Y. Ozgur and N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc., (2017).
18. S. Mohamad, Global exponential stability in DCNNs with distributed delays and unbounded activations, J. Comput. Appl. Math., 205 (2007), pp. 161-173.
19. Y. Zhang and Q.G. Wang, Stationary oscillation for high-order Hopfield neural networks with time delays and impulses, J. Comput. Appl. Math., 231 (2009), pp. 473-477.
20. N.Y. Ozgur and N. Tas, Some fixed-circle theorems and discontinuity at fixed circle, AIP Conf. Proc., 1926, 020048, (2018).
21. N.Y. Ozgur and N. Tas, A new contribution to discontinuity at fixed point, arxiv:1705.03699v2[math.MG], (2018).
22. N. Ozdemir, B.B. Iskender and N.Y. Ozgur, Complex valued neural network with Mobius activation function, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011), pp. 4698-4703.
23. N.Y. Ozgur and N. Tas, New fixed-circle results on S-metric spaces, Bull. Math. Anal. Appl., 9 (2017), pp. 10-23.
24. N.Y. Ozgur and N. Tas, Fixed-circle problem on S-metric spaces with a geometric viewpoint, arxiv:1704.08838v3[math.MG], (2018).
25. D. Gopal, J. M. Moreno and N. Ozgur, On fixed figure problems in fuzzy metric spaces, Kybernetika, 59 (2023), pp. 110-129.
26. A. Kamal, A geometric approach to fixed point theorems for mappings satisfying implicit relations in quasi multiplicative metric spaces, DOI: 10.21203/