On closedness of fixed point sets of a new class of Suzuki type mappings in metric spaces

Abstract

This paper introduces a new class of Suzuki-type self-mappings in metricspaces, and investigates the geometric and topological properties of theirfixed point sets. We establish sufficient conditions under which the fixedpoint set is closed and examine the relationship between this new class andquasi-nonexpansive mappings. Necessary conditions for a mapping to bequasi-nonexpansive are also provided. These results are then applied tocharacterize the newly introduced class of Suzuki-type mappings. Severalillustrative examples are included, with particular attention to anapplication involving the design of a general class of activation functions.

Keywords

• Suzuki-type contractive mapping
• quasi-nonexpansive self map
• fixed point
• fixed point set

References

1. [1] B. Ali, H. Ali, T. Nazir and Z. Ali, Existence of fixed points of Suzuki-type contractions of quasi-metric spaces, Mathematics 11 (21), 4445, 2023.
2. [2] İ. Altun, Suzuki type P-contractive mappings, Proc. Math. Sci. 5 (1), 1-4, 2023.
3. [3] A. Betiuk-Pilarska, T. Domínguez Benavides and P. Lorenzo Ramírez, Fixed points for Suzuki type mappings by means of “ultra”-techniques, J. Nonlinear Convex Anal. 18 (10), 1753-1770, 2017.
4. [4] R.K. Bisht and N. Özgür, Discontinuous convex contractions and their applications in neural networks, Comput. Appl. Math. 40 (1), Paper No. 11, 11 pp, 2021.
5. [5] R.K. Bisht and N. Özgür, Geometric properties of discontinuous fixed point set of $(\epsilon -\delta )$ contractions and applications to neural networks, Aequationes Math. 94 (5), 847-863, 2020.
6. [6] U. Çelik and N. Özgür, Fixed point sets of self-mappings with a geometric viewpoint, Facta Univ. Ser. Math. Inform. 37 (5), 891-915, 2022.
7. [7] P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl. 320 (2), 983-987, 2006.
8. [8] P. Chaoha and W. Sudprakhon, Fixed point sets of Subrahmanyam maps, Linear Nonlinear Anal. 3 (2), 149-154, 2017.

9. [9] P. Chaoha, Fixed point sets of $\psi $-iterated nonexpansive maps, J. Nonlinear Convex Anal. 23 (1), 59-66, 2022.
10. [10] T. Domínguez Benavides and E. Llorens-Fuster, Iterated nonexpansive mappings, J. Fixed Point Theory Appl. 20 (3), Paper No. 104, 18 pp, 2018.
11. [11] S.R. Dubey, S.K. Singh and B.B. Chaudhuri, Activation functions in deep learning: A comprehensive survey and benchmark, Neurocomputing 503, 92-108, 2022.
12. [12] J. Gao, L. Dai and M. Xiao, Dynamical analysis of almost periodic solutions for delayed octonion-valued BAM neural networks, Comput. Appl. Math. 44, Paper No. 165, 28 pp, 2025.
13. [13] M. Hasanuzzaman and M. Imdad, Relation theoretic metrical fixed point results for Suzuki type $Z_{\mathcal{R}}$-contraction with an application, Aims Math. 5 (3), 2071-2087, 2020.
14. [14] A. Kaewcharoen and B. Panyanak, Fixed points for multivalued mappings in uniformly convex metric spaces, Int. J. Math. Math. Sci. Art. ID 163580, 9 pp, 2008.
15. [15] V. Kunc and J. Kléma, Three Decades of Activations: A comprehensive survey of 400 activation functions for neural networks, arXiv:2402.09092 [cs.LG].
16. [16] Y. Liu Y, Z. Wang and M. Xiao, $\mu$-stability and instability of multiple equilibrium points in delayed neural networks with general discontinuous activation functions, Information Sciences 660, 120129, 2024.
17. [17] C.E. Nwankpa, W. Ijomah, A. Gachagan and S. Marshall, Activation functions: Comparison of trends in practice and research for deep learning, In: 2nd International Conference on Computational Sciences and Technology, Jamshoro, Pakistan, 124- 133, 2021.
18. [18] N. Özgür, Fixed-disc results via simulation functions, Turkish J. Math. 43 (6), 2794- 2805, 2019.
19. [19] W. Qian, W. Xing and S. Fei, $H_{\infty }$ state estimation for neural networks with general activation function and mixed time-varying delays, IEEE Trans. Neural Netw. Learn. Syst. 32 (9), 3909-3918, 2021.
20. [20] S. Park, Comments on the Suzuki type fixed point theorems, Advances in the theory of nonlinear analysis and its application 7 (3), 67-78, 2023.
21. [21] T. Peng, H. Peng and F. Liu, Guided Deep Reinforcement Learning based on RBFARX Pseudo LQR in Single Stage Inverted Pendulum, In: 2022 International Conference on Intelligent Systems and Computational Intelligence (ICISCI), Changsha, China, 62-67, 2022.
22. [22] T.J. Piotrowski, R.L. Cavalcante and M. Gabor, Fixed points of nonnegative neural networks, J. Mach. Learn. Res. 25 (139), 1-40, 2024.
23. [23] S. Romaguera S. Basic contractions of Suzuki-type on quasi-metric spaces and fixed point results, Mathematics 10 (21), 3931, 2022.
24. [24] J. Xu, Y-Y. Cao, Y. Sun and J. Tang, Absolute exponential stability of recurrent neural networks with generalized activation function, IEEE transactions on neural networks 19 (6), 1075-1089, 2008.
25. [25] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2), 1088-1095, 2008.
26. [26] T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal. 71 (11), 5313-5317, 2009.
27. [27] L.E. Ward, Fixed point sets, Pacific J. Math. 47 (2), 553-565, 1973.