A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor

Haroon Ahmad; Mohammad Azhar Peerzada; Wardah Wardah

A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor

Abstract

In this paper, we study a generalized contraction in a complete metric space using a different approach from classical methods, providing a more flexible framework than Banach, Kannan, or Chatterjea contractions. We present two illustrative examples, one involving a continuous mapping and the other a discontinuous mapping, which satisfy the generalized contraction but fail under classical contraction conditions. As an application, we employ this generalized contraction to establish the existence and uniqueness of solutions for a fractional-order Thomas cyclically symmetric attractor via the Caputo-Fabrizio derivative. The results reveal the complex dynamical behavior and sensitivity to initial conditions, demonstrating the effectiveness of the generalized contraction approach in analyzing nonlinear and chaotic systems.

Keywords

Fixed points, Caputo-Fabrizio, Thomas cyclically symmetric attractor

References

[1] S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[2] R. Kannan, Some results on fixed points II, Bull. Calcutta Math. Soc., 60(1) (1968), 71–76.
[3] S. K. Chatterjee, Fixed point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727–730.
[4] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971), 121–124.
[5] L. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc., 45(2) (1974), 267–273.
[6] G. E. Hardy, T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16(2) (1973), 201–206.
[7] M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 1–74.
[8] I. A. Bakhtin, The contraction mapping principle in quasi-metric spaces, Funct. Anal. Ulyanovsk Gos. Ped. Inst., 30 (1989), 26–37.
[9] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5–11.
[10] T. Kamran, M. Samreen, Q. Ul Ain, A generalization of b-metric space and some fixed point theorems, Mathematics, 5(2) (2017), 19. https://doi.org/10.3390/math5020019

[11] N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics, 6(10) (2018), 194. https://doi.org/10.3390/math6100194
[12] T. Abdeljawad, N. Mlaiki, H. Aydi, N. Souayah, Double controlled metric type spaces and some fixed point results, Mathematics, 6(12) (2018), 320. https://doi.org/10.3390/math6120320
[13] H. Ahmad, M. Younis, M. E. Köksal, Double controlled partial metric type spaces and convergence results, J. Math., 2021 (2021), Article ID 7008737. https://doi.org/10.1155/2021/7008737
[14] M. Younis, H. Ahmad, L. Chen, M. Han, Computation and convergence of fixed points in graphical spaces with an application to elastic beam deformations, J. Geom. Phys., 192 (2023), Article ID 104955. https://doi.org/10.1016/j.geomphys.2023.104955
[15] H. Ahmad, F. U. Din, M. Younis, A fixed point analysis of fractional dynamics of heat transfer in chaotic fluid layers, J. Comput. Appl. Math., 453 (2025), Article ID 116144. https://doi.org/10.1016/j.cam.2024.116144
[16] M. Younis, H. Ahmad, M. Öztürk, D. Singh, A novel approach to the convergence analysis of chaotic dynamics in fractional order Chua’s attractor model employing fixed points, Alexandria Eng. J., 110 (2025), 363–375. https://doi.org/10.1016/j.aej.2024.10.001
[17] H. Ahmad, Analysis of fixed points in controlled metric type spaces with application, Recent Developments in Fixed-Point Theory, Springer, (2024), 225–240.
[18] H. U. Rehman, D. Gopal, P. Kumam, Generalizations of Darbo’s fixed point theorem for new condensing operators with application to a functional integral equation, Demonstratio Math., 52(1) (2019), 166–182. https://doi.org/10.1515/dema-2019-0012
[19] H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta Math., 13(1) (1890), 1–270.
[20] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20(2) (1963), 130–141. https://doi.org/10.1177/0309133308091948
[21] R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459–467. https://doi.org/10.1038/261459a0
[22] M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 19 (1978), 25–52. https://doi.org/10.1007/BF01020332
[23] B. B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Company, San Francisco, 1982.
[24] J. Liouville, Mémoire sur quelques questions de géométrie et de mécanique, J. École Polytech., 13 (1832), 1–69.
[25] M. Caputo, Linear models of dissipation whose Q is almost frequency independent. II, Geophys. J. Int., 13(5) (1967), 529–539.
[26] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1(2) (2015), 1–3.
[27] A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel, theory and application to heat transfer model, Thermal Sci., 20(2) (2016), 763–769. https://doi.org/10.2298/TSCI160111018A
[28] J. Matkowski, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc., 62(2) (1977), 344–348.
[29] R. Thomas, Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, labyrinth chaos, Int. J. Bifurcation Chaos, 9(10) (1999), 1889-1905.
[30] H. Ahmad, F. U. Din, M. Younis, A novel Ćirić–Reich–Rus fixed point approach for the existence and uniqueness criterion of a fractional-order Aizawa chaotic system, Chaos Solitons Fractals, 200 (2025), Article ID 116932. https://doi.org/10.1016/j.chaos.2025.116932
[31] H. Ahmad, F. U. Din, M. Younis, L. Chen, A simple interpolative contraction approach for analyzing existence and uniqueness of solutions in the fractional-order King Cobra model, Chaos Solitons Fractals, 199 (2025), Article ID 116578. https://doi.org/10.1016/j.chaos.2025.116578
[32] M. Younis, H. Ahmad, M. Öztürk, F. U. Din, M. Qasim, Unveiling fractional-order dynamics: A new method for analyzing Rossler chaos, J. Comput. Appl. Math., 468 (2025), 116639. https://doi.org/10.1016/j.cam.2025.116639
[33] H. Ahmad, F. U. Din, M. Younis, Fractional order Lorenz dynamics: Investigating existence and uniqueness via basic contraction, J. Appl. Math. Comput., 71 (2025), 6827–6858. https://doi.org/10.1007/s12190-025-02550-9

Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis, Dynamical Systems in Applications

Journal Section

Research Article

Authors

Haroon Ahmad ^*
0000-0002-3907-0554
Pakistan

Mohammad Azhar Peerzada
0000-0002-3907-0555
Türkiye

Wardah Wardah This is me
0009-0005-7529-8432
Pakistan

Early Pub Date

December 29, 2025

Publication Date

December 29, 2025

Submission Date

November 30, 2025

Acceptance Date

December 23, 2025

Published in Issue

Year 2025 Volume: 1 Number: 1

IZ

https://izlik.org/JA37ZD52XS

APA

Ahmad, H., Peerzada, M. A., & Wardah, W. (2025). A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor. Sakarya Journal of Mathematics, 1(1), 22-36. https://izlik.org/JA37ZD52XS

AMA

1.Ahmad H, Peerzada MA, Wardah W. A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor. Sakarya Journal of Mathematics. 2025;1(1):22-36. https://izlik.org/JA37ZD52XS

Chicago

Ahmad, Haroon, Mohammad Azhar Peerzada, and Wardah Wardah. 2025. “A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor”. Sakarya Journal of Mathematics 1 (1): 22-36. https://izlik.org/JA37ZD52XS.

EndNote

Ahmad H, Peerzada MA, Wardah W (December 1, 2025) A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor. Sakarya Journal of Mathematics 1 1 22–36.

IEEE

[1]H. Ahmad, M. A. Peerzada, and W. Wardah, “A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor”, Sakarya Journal of Mathematics, vol. 1, no. 1, pp. 22–36, Dec. 2025, [Online]. Available: https://izlik.org/JA37ZD52XS

ISNAD

Ahmad, Haroon - Peerzada, Mohammad Azhar - Wardah, Wardah. “A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor”. Sakarya Journal of Mathematics 1/1 (December 1, 2025): 22-36. https://izlik.org/JA37ZD52XS.

JAMA

1.Ahmad H, Peerzada MA, Wardah W. A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor. Sakarya Journal of Mathematics. 2025;1:22–36.

MLA

Ahmad, Haroon, et al. “A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor”. Sakarya Journal of Mathematics, vol. 1, no. 1, Dec. 2025, pp. 22-36, https://izlik.org/JA37ZD52XS.

Vancouver

1.Haroon Ahmad, Mohammad Azhar Peerzada, Wardah Wardah. A Fixed Point Approach to the Existence and Uniqueness of Solutions for Thomas Cyclically Symmetric Attractor. Sakarya Journal of Mathematics [Internet]. 2025 Dec. 1;1(1):22-36. Available from: https://izlik.org/JA37ZD52XS