Fractal Attractor Associated with Banach and Reich Contractions on Controlled Strong \(b-\)Metric Space

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DOI: https://doi.org/10.28924/2291-8639-24-2026-297
Published: Sep 21, 2026
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M. Dhanzeem Ahmed, D. Easwaramoorthy, Fractal Attractor Associated with Banach and Reich Contractions on Controlled Strong \(b-\)Metric Space, Int. J. Anal. Appl., 24 (2026), 297.

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M. Dhanzeem Ahmed, D. Easwaramoorthy

Abstract

This article aims to derive the controlled strong \(b-\)Banach fractal (CSbB-Fractal) by Banach contraction and the controlled strong \(b-\)Reich fractal (CSbR-Fractal) by Reich contraction on the controlled strong \(b-\)metric space (CSbMS). In this context, a new class of Iterated function system (IFS) and its fractal attractors are constructed as a general case of certain existing systems by the Hutchinson-Barnsley (HB) theory on CSbMS and illustrated with examples. Also, the Reich fixed point theorem is discussed with examples and applications. Furthermore, the findings of the study in generalized spaces can offer a new approach to generate a novel type of fractal attractor.

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References

  1. B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, New York, 1983.
  2. J.E. Hutchinson, Fractals and Self Similarity, Indiana Univ. Math. J. 30 (1981), 713–747. https://doi.org/10.1512/iumj.1981.30.30055.
  3. M.F. Barnsley, Fractals Everywhere, Academic Press, 1993.
  4. M.F. Barnsley, SuperFractals, Cambridge University Press, New York, 2006.
  5. K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, Hoboken, 2003.
  6. G. Edgar, Measure, Topology, and Fractal Geometry, Springer New York, 2008. https://doi.org/10.1007/978-0-387-74749-1.
  7. S. Banerjee, D. Easwaramoorthy, A. Gowrisankar, Fractal Functions, Dimensions and Signal Analysis, Springer International Publishing, 2021. https://doi.org/10.1007/978-3-030-62672-3.
  8. Z. Li, B. Selmi, H. Zyoudi, A Comprehensive Approach to Multifractal Analysis, Expos. Math. 43 (2025), 125690. https://doi.org/10.1016/j.exmath.2025.125690.
  9. H. Jebali, On the Study of a New Class of Local Fractal Functions, J. Anal. 33 (2025), 2225–2244. https://doi.org/10.1007/s41478-025-00917-6.
  10. D. Wójcik, I. Białynicki-Birula, K. Życzkowski, Time Evolution of Quantum Fractals, Phys. Rev. Lett. 85 (2000), 5022–5025. https://doi.org/10.1103/PhysRevLett.85.5022.
  11. C. Shaju, Kamal, A. Panwar, Seismic Trends in Indian Plate: A Study on Epicentral Characteristics by Using a Driven Iterated Function System, Chaos Solitons Fractals 173 (2023), 113680. https://doi.org/10.1016/j.chaos.2023.113680.
  12. S. Abdulla, K.M. Reddy, Optimizing the Neural Network and Iterated Function System Parameters for Fractal Approximation Using a Modified Evolutionary Algorithm, Sci. Rep. 15 (2025), 13720. https://doi.org/10.1038/s41598-025-94821-5.
  13. T. Nazir, A. Zakaria Idriss, Iterated Function Systems of Generalized Multivalued Mappings in Partial Metric Spaces, Contemp. Math. 6 (2025), 5368–5387. https://doi.org/10.37256/cm.6520256370.
  14. I. Abraham, R. Miculescu, A. Mihail, Relational Generalized Iterated Function Systems, Chaos Solitons Fractals 182 (2024), 114823. https://doi.org/10.1016/j.chaos.2024.114823.
  15. C. Thangaraj, D. Easwaramoorthy, Fractals via Controlled Fisher Iterated Function System, Fractal Fract. 6 (2022), 746. https://doi.org/10.3390/fractalfract6120746.
  16. I. Savu, New Aspects Concerning IFSs Consisting of Continuous Functions Satisfying Banach’s Orbital Condition, J. Fixed Point Theory Appl. 21 (2019), 62. https://doi.org/10.1007/s11784-019-0700-4.
  17. C. Thangaraj, D. Easwaramoorthy, B. Selmi, B.P. Chamola, Generation of Fractals via Iterated Function System of Kannan Contractions in Controlled Metric Space, Math. Comput. Simul. 222 (2024), 188–198. https://doi.org/10.1016/j.matcom.2023.08.017.
  18. P. Athul, D. Ramesh Kumar, Some Coupled Fixed Point Theorems for (ψ,φ)-Contraction with Applications to Fractals, Filomat 38 (2024), 9305–9320. https://doi.org/10.2298/FIL2426305P.
  19. L.F. Barnsley, M.F. Barnsley, A. Vince, Tiling Iterated Function Systems, Chaos Solitons Fractals 182 (2024), 114807. https://doi.org/10.1016/j.chaos.2024.114807.
  20. D. Easwaramoorthy, R. Uthayakumar, Analysis on Fractals in Fuzzy Metric Spaces, Fractals 19 (2011), 379–386. https://doi.org/10.1142/S0218348X11005543.
  21. R. Pasupathi, A.K.B. Chand, M.A. Navascués, Cyclic Iterated Function Systems, J. Fixed Point Theory Appl. 22 (2020), 58. https://doi.org/10.1007/s11784-020-00790-9.
  22. M. Abbas, R. Anjum, H. Iqbal, Generalized Enriched Cyclic Contractions with Application to Generalized Iterated Function System, Chaos Solitons Fractals 154 (2022), 111591. https://doi.org/10.1016/j.chaos.2021.111591.
  23. R. Uthayakumar, D. Easwaramoorthy, Hutchinson-Barnsley Operator in Fuzzy Metric Spaces, Int. J. Math. Comput. Sci. 5 (2011), 1418–1422. https://doi.org/10.5281/zenodo.1074795.
  24. N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled Metric Type Spaces and the Related Contraction Principle, Mathematics 6 (2018), 194. https://doi.org/10.3390/math6100194.
  25. W. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces, Springer International Publishing, 2014. https://doi.org/10.1007/978-3-319-10927-5.
  26. D. Santina, W.A. Mior Othman, K.B. Wong, N. Mlaiki, New Generalization of Metric-Type Spaces—Strong Controlled, Symmetry 15 (2023), 416. https://doi.org/10.3390/sym15020416.
  27. D. Santina, W.A. Mior Othman, K.B. Wong, N. Mlaiki, Exploring Strong Controlled Partial Metric Type Spaces: Analysis of Fixed Points and Theoretical Contributions, Heliyon 10 (2024), e39525. https://doi.org/10.1016/j.heliyon.2024.e39525.
  28. C. Thangaraj, R. Valarmathi, D. Easwaramoorthy, D. Ramesh Kumar, B.P. Chamola, Generation of Fractal Attractor for Controlled Metric Based Dynamical Systems, Contemp. Math. 5 (2024), 6165–6188. https://doi.org/10.37256/cm.5420245323.
  29. M. Dhanzeem Ahmed, D. Easwaramoorthy, Fractal Attractor via Controlled Strong b-Kannan Iterated Function System, Int. J. Anal. Appl. 23 (2025), 155. https://doi.org/10.28924/2291-8639-23-2025-155.
  30. S. Reich, Some Remarks Concerning Contraction Mappings, Can. Math. Bull. 14 (1971), 121–124. https://doi.org/10.4153/CMB-1971-024-9.
  31. T. Kamran, M. Samreen, Q. UL Ain, A Generalization of b-Metric Space and Some Fixed Point Theorems, Mathematics 5 (2017), 19. https://doi.org/10.3390/math5020019.