Honam Mathematical Journal (호남수학학술지)

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ON M-ITERATIVE SCHEME FOR MAPPINGS WITH ENRICHED (C) CONDITION IN BANACH SPACES

  • Junaid Ahmad (Department of Mathematics and Statistics, International Islamic University)
  • Received : 2023.09.28
  • Accepted : 2024.06.29
  • Published : 2024.12.20
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Abstract

In this paper, we approximate fixed points of mappings satisfying enriched (C) condition under a modified three-step M-iterative scheme in a Banach space setting. First, we establish a weak convergence theorem and then obtain several strong convergence theorems for our iterative scheme under some mild conditions. A numerical example of mappings satisfying enriched (C) condition is used that does not satisfy the ordinary (C) condition to support our main outcome. Numerical computations and graphs obtained from different iterative schemes show that the studied scheme provides a better rate of convergence as compared to some other schemes of the literature. As an application of our main result, we provide a projection type iterative scheme for solving split feasibility problems (SFP) on a Hilbert space setting.

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Acknowledgement

The author would like to thank the unknown reviewers for useful suggestions that improved the first version of the paper.

References

  1. M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Math. Vesnik 66, (2014), 223-234.
  2. T. Abdeljawad, K. Ullah, J. Ahmad, and Z. Ma, On the convergence of an iterative process for enriched Suzuki nonexpansive mappings in Banach spaces, AIMS Math., 7 (2022), 20247-20258. https://doi.org/10.3934/math.20221108
  3. R. P. Agarwal, D. O'Regon, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications Series, Topological Fixed Point Theory and Its Applications, Springer: New York, NY, USA, 2009.
  4. R. P. Agarwal, D. O'Regon, and D. R. Sahu, Iterative construction of fixed points of nearly asymtotically non-expansive mappings, J. Nonlinear Convex Anal. 8 (2007), 61-79.
  5. S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3, (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181
  6. M. Basarir and A. Sahin, On the strong and ∆-convergence of S-iteration process for generalized nonexpansive mappings on CAT(0) space, Thai J. Math. 12 (2014), no. 3, 549-559.
  7. V. Berinde, Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math. 35 (2019), 293-304. https://doi.org/10.37193/CJM.2019.03.04
  8. F. E. Browder, onexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54, (1965), 1041-1044. https://doi.org/10.1073/pnas.54.4.1041
  9. C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Prob. 20 (2004), 103-120. https://doi.org/10.1088/0266-5611/20/1/006
  10. Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algor. 8 (1994), 221-239. https://doi.org/10.1007/BF02142692
  11. J. A. Clarkson, Uniformly convex spaces, Trans. Am. Math. Soc. 40 (1936), 396-414. https://doi.org/10.1090/S0002-9947-1936-1501880-4
  12. D. Gohde, Zum Prinzip der Kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258. https://doi.org/10.1002/mana.19650300312
  13. S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1976), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
  14. M. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  15. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042
  16. Z. Opial, Weak and strong convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc. 73 (1967), 591-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
  17. E. M. Picard, Memorie sur la theorie des equations aux derivees partielles et la method des approximation ssuccessives, J. Math. Pure Appl. 6 (1980), 145-210.
  18. A. Sahin, Some new results of M-iteration process in hyperbolic spaces, Carpathian J. Math. 35 (2019), no. 2, 221-232. https://doi.org/10.37193/CJM.2019.02.10
  19. J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153-159. https://doi.org/10.1017/S0004972700028884
  20. H. F. Senter and W. G. Dotson, Approximating fixed points of non-expansive mappings, Proc. Am. Math. Soc. 44 (1976), 375-380. https://doi.org/10.1090/S0002-9939-1974-0346608-8
  21. T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), 1088-1095. https://doi.org/10.1016/j.jmaa.2007.09.023
  22. W. Takahashi, Nonlinear Functional Analysis, Yokohoma Publishers: Yokohoma, Japan, 2000.
  23. B. S. Thakur, D. Thakur, and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comput. 275, (2016), 147-155. https://doi.org/10.1016/j.amc.2015.11.065
  24. K. Ullah, J. Ahmad, M. Arshad, and Z. Ma, Approximation of fixed points for enriched Suzuki nonexpansive operators with an application in Hilbert spaces, Axioms 11 (2022), no. 1, 14.
  25. K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki's generalized nonexpansive mappings via new iteration process, Filomat 32 (2018), 187-196. https://doi.org/10.2298/FIL1801187U
  26. K. Ullah, J. Ahmad, and M. de la Sen, Some new results on a three-step iteration process, Axioms 9 (2020), no. 3, 110.