Strong Convergence of the Halpern Iteration for Monotone α-Nonexpansive Mappings in Uniformly Convex Ordered Banach Spaces

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-129
Published: Apr 30, 2026
Cite as:
Inemesit D. Edem, Joel Akpakwa, Marshal Sampson, Reny George, Mohammed M. Awad, Hossam A. Nabwey, Strong Convergence of the Halpern Iteration for Monotone α-Nonexpansive Mappings in Uniformly Convex Ordered Banach Spaces, Int. J. Anal. Appl., 24 (2026), 129.

Main Article Content

Inemesit D. Edem, Joel Akpakwa, Marshal Sampson, Reny George, Mohammed M. Awad, Hossam A. Nabwey

Abstract

We investigate the strong convergence of the Halpern iteration for monotone α-nonexpansive mappings in uniformly convex Banach spaces endowed with a partial order induced by a normal cone. By establishing new demiclosedness principles, analyzing boundedness and asymptotic regularity, and exploiting order-preserving properties, we prove that the Halpern sequence converges strongly to an extremal fixed point of the operator. The framework is further extended to hybrid iterative schemes, modular and Orlicz function spaces, and applications to variational inequalities, monotone operators in partial differential equations, and equilibrium problems in optimization. Illustrative examples in classical and modular Banach spaces are provided, and convergence is shown to hold under relaxed geometric conditions, such as strict convexity or smoothness, thereby unifying and generalizing existing theories for nonexpansive and α-nonexpansive mappings.

Article Details

References

  1. M.A. Krasnoselskii, Y.B. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961. https://archive.org/details/convexfunctionso0000makr.
  2. B. Halpern, Fixed Points of Nonexpanding Maps, Bull. Am. Math. Soc. 73 (1967), 957–961. https://doi.org/10.1090/s0002-9904-1967-11864-0.
  3. R. Wittmann, Approximation of Fixed Points of Nonexpansive Mappings, Arch. Math. 58 (1992), 486–491. https://doi.org/10.1007/bf01190119.
  4. Y. Alber, S. Guerre-Delabriere, On the Projection Methods for Fixed Point Problems, Analysis 21 (2001), 17–40. https://doi.org/10.1524/anly.2001.21.1.17.
  5. F. Kohsaka, W. Takahashi, Strong Convergence of an Iterative Sequence for Maximal Monotone Operators in a Banach Space, Abstr. Appl. Anal. 2004 (2004), 239–249. https://doi.org/10.1155/S1085337504309036.
  6. H.K. Xu, Iterative Algorithms for Nonlinear Operators, J. Lond. Math. Soc. 66 (2002), 240–256. https://doi.org/10.1112/S0024610702003332.
  7. C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer, 2009. https://doi.org/10.1007/978-1-84882-190-3.
  8. D. Ariza-Ruiz, C.H. Linares, E. Llorens-Fuster, E. Moreno-Gálvez, On $alpha$-Nonexpansive Mappings in Banach Spaces, Carpathian J. Math. 32 (2016), 13–28. https://www.jstor.org/stable/44000085.
  9. R. Shukla, R. Pant, M. De la Sen, Generalized $alpha$-Nonexpansive Mappings in Banach Spaces, Fixed Point Theory Appl. 2017 (2017), 4. https://doi.org/10.1186/s13663-017-0597-9.
  10. Y. Song, K. Promluang, P. Kumam, Y. Je Cho, Some Convergence Theorems of the Mann Iteration for Monotone $alpha$-Nonexpansive Mappings, Appl. Math. Comput. 287-288 (2016), 74–82. https://doi.org/10.1016/j.amc.2016.04.011.
  11. Y. Song, P. Kumam, Y.J. Cho, Fixed Point Theorems and Iterative Approximations for Monotone Nonexpansive Mappings in Ordered Banach Spaces, Fixed Point Theory Appl. 2016 (2016), 73. https://doi.org/10.1186/s13663-016-0563-y.
  12. C.E. Chidume, M.O. Nnakwe, A. Adamu, A Strong Convergence Theorem for Generalized-$Phi$-Strongly Monotone Maps, with Applications, Fixed Point Theory Appl. 2019 (2019), 11. https://doi.org/10.1186/s13663-019-0660-9.
  13. U. Kohlenbach, L. Leuştean, Effective Metastability of Halpern Iterates in CAT(0) Spaces, Adv. Math. 231 (2012), 2526–2556. https://doi.org/10.1016/j.aim.2012.06.028.
  14. Y. Censor, A. Gibali, S. Reich, The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space, J. Optim. Theory Appl. 148 (2010), 318–335. https://doi.org/10.1007/s10957-010-9757-3.
  15. M.M. Matar, S. Ayadi, J. Alzabut, A. Salim, Fixed Point Approach for Nonlinear $psi$-Caputo ­Fractional Differential Hybrid Coupled System With ­Periodic Boundary Conditions, Results Nonlinear Anal. 6 (2023), 13–29.
  16. I.D. Edem, J. Akpakwa, U.E. Udofia, M.I. Sampson, Improved Convergence Theorems for Monotone $alpha$-Nonexpansive Mappings in Uniformly Convex Banach Spaces, Int. J. Math. Anal. Model. 8 (2025), 291–309.
  17. E. Karapınar, A. Fulga, R.P. Agarwal, A Survey: F-Contractions with Related Fixed Point Results, J. Fixed Point Theory Appl. 22 (2020), 69. https://doi.org/10.1007/s11784-020-00803-7.
  18. M. Bachar, M.A. Khamsi, On Common Approximate Fixed Points of Monotone Nonexpansive Semigroups in Banach Spaces, Fixed Point Theory Appl. 2015 (2015), 160. https://doi.org/10.1186/s13663-015-0405-3.
  19. B.A.B. Dehaish, M.A. Khamsi, On Monotone Mappings in Modular Function Spaces, J. Nonlinear Sci. Appl. 09 (2016), 5219–5228. https://doi.org/10.22436/jnsa.009.08.07.