A Hybrid Iterative Algorithm for Fixed Point Approximation in CAT (0) Spaces

References

[1] Kadhim, A. J. (2021). New common fixed points for total asymptotically nonexpansive mappings in CAT(0) space. Baghdad Science Journal, 18(4), Article 10, 1286–1293. https://doi.org/10.21123/bsj.2021.18.4.1286

[2] Panwar, A., Lamba, P., & Kumar, S. (2022). Some convergence results using a new iterative algorithm in CAT(0) space. Results in Nonlinear Analysis, 5(3), 263–272. https://doi.org/10.53006/rna.1097678

[3] Kumar, M., & Pathak, H. K. (2022). Numerical reckoning of fixed points for generalized nonexpansive mappings in CAT(0) spaces with applications. U.P.B. Scientific Bulletin, Series A, 84(2), 117–128.

[4] Ali, F., Ali, J., & Nieto, J. J. (2020). Some observations on generalized non-expansive mappings with an application. Computational and Applied Mathematics, 39, Article 74. https://doi.org/10.1007/s40314-020-1101-4

[5] Sanboonsiri, P., & Wongsasinchai, P. (2020). Generalized nonexpansive mappings in CAT(0) spaces. Progress in Applied Science and Technology, 10(1), 148–160.

[6] Nanjaras, B., & Panyanak, B. (2010). Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory and Applications, 2010, Article 268780. https://doi.org/10.1155/2010/268780

[7] Niwongsa, Y., & Panyanak, B. (2010). Noor iterations for asymptotically nonexpansive mappings in CAT(0) spaces. International Journal of Mathematical Analysis, 4(13), 645–656.

[8] Thakur, B. S., Thakur, D., & Postolache, R. (2015). Modified Picard-Mann hybrid iteration process for total asymptotically nonexpansive mappings. Fixed Point Theory and Applications, 2015, Article 140. https://doi.org/10.1186/s13663-015-0391-5

[9] Pansuma, A., & Sintunavarat, W. (2016). A new iterative scheme for numerical reckoning fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory and Applications, 2016, Article 83. https://doi.org/10.1186/s13663-016-0573-9

[10] Kumam, W., Pakkaranang, N., Kumam, P., & Cholamjiak, P. (2020). Convergence analysis of modified Picard-S hybrid iterative algorithms for total asymptotically nonexpansive mappings in Hadamard spaces. International Journal of Computer Mathematics, 97(1–2), 175–188. https://doi.org/10.1080/00207160.2018.1476685

[11] Kadhim, A. J. (2022). Convergence theorems of three-step iteration algorithm in CAT(0) spaces. Iraqi Journal of Science, 63(4), 1642–1652. https://doi.org/10.24996/ijs.2022.63.4.22

[12] Padcharoen, A., & Sukprasert, P. (2023). Convergence of iterative scheme for asymptotically nonexpansive mappings in Hadamard spaces. WSEAS Transactions on Mathematics, 22, 47–54. https://doi.org/10.37394/23206.2023.22.6

[13] Verma, A., & Tripathi, B. P. (2023). Approximation of fixed points in CAT(0) spaces using a new three-step iteration process for asymptotically quasi-nonexpansive mapping. International Journal of Membrane Science and Technology, 10(4), 2577–2584. https://doi.org/10.15379/ijmst.v10i4.3677

@article{1723726,
      author = {Ibrahim Usman Garba, M. I. Bello, M S Adamu, Daben Bitrus Danladi},
      title = {A Hybrid Iterative Algorithm for Fixed Point Approximation in CAT (0) Spaces},
      journal = {Iconic Research And Engineering Journals},
      year = {2026},
      volume = {10},
      number = {4},
      pages = {511-516},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1723726.pdf},
      abstract = {Fixed-point approximation for nonlinear mappings in geodesic metric spaces has attracted considerable attention because of its importance in nonlinear analysis, optimization, and related areas. This paper proposes a hybrid iterative algorithm for approximating fixed points of total asymptotically nonexpansive mappings in CAT (0) spaces. The proposed algorithm is developed for a nonempty closed convex subset of a CAT (0) space and incorporates suitable control parameters to generate successive approximations to a fixed point of the underlying mapping. The convergence analysis is established by exploiting the fundamental geometric properties of CAT (0) spaces, particularly the CAT (0) convexity inequality, together with the defining properties of total asymptotically nonexpansive mappings. Under appropriate conditions on the control parameters and asymptotic error sequences, it is proved that the sequence generated by the proposed hybrid algorithm converges strongly to a fixed point of the mapping. The obtained result demonstrates the effectiveness of the hybrid approach within the framework of nonlinear metric spaces and contributes to the existing theory of iterative fixed-point approximation. The study also discusses the limitations of the proposed scheme and identifies possible directions for further investigations, including extensions to other classes of nonlinear mappings and metric spaces, as well as applications to optimization, variational inequalities, and equilibrium problems.},
      keywords = {CAT(0) spaces; total asymptotically nonexpansive mappings; hybrid iterative algorithm; fixed-point approximation; strong convergence; nonlinear metric spaces; iterative methods.},
      month = {October},
  }