Weak and Strong Convergence Theorems for Krasnoselski-Mann- Type Method for Solving Hierarchical Fixed-Point Problems and Split Generalized Mixed Equilibrium Problems

[1] Chuasuk, P., & Kaewcharoen, A. (2021). Krasnoselski–Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems. Journal of Inequalities and Applications, 2021(1). doi:10.1186/s13660-021-02632-9 [2] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems.Math. Stud. 63, 123–145 (1994). [3] Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011). [4] Kazmi, K.R., Rizvi, S.H.: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egyptian Math. Soc. 21, 44–51 (2013). [5] Bnouhachem, A.: Algorithms of common solution for a variational inequality, a split equilibrium problem and a hierarchical fixed point problem. Fixed Point Theory Appl. 2013, Article ID 278 (2013). [6] Nwawuru, F.O., Ezeora, J.N., ur Rehman, H. Jen-Chih Yao. Self-adaptive subgradient extragradient algorithm for solving equilibrium and fixed point problems in Hilbert spaces. Numer Algor (2025). https://doi.org/10.1007/s11075-025-02055-3 [7] Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59(2), 301–323 (2012) [8] Nwawuru, F.O., Echezona, G.N. and Okeke, C.C. Finding a common solution of variational inequality and fixed point problems using subgradient extragradient techniques. Rend. Circ. Mat. Palermo, II. Ser (2023). https://doi.org/10.1007/s12215-023-00978-1, click here to read https://rdcu.be/dtyge [9] Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006) [10] Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imag. Electron Phys.95, 155–453 (1996) [11] Cyril Dennis Enyi, Jeremiah Nkwegu Ezeora, Godwin Chidi Ugwunnadi, Francis O. Nwawuru, and Soh Edwin Mukiawa: (2024): Generalized Split Feasibility Problem: Solution by Iteration. Online version at https://www.carpathian.cunbm.utcluj.ro/ Print Edition: ISSN 1584 - 2851; Online Edition: ISSN 1843 - 4401 DOI: https://doi.org/10.37193/CJM.2024.03.08 G, pp. 655–679 [12] Francis O. Nwawuru, Ojen Narian, Mohammad Dilshad and Jeremiah N. Ezeora. Splitting method involving two-step inertial for solving inclusion and fixed point problems with applications. Fixed Point Theory Algorithms Sci Eng 2025, 8 (2025). https://doi.org/10.1186/s13663-025-00781-w [13] Jolaoso, L.O.: An inertial projection and contraction method with a line search technique for variational inequality and fixed point problems. Optimization (2021). https://doi.org/10.1080/02331934.2021.1901289 [14] Censor Y, Elfving T. A multiprojection algorithm using Bregman projections in product space. Numer Algorithms. 1994;8:221–239. [15] Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004) [16] Nwawuru, F.O., Ezeora, J.N. Inertial-based extragradient algorithm for approximating a common solution of split-equilibrium problems and fixed-point problems of nonexpansive semigroups. J Inequal Appl 2023, 22 (2023). https://doi.org/10.1186/s13660-023-02923-3. [17] Ezeora, J.N., Enyi, C.D., Nwawuru, F.O. and Richard C. Ogbonna. An algorithm for split equilibrium and fixed-point problems using inertial extragradient techniques. Comp. Appl. Math. 42, 103 (2023). https://doi.org/10.1007/s40314- 023-02244-7 [18] Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006) [19] Jolaoso, L.O., Oyewole, K.O., Okeke, C.C., Mewomo, O.T.: A unified algorithm for solving split generalized mixed equilibrium problem and for finding fixed point of nonspreading mapping in Hilbert spaces. Demonstr. Math. 51, 211–232 (2018). [20] Francis O. Nwawuru. Approximation of solutions of split monotone variational inclusion problems and fixed point problems. Pan-American Journal of Mathematics 2 (2023), 1 https://doi.org/10.28919/cpr-pajm/2-1 [21] Yamada, I., Ogura, N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004) [22] Francis O Nwawuru, Grace N Echezona "Forward-Backward Splitting Method with Viscosity Iteration for Solving Monotone Inclusion Problems" Iconic Research And Engineering Journals Volume 6 Issue 10 2023 Page 588-594. [23] Jeremiah N. Ezeora and Francis O. Nwawuru: An inertial-based hybrid and shrinking projection methods for solving split common fixed point problems in real reflexive spaces. Int. J. Nonlinear Anal. Appl. 14 (2023) 1, 2541–2556 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2022.24912.2852. [24] Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 23, 1635–1640 (2007 [25] Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009) [26] Olona, M.A., Alakoya, T.O., Owolabi, A.O.-E., Mewomo, O.T.: Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strictly pseudocontractive mappings. J. Nonlinear Funct. Anal. 2, Article ID 10 (2021) [27] Lorenzand, D.A., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51(2), 311–325 (2015) [28] Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472–487 (2015) [29] Moudafi, A., Oilny, M.: Convergence of a splitting inertial proximal method for monotone operator. J. Comput. Appl. Math. 155, 447–454 (2003) [30] Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space. Optimization 70(2), 387–412 (2021) [31] Francis O. Nwawuru and Blessing E. Chukwuemeka. A modified iterative Method for Solving Split variational Inclusion Problems and Fixed-Point Problems for Nonexpansive Semigroup. International Journal of Mathematical Sciences and Optimization: Theory and Applications Vol. 8, No. 2, pp. 109 - 130 https://doi.org/10.6084/m9.figshare.22347139 [32] Olona, M.A., Alakoya, T.O., Owolabi, A.O.-E., Mewomo, O.T.: Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings. Demonstr. Math. (2021). https://doi.org/10.1515/dema-2021-0006 [33] Preeyanuch Chuasuk;Anchalee Kaewcharoen; (2021). Krasnoselski–Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems . Journal of Inequalities and Applications, (), –. doi:10.1186/s13660-021-02632- 9 [34] Grace N. Echezona and Francis O. Nwawuru. A Summary Result on Second Order Linear Differential Equations with Variable Coefficients using Adomian Decomposition Method and Power Series Method. International Journal of Applied Science and Mathematical Theory E- ISSN 2489-009X P-ISSN 2695- 1908, Vol. 8 No. 4 2022 www.iiardjournals.org. [35] Kazmi, K. R., Ali, R., & Furkan, M. (2017). Krasnoselski-Mann type iterative method for hierarchical fixed point problem and split mixed equilibrium problem. Numerical Algorithms, 77(1), 289–308. doi:10.1007/s11075-017-0316-y [36] Patrick, L: Combettes, Jean-Christophe Pesquet: Deep Neural Network Structures Solving Variational