Approximating fixed points of nonexpansive-type mappings in Hadamard spaces

Volume 39, Issue 4, pp 522--540 https://dx.doi.org/10.22436/jmcs.039.04.07

Publication Date: May 22, 2025 Submission Date: December 14, 2024 Revision Date: January 08, 2025 Accteptance Date: February 13, 2025

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Authors

W. Worapitpong - Departments of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. P. Chaipunya - Departments of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. - Center of Excellence in Theoretical and Computational Science (TaCS-CoE) & KMUTT Fixed Point Research Laboratory, Room SCL 802, Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. P. Kumam - Center of Excellence in Theoretical and Computational Science (TaCS-CoE) & KMUTT Fixed Point Research Laboratory, Room SCL 802, Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. S. Salisu - Center of Excellence in Theoretical and Computational Science (TaCS-CoE) & KMUTT Fixed Point Research Laboratory, Room SCL 802, Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. - Department of Mathematics, Faculty of Natural and Applied Sciences, Sule Lamido University Kafin Hausa, Jigawa, Nigeria.

Abstract

Certain essential properties of a generalized nonexpansive mapping, showcasing an inclusion relation to known mappings within the framework of Hadamard spaces, are investigated in this work. Additionally, a viscosity-type algorithm is proposed for approximating fixed points of such mappings. The strong convergence of the sequences generated by this algorithm is demonstrated under appropriate assumptions. As an application of the findings, the existence of a solution to the variational inequality problem involving the mapping is established. Furthermore, an iterative scheme is derived, exhibiting strong convergence towards solving the variational inequality problem. To demonstrate the implementation of the proposed algorithm, a numerical example is provided in a non-Hilbert CAT(0) space.

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Keywords

• Convergence analysis
• fixed point
• Hadamard space
• inverse strongly monotone
• nonexpansive mapping
• variational inequality
• viscosity iteration

MSC

•  47H09
•  47H10
•  47J05
•  47J25
•  49J40
•  53C23

References

• [1] B. Ahmadi Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc., 141 (2013), 1029–1039
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] S. Alizadeh, H. Dehghan, F. Moradlou, Δ-convergence theorems for inverse-strongly monotone mappings in CAT(0) spaces, Fixed Point Theory, 19 (2018), 45–56
  • View Article
  • Google Scholar
  • MATH


• [3] K. O. Aremu, C. Izuchukwu, H. A. Abass, O. T. Mewomo, On a viscosity iterative method for solving variational inequality problems in Hadamard spaces, Axioms, 9 (2020), 14 pages
  • View Article
  • Google Scholar


• [4] M. Baˇcák, Old and new challenges in Hadamard spaces, Jpn. J. Math., 18 (2023), 115–168
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] C. Baiocchi, A. Capelo, Variational and quasivariational inequalities, John Wiley & Sons, Inc., New York (1984)
  • View Article
  • Google Scholar


• [6] I. D. Berg, I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] V. Berinde, Iterative approximation of fixed points, Springer, Berlin (2007)
  • View Article
  • Google Scholar


• [8] V. Berinde, Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retractiondisplacement condition, Carpathian J. Math., 36 (2020), 27–34
  • View Article
  • Google Scholar
  • MATH


• [9] V. Berinde, A modified Krasnosel’ski˘ı-Mann iterative algorithm for approximating fixed points of enriched nonexpansive mappings, Symmetry, 14 (2022), 12 pages
  • View Article
  • Google Scholar


• [10] V. Berinde, M. P˘acurar, Recent developments in the fixed point theory of enriched contractive mappings. A survey, Creat. Math. Inform., 33 (2024), 137–159
  • View Article
  • Google Scholar
  • MATH


• [11] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin (1999)
  • View Article
  • MathSciNet
  • Google Scholar


• [12] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] R. E. Bruck, Asymptotic behavior of nonexpansive mappings, Amer. Math. Soc., Providence, RI, 18 (1983), 1–47
  • View Article
  • MathSciNet
  • Google Scholar


• [14] F. Bruhat, J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., 41 (1972), 5–251
  • View Article
  • MathSciNet
  • Google Scholar


• [15] C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Springer-Verlag London, Ltd., London (2009)
  • View Article
  • Google Scholar
  • MATH


• [16] C. E. Chidume, A. U. Bello, P. Ndambomve, Strong and Δ-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr. Appl. Anal., 2014 (2014), 6 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [17] H. Dehghan, J. Rooin, A characterization of metric projection in CAT (0) spaces, arXiv preprint arXiv:1311.4174, (2013), 3 pages
  • View Article
  • Google Scholar


• [18] S. Dhompongsa, A. Kaewkhao, B. Panyanak, On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces, Nonlinear Anal., 75 (2012), 459–468
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] S. Dhompongsa, W. A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 35–45
  • View Article
  • Google Scholar
  • MATH


• [20] S. Dhompongsa, W. A. Kirk, B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), 762–772
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] S. Dhompongsa, B. Panyanak, On Δ-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572–2579
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] G. Z. Eskandani, M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algorithms, 80 (2019), 1155–1179
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] A. Genel, J. Lindenstrauss, An example concerning fixed points, Israel J. Math, 22 (1975), 81–86
  • View Article
  • MathSciNet
  • Google Scholar


• [24] A. Gharajelo, H. Dehghan, Convergence theorems for strict pseudo-contractions in CAT(0) metric spaces, Filomat, 31 (2017), 1967–1971
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150
  • View Article
  • MathSciNet
  • Google Scholar


• [26] Z. Jiao, I. Jadlovská, T. Li, Global existence in a fully parabolic attraction-repulsion chemotaxis system with singular sensitivities and proliferation, J. Differ. Equ., 411 (2024), 227–267
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv., 70 (1995), 659–673
  • View Article
  • MathSciNet
  • Google Scholar


• [28] K. Khammahawong, P. Kumam, Generalized Projection Algorithm for Convex Feasibility Problems on Hadamard Manifolds, Nonlinear Convex Anal. Optim., 1 (2022), 25–45
  • View Article
  • Google Scholar


• [29] S. H. Khan, M. Abbas, Strong and Δ-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl., 61 (2011), 109–116
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] W. A. Kirk, Geodesic geometry and fixed point theory, Univ. Sevilla Secr. Publ., Seville, 64 (2003), 195–225
  • View Article
  • Google Scholar


• [31] W. A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl., 2004 (2004), 309–316
  • View Article
  • Google Scholar


• [32] W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689–3696
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] W. A. Kirk, N. Shahzad, Fixed point theory in distance spaces, Springer, Cham (2014)
  • View Article
  • MathSciNet
  • Google Scholar


• [34] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510
  • View Article
  • MathSciNet
  • Google Scholar


• [35] G. Marino, H.-K. Xu, Explicit hierarchical fixed point approach to variational inequalities, J. Optim. Theory Appl., 149 (2011), 61–78
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [36] M. S. Minjibir, S. Salisu, Strong and Δ-convergence theorems for a countable family of multi-valued demicontractive maps in Hadamard spaces, Nonlinear Funct. Anal. Appl., 27 (2022), 45–58
  • View Article
  • Google Scholar
  • MATH


• [37] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] S. Saejung, Halpern’s iteration in CAT(0) spaces, Fixed Point Theory Appl., 2010 (2010), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [39] S. Salisu, V. Berinde, S. Sriwongsa, P. Kumam, Approximating fixed points of demicontractive mappings in metric spaces by geodesic averaged perturbation techniques, AIMS Math., 8 (2023), 28582–28600
  • View Article
  • MathSciNet
  • Google Scholar


• [40] S. Salisu, V. Berinde, S. Sriwongsa, P. Kumam, On approximating fixed points of strictly pseudocontractive mappings in metric spaces, Carpathian J. Math., 40 (2024), 419–430
  • View Article
  • Google Scholar


• [41] S. Salisu, L. Hashim, A. Y. Inuwa, A. U. Saje, Implicit Midpoint Scheme for Enriched Nonexpansive Mappings, Nonlinear Convex Anal. Optim., 1 (2022), 211–225
  • View Article
  • Google Scholar


• [42] S. Salisu, P. Kumam, S. Sriwongsa, One step proximal point schemes for monotone vector field inclusion problems, AIMS Math., 7 (2022), 7385–7402
  • View Article
  • MathSciNet
  • Google Scholar


• [43] S. Salisu, P. Kumam, S. Sriwongsa, On fixed points of enriched contractions and enriched nonexpansive mappings, Carpathian J. Math., 39 (2023), 237–254
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [44] S. Salisu, P. Kumam, S. Sriwongsa, J. Abubakar, On minimization and fixed point problems in Hadamard spaces, Comput. Appl. Math., 41 (2022), 22 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [45] S. Salisu, P. Kumam, S. Sriwongsa, A. Y. Inuwa, Enriched multi-valued nonexpansive mappings in geodesic spaces, Rend. Circ. Mat. Palermo (2), 74 (2023), 1435–1451
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [46] S. Salisu, S. Sriwongsa, P. Kumam, V. Berinde, ariational inequality and proximal scheme for enriched nonexpansive mappings in CAT(0) spaces, J. Nonlinear Convex Anal., 25 (2024), 1759–1776
  • View Article
  • Google Scholar
  • MATH


• [47] C. Suanoom, B. Tanut, T. Grace, The proposition for an CAK-generalized nonexpansive mapping in Hilbert spaces, Bangmod Int. J. Math. Comput. Sci., 8 (2022), 100–108
  • View Article
  • Google Scholar


• [48] P. Thammasiri, K. Ungchittrakool, Accelerated hybrid Mann-type algorithm for fixed point and variational inequality problems, Nonlinear Convex Anal. Optim., 1 (2022), 97–111
  • View Article
  • Google Scholar


• [49] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 6 (2002), 240–256
  • View Article
  • MathSciNet
  • Google Scholar


• [50] Y. Yao, H. Zhou, Y.-C. Liou, Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput., 29 (2009), 383–389
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [51] E. Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York (1986)
  • View Article
  • MathSciNet
  • Google Scholar