Approximating common fixed points of non-expansive type multi-valued mappings in convex metric space

Tanveer Hussain; Abdul Rahim Khan; Kiran Riaz; Hind Alamri; Tanveer Hussain; Abdul Rahim Khan; Kiran Riaz; Hind Alamri

doi:10.3934/math.20251321

AIMS Mathematics

2025, Volume 10, Issue 12: 30053-30067. doi: 10.3934/math.20251321

Previous Article Next Article

Research article

Approximating common fixed points of non-expansive type multi-valued mappings in convex metric space

1.
Department of Mathematics and Statistics, University of Southern Punjab, Multan, Pakistan
2.
Department of Mathematics, College of Science, Taif University, Taif 21944, Saudi Arabia

Received: 23 September 2025 Revised: 01 December 2025 Accepted: 09 December 2025 Published: 22 December 2025
MSC : 47H09, 47H10, 54E35

We propose a one-step iterative scheme to approximate common fixed points of multi-valued non-expansive mappings in a convex metric space $ \varUpsilon $, under certain mild boundary conditions. We present a numerical example to verify our convergence result obtained herein. We also establish some convergence results for multi-valued asymptotically non-expansive mappings in this context, which either improve or generalize results obtained for multi-valued non-expansive mappings on $ \varUpsilon $. Furthermore, we demonstrate how these findings can be applied to solve a system of two equations defined by multi-valued asymptotically non-expansive mappings on a convex metric space.
- common fixed point,
- multi-valued mappings,
- non-expansive mappings,
- convex metric space,
- fixed point iteration,
- asymptotically non-expansive
Citation: Tanveer Hussain, Abdul Rahim Khan, Kiran Riaz, Hind Alamri. Approximating common fixed points of non-expansive type multi-valued mappings in convex metric space[J]. AIMS Mathematics, 2025, 10(12): 30053-30067. doi: 10.3934/math.20251321

Related Papers:

Abstract

We propose a one-step iterative scheme to approximate common fixed points of multi-valued non-expansive mappings in a convex metric space $ \varUpsilon $, under certain mild boundary conditions. We present a numerical example to verify our convergence result obtained herein. We also establish some convergence results for multi-valued asymptotically non-expansive mappings in this context, which either improve or generalize results obtained for multi-valued non-expansive mappings on $ \varUpsilon $. Furthermore, we demonstrate how these findings can be applied to solve a system of two equations defined by multi-valued asymptotically non-expansive mappings on a convex metric space.

References

[1]	W. Takahashi, A convexity in metric space and nonexpansive mappings, Ⅰ, Kodai Math. Sem. Rep., 22 (1970), 142–149. https://doi.org/10.2996/kmj/1138846111 doi: 10.2996/kmj/1138846111
[2]	A. Kaewcharoen, B. Panyanak, Fixed points for multivalued mappings in uniformly convex metric spaces, Int. J. Math. Math. Sci., 2008 (2008), 163580. https://doi.org/10.1155/2008/163580 doi: 10.1155/2008/163580
[3]	S. Nawaz, K. Rafique, A. Batool, Z. Mahmood, T. Muhammad, Fixed point approximation of multi-valued non-expansive mappings in uniformly convex Banach spaces via AR-iteration, Modern Phys. Lett. B, 39 (2025), 2450495. https://doi.org/10.1142/s0217984924504955 doi: 10.1142/s0217984924504955
[4]	B. B. Salman, S. S. Abed, A new iterative sequence of ($\lambda, \rho$)-firmly nonexpansive multi-valued mappings in modular function spaces with applications, Math. Model. Eng. Probl., 10 (2023), 212–219. https://doi.org/10.18280/mmep.100124 doi: 10.18280/mmep.100124
[5]	A. Azam, M. Rashid, A. Kalsoom, F. Ali, Fixed-point convergence of multi-valued non-expansive mappings with applications, Axioms, 12 (2023), 1020. https://doi.org/10.3390/axioms12111020 doi: 10.3390/axioms12111020
[6]	A. R. Khan, V. Kumar, N. Hussain, Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput., 231 (2014), 521–535. https://doi.org/10.1016/j.amc.2013.12.150 doi: 10.1016/j.amc.2013.12.150
[7]	S. H. Khan, M. Abbas, B. E. Rhoades, A new one-step iterative scheme for approximating common fixed points of two multivalued nonexpansive mappings, Rend. Circ. Mat. Palermo, 59 (2010), 151–159. https://doi.org/10.1007/s12215-010-0012-4 doi: 10.1007/s12215-010-0012-4
[8]	A. R. Khan, M. A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Comput. Math. Appl., 59 (2010), 2990–2995. https://doi.org/10.1016/j.camwa.2010.02.017 doi: 10.1016/j.camwa.2010.02.017
[9]	M. Abbas, S. H. Khan, A. R. Khan, R. P. Agarwal, Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme, Appl. Math. Lett., 24 (2011), 97–102. https://doi.org/10.1016/j.aml.2010.08.025 doi: 10.1016/j.aml.2010.08.025
[10]	J. Ahmad, I. A. Kallel, A. Aloqaily, N. Mlaiki, On multi-valued generalized $\alpha$-nonexpansive mappings and an application to two-point BVPs, AIMS Math., 10 (2025), 403–419. https://doi.org/10.3934/math.2025019 doi: 10.3934/math.2025019
[11]	H. Fukhar-ud-Din, One step iterative scheme for a pair of nonexpansive mappings in a convex metric space, Hacet. J. Math. Stat., 44 (2015), 1023–1031.
[12]	Y. H. Yao, R. D. Chen, Weak and strong convergence of a modified Mann iteration for asymptotically nonexpansive mappings, Nonlinear Funct. Anal. Appl., 12 (2007), 307–315.
[13]	I. Uddin, J. J. Nieto, J. Ali, One-step iteration scheme for multivalued nonexpansive mappings in CAT(0) spaces, Mediterr. J. Math., 13 (2016), 1211–1225. https://doi.org/10.1007/s00009-015-0531-5 doi: 10.1007/s00009-015-0531-5
[14]	A. Tassaddiq, S. Kanwal, S. Perveen, R. Srivastava, Fixed points of single-valued and multi-valued mappings in sb-metric spaces, J. Inequal. Appl., 2022 (2022), 85. https://doi.org/10.1186/s13660-022-02814-z doi: 10.1186/s13660-022-02814-z
[15]	N. Hussain, H. Alamri, S. Alsulami, Fixed point approximation for a class of generalized nonexpansive multi-valued mappings in Banach spaces, Arab. J. Math., 12 (2023), 363–377. https://doi.org/10.1007/s40065-022-00403-y doi: 10.1007/s40065-022-00403-y
[16]	B. Nuntadilok, P. Kingkam, J. Nantadilok, Common fixed point theorems of two finite families of asymptotically quasi-nonexpansive mappings in hyperbolic spaces, J. Nonlinear Funct. Anal., 2023 (2023), 1–15. https://doi.org/10.23952/jnfa.2023.27 doi: 10.23952/jnfa.2023.27
[17]	A. J. Zaslavski, Convergence of inexact orbits of nonexpansive mappings in complete metric spaces, Commun. Optim. Theory, 2024 (2024), 1–10.
[18]	A. Latif, A. H. Alotaibi, M. Noorwali, Fixed point results via multivalued contractive type mappings involving a generalized distance on metric type spaces, J. Nonlinear Var. Anal., 8 (2024), 787–798. https://doi.org/10.23952/jnva.8.2024.5.06 doi: 10.23952/jnva.8.2024.5.06
[19]	K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171–174. https://doi.org/10.1090/s0002-9939-1972-0298500-3 doi: 10.1090/s0002-9939-1972-0298500-3
[20]	W. A. Kirk, H. K. Xu, Asymptotic pointwise contractions, Nonlinear Anal., 69 (2008), 4706–4712. https://doi.org/10.1016/j.na.2007.11.023 doi: 10.1016/j.na.2007.11.023
[21]	N. Hussain, M. A. Khamsi, On asymptotic pointwise contractions in metric spaces, Nonlinear Anal., 71 (2009), 4423–4429. https://doi.org/10.1016/j.na.2009.02.126 doi: 10.1016/j.na.2009.02.126
[22]	M. A. Khamsi, W. M. Kozlowski, On asymptotic pointwise nonexpansive mappings in modular function spaces, J. Math. Anal. Appl., 380 (2011), 697–708. https://doi.org/10.1016/j.jmaa.2011.03.031 doi: 10.1016/j.jmaa.2011.03.031
[23]	B. E. Rhoades, Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183 (1994), 118–120. https://doi.org/10.1006/jmaa.1994.1135 doi: 10.1006/jmaa.1994.1135
[24]	P. K. F. Kuhfittig, Common fixed points of nonexpansive mappings by iteration, Pac. J. Math., 97 (1981), 137–139. https://doi.org/10.2140/pjm.1981.97.137 doi: 10.2140/pjm.1981.97.137

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)