Stability and convergence of common fixed point algorithms for a countable infinite family of enriched nonexpansive mappings

Muhammad Jabir Khan; Somayya Komal; Athar Abbas; Muhammad Jabir Khan; Somayya Komal; Athar Abbas

doi:10.3934/math.2026262

AIMS Mathematics

2026, Volume 11, Issue 3: 6350-6373. doi: 10.3934/math.2026262

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Stability and convergence of common fixed point algorithms for a countable infinite family of enriched nonexpansive mappings

1.
School of Artificial Intelligence and Computer Science, Nantong University, Nantong 226019, Jiangsu, China
2.
Department of Mathematics, Faculty of Sciences, University of Mianwali, Punjab, Pakistan
3.
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

Received: 16 January 2026 Revised: 03 March 2026 Accepted: 05 March 2026 Published: 12 March 2026
MSC : 47H09, 47H10, 47J25, 54E40, 54H25

This work intoduced a modified Halpern iterate for a countably infinite family of enriched nonexpansive mappings within convex metric spaces. Under the Aoyama-Kimura-Takahashi-Toyoda (AKTT) condition, we established that the generated sequence serves as an approximating common fixed point sequence of enriched nonexpansive mappings. Furthermore, strong convergence theorems were presented, ensuring that the iterative sequence converges to a common fixed point of the countably infinite family of enriched nonexpansive mappings in convex metric spaces, provided that the AKTT and the Song-Zheng (SZ) conditions are satisfied. Additionally, the concept of a $ \texttt{W} $-mapping was extended from Banach spaces to the convex metric spaces, thereby broadening and refining existing results in the literature. We also gave a numerical illustration in the framework of convex metric spaces to show the efficiency of our proposed algorithm.
- enriched nonexpansive mapping,
- convex metric space,
- common fixed point,
- AKTT and SZ conditions,
- $ \Delta- $convergence,
- strong convergence
Citation: Muhammad Jabir Khan, Somayya Komal, Athar Abbas. Stability and convergence of common fixed point algorithms for a countable infinite family of enriched nonexpansive mappings[J]. AIMS Mathematics, 2026, 11(3): 6350-6373. doi: 10.3934/math.2026262

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Abstract

This work intoduced a modified Halpern iterate for a countably infinite family of enriched nonexpansive mappings within convex metric spaces. Under the Aoyama-Kimura-Takahashi-Toyoda (AKTT) condition, we established that the generated sequence serves as an approximating common fixed point sequence of enriched nonexpansive mappings. Furthermore, strong convergence theorems were presented, ensuring that the iterative sequence converges to a common fixed point of the countably infinite family of enriched nonexpansive mappings in convex metric spaces, provided that the AKTT and the Song-Zheng (SZ) conditions are satisfied. Additionally, the concept of a $ \texttt{W} $-mapping was extended from Banach spaces to the convex metric spaces, thereby broadening and refining existing results in the literature. We also gave a numerical illustration in the framework of convex metric spaces to show the efficiency of our proposed algorithm.

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