A novel modified one-step iterative method for common fixed points of three nonexpansive mappings

Athar Abbas; Somayya Komal; Hafiz Fukhar-ud-din; Muhammad Aqeel Ahmad Khan; Muhammad Jabir Khan; Athar Abbas; Somayya Komal; Hafiz Fukhar-ud-din; Muhammad Aqeel Ahmad Khan; Muhammad Jabir Khan

doi:10.3934/math.2026496

AIMS Mathematics

2026, Volume 11, Issue 4: 12094-12107. doi: 10.3934/math.2026496

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A novel modified one-step iterative method for common fixed points of three nonexpansive mappings

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

Received: 16 January 2026 Revised: 07 April 2026 Accepted: 16 April 2026 Published: 29 April 2026
MSC : 46B20, 47H10, 47H09, 47J25

This paper proposes a novel modified one-step iterative scheme for approximating common fixed points of three nonexpansive mappings in uniformly convex Banach spaces. The proposed scheme extends the classical one-step iteration associated with two mappings, which can be recovered as a particular case by an appropriate choice of the third mapping. Weak convergence of the generated sequence is established under both the Kadec-Klee property (KKP) and the Opial condition, while strong convergence is obtained by assuming a modified condition (B). Moreover, we introduce a new Condition (P) tailored for three mappings, which reduces to the well-known Condition (S) in the case of two mappings. The results presented here broaden and unify several existing contributions in the theory of common fixed point approximation.
- modified one-step iteration,
- common fixed point approximation,
- Kadec-Klee property,
- condition $ (P) $,
- uniformly convex Banach spaces,
- convergence
Citation: Athar Abbas, Somayya Komal, Hafiz Fukhar-ud-din, Muhammad Aqeel Ahmad Khan, Muhammad Jabir Khan. A novel modified one-step iterative method for common fixed points of three nonexpansive mappings[J]. AIMS Mathematics, 2026, 11(4): 12094-12107. doi: 10.3934/math.2026496

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Abstract

This paper proposes a novel modified one-step iterative scheme for approximating common fixed points of three nonexpansive mappings in uniformly convex Banach spaces. The proposed scheme extends the classical one-step iteration associated with two mappings, which can be recovered as a particular case by an appropriate choice of the third mapping. Weak convergence of the generated sequence is established under both the Kadec-Klee property (KKP) and the Opial condition, while strong convergence is obtained by assuming a modified condition (B). Moreover, we introduce a new Condition (P) tailored for three mappings, which reduces to the well-known Condition (S) in the case of two mappings. The results presented here broaden and unify several existing contributions in the theory of common fixed point approximation.

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