$ g $-Averaged Halpern-type convergence theorems for enriched $ g $-nonexpansive mappings in Hilbert spaces

Naseer Shahzad; Amer Hassan Albargi; Manahell Alsosui; Naseer Shahzad; Amer Hassan Albargi; Manahell Alsosui

doi:10.3934/math.2026536

AIMS Mathematics

2026, Volume 11, Issue 5: 13023-13041. doi: 10.3934/math.2026536

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$ g $-Averaged Halpern-type convergence theorems for enriched $ g $-nonexpansive mappings in Hilbert spaces

1.
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Taibah University, Madinah, Saudi Arabia

Received: 14 February 2026 Revised: 16 April 2026 Accepted: 23 April 2026 Published: 11 May 2026
MSC : 47H09, 47H10, 47J25

We introduce the set of $g$-attractive points of a mapping $ f $ and study a $g$-averaged Halpern-type iterative scheme for $ b $-enriched $g$-nonexpansive mappings in real Hilbert spaces. For $0 < \; \lambda\leq \; 1 /(b+1)$, we prove that every sequence generated by the scheme converges strongly-after applying $g$-to the metric projection of the anchor onto the set $A^{g}(f_{g, \lambda})$, provided this set is nonempty. As a consequence, we obtain the corresponding strong convergence theorem for $ b $-enriched nonexpansive mappings. A numerical example is given to illustrate the method and the role of the averaging parameter.
- fixed point,
- attractive point,
- enriched $ g $-nonexpansive mapping,
- metric projection,
- Halpern-type iteration,
- averaged $ g $-mapping
Citation: Naseer Shahzad, Amer Hassan Albargi, Manahell Alsosui. $ g $-Averaged Halpern-type convergence theorems for enriched $ g $-nonexpansive mappings in Hilbert spaces[J]. AIMS Mathematics, 2026, 11(5): 13023-13041. doi: 10.3934/math.2026536

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Abstract

We introduce the set of $g$-attractive points of a mapping $ f $ and study a $g$-averaged Halpern-type iterative scheme for $ b $-enriched $g$-nonexpansive mappings in real Hilbert spaces. For $0 < \; \lambda\leq \; 1 /(b+1)$, we prove that every sequence generated by the scheme converges strongly-after applying $g$-to the metric projection of the anchor onto the set $A^{g}(f_{g, \lambda})$, provided this set is nonempty. As a consequence, we obtain the corresponding strong convergence theorem for $ b $-enriched nonexpansive mappings. A numerical example is given to illustrate the method and the role of the averaging parameter.

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