European Journal of Mathematics and Applications

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Volume 6 (2026), Article ID 15

https://doi.org/10.28919/ejma.2026.6.15

Published: 30/10/2026

Abstract:

We study an anchored general Picard–Mann method for nonexpansive mappings with fixed points on nonempty closed convex subsets of real Hilbert spaces. The method combines an anchor, a Mann average, and a fixed number of Picard steps. We prove that, when the anchor weights lie strictly between zero and one, tend to zero, and have a divergent sum, all iterates and intermediate sequences converge strongly to the projection of the anchor onto the fixed-point set, provided the Mann parameters stay uniformly away from zero and one. This result requires no compactness assumption or bounded-variation condition on the controls. If the anchor weights are summable, the iterates converge weakly; they converge strongly if the image set is relatively compact, if the iterates approach the fixed-point set along a subsequence, or if the mapping satisfies condition (I). We also study a reduced method that omits the Mann step. Its auxiliary sequence follows Halpern iteration for a positive integer power of the mapping. When the anchor weights tend to zero, have a divergent sum, and have bounded variation, both sequences converge strongly to the projection onto that power’s fixed-point set, which may differ from the original fixed-point set. Equality of the two fixed-point sets is necessary and sufficient for convergence to the projection onto the original set for every anchor; this equality holds when there is one Picard step or when the mapping is averaged. Examples and a higher-dimensional numerical comparison illustrate fixed-point selection, periodic-point obstructions, and the distinction between iteration counts and operator-evaluation costs.

How to Cite:

Rahul Shukla, Strong convergence of anchored general Picard–Mann and Halpern iterations for nonexpansive mappings, Eur. J. Math. Appl. 6 (2026), Article ID 15. https://doi.org/10.28919/ejma.2026.6.15