Double-Inertial Halpern Approximation for Nonexpansive and Strongly Quasi-Nonexpansive Mappings

Rahul Shukla

Abstract and Applied Analysis, 2026, vol. 2026, 1-10

Abstract: Let C be a nonempty closed convex subset of a real Hilbert space, and let T,S:C→C, where T is nonexpansive and S is qualitatively strongly quasi-nonexpansive with Id−S demiclosed at zero. We propose a projected double-inertial Halpern method in which a convex combination of T and S is evaluated at a two-step extrapolated point constructed from the two most recent displacement vectors. The inertial parameters are governed by a computable bound on the effective extrapolation error. The projection ensures that the method is well defined on a general closed convex domain, while the bound controls the effective inertial perturbation. Strong convergence is established to the metric projection of the anchor onto the appropriate fixed-point set in three parameter regimes: the common fixed point regime, the T-dominant regime, and the S-dominant regime. The convergence analysis is based on a mixed defect functional that simultaneously captures the norm losses associated with both mappings and their discrepancy. This approach preserves the qualitative strong quasi-nonexpansivity assumption and does not require a uniform quadratic modulus. The original noninertial Halpern iteration and single-inertial variants are recovered as special cases. An application to the best-approximation formulation of convex feasibility is presented, together with numerical experiments illustrating the computational behavior of the proposed method.MSC2020 Classification: Primary 47H10, 47H09, 47J25, 47J26, 46B20

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:8366577

DOI: 10.1155/aaa/8366577

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