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

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 I d − 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