A Two-Step Inertial Picard–CR Framework for Fixed Point Problems with Applications to Optimization and Image Recovery

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DOI: https://doi.org/10.28924/2291-8639-24-2026-276
Published: Sep 2, 2026
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Austine Efut Ofem, Seithuti Philemon Moshokoa, Malesela Clifford Kekana, Adhir Maharaj, A Two-Step Inertial Picard–CR Framework for Fixed Point Problems with Applications to Optimization and Image Recovery, Int. J. Anal. Appl., 24 (2026), 276.

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Austine Efut Ofem, Seithuti Philemon Moshokoa, Malesela Clifford Kekana, Adhir Maharaj

Abstract

This paper proposes a two-step inertial Picard–CR iterative framework for approximating common fixed points of a countable family of nonexpansive mappings in real Hilbert spaces. The method combines a viscosity approximation with two inertial corrections and a Picard–CR-type fixed-point update. Under suitable control conditions and an asymptotic compatibility requirement on the intermediate iterates, strong convergence to the viscosity-selected common fixed point is established. The abstract fixed-point result is then specialized to a convex bilevel optimization problem through a family of forward–backward operators. An image-restoration experiment based on a regularized inverse problem is also presented. Numerical comparisons with several forward–backward and accelerated schemes illustrate the computational behavior of the proposed approach.

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