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

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.