A novel inertial viscosity Z-algorithm for common fixed point problems with an application to image deblurring

Source abstract

Abstract Fixed point theory plays a central role in functional analysis and provides a unifying framework for the analysis of iterative methods arising in differential equations, dynamical systems, and inverse problems. Motivated by recent developments in inertial and viscosity-based schemes, this paper introduces a novel inertial viscosity Z -algorithm for approximating a common fixed point of an infinite collection of nonexpansive maps in a real Hilbert space. The algorithm effectively combines the inertial term with a viscosity approximation, thereby guaranteeing strong convergence via a contraction map. Under mild and verifiable control conditions on the algorithmic parameters, strong convergence of the generated sequence is rigorously established. The framework is then extended to solve monotone inclusion problems and convex minimization problems of the form arg min x { h ( x ) + g ( x )}, leading to a specialized forward–backward algorithm for image deblurring. To highlight the practical relevance of the theoretical findings, the proposed algorithm is applied to image deblurring problems formulated via variational and fixed point models. Numerical examples demonstrate that incorporating both inertial and viscosity effects leads to accelerated convergence and improved reconstruction quality when compared with existing methods. These results confirm that the proposed scheme provides an effective bridge between abstract fixed point theory and concrete applications in image restoration.