Self-adaptive Iterative Proximal Method for Split Minimization Problems with Applications to Image and Signal Processing

In this paper, we study the proximal split fixed point problem (PSFPP) and propose a viscosity-type iterative algorithm that incorporates both inertial effects and a self-adaptive stepsize strategy. The proposed scheme is intended to address limitations of existing methods that typically require prior knowledge of the operator norm, which may be difficult to estimate in practice. We establish strong convergence of the generated sequence to a solution associated with a nonexpansive mapping under standard assumptions in Hilbert spaces. To illustrate the effectiveness and practical applicability of the proposed algorithm, we present numerical experiments on representative inverse problems, including image restoration and sparse signal recovery. The results demonstrate the robustness and computational efficiency of the method in practical settings.