An S-Iteration Tseng Splitting Method for Solving Monotone Inclusion and Fixed Point Problems with Applications

In recent years, the monotone inclusion problem has attracted significant research attention due to its wide range of applications. However, many existing results in the literature require the single-valued operator in the problem formulation to be co-coercive (inverse strongly monotone) and are often restricted to Hilbert spaces. This limitation reduces the applicability of such results, as real-world problems involving monotone operators are not uncommon. In this paper, we study a class of monotone inclusion problems where the single-valued operator is not necessarily co-coercive, together with the common fixed point problem for relatively nonexpansive multivalued mappings. We propose a new iterative method for solving this common solution problem in the framework of Banach spaces. Our method incorporates several techniques to achieve high computational efficiency. One key component is the S-iteration process, which is known to outperform many classical methods. In addition, the proposed algorithm employs an inertial technique and a non-monotonic self-adaptive step size to ensure a high rate of convergence and ease of implementation. We establish strong convergence results for the proposed method under mild conditions and apply our findings to study certain optimization problems. Finally, we demonstrate the practical efficiency of the proposed method through extensive numerical experiments on real-world applications, including image restoration, economic modelling via price adjustment under uncertainty, and robust consensus in networked control systems. Our results extend and improve upon several recently published works in this area.