Strong convergence theorem for split feasibility problems and variational inclusion problems in real Banach spaces

The purpose of this paper is to study and analyze an iterative method for split feasibility problem and variational inclusion problem (also known as the problem of finding a zero of the sum of two monotone operators) in the framework of real Banach spaces. By combining Mann's and Halpern's approximation methods, we propose an iterative algorithm for approximating a common solution of the aforementioned problems. Furthermore, we derive the strong convergence of the proposed algorithm under appropriate conditions. In all our results, we use the new way introduced by Suanti et al. to select the step-size which ensures the convergence of the sequences generated by our scheme. We also gave an application of our results and a numerical example of the proposed algorithm in comparison with the algorithm of Suanti et al. to show the efficiency and advantage of our algorithm. Our results extend and complement many known related results in the literature.