Convergence analysis of the generalized viscosity explicit method for solving variational inclusion problems in Banach spaces with applications

Abstract

In this paper, we develop the viscosity explicit method for finding a solution of the variational inclusion problem, which is a zero of the sum of two accretive operators, in Banach spaces which integrates the algorithm defined by Cholamjiak, Pholasa, Suantai, and Sunthrayuth [The generalized viscosity explicit rules for solving variational inclusion problems in Banach spaces. Optimization. 2021;70(12):2607–2633. doi: 10.1080/02331934.2020.1789131] and the algorithm defined by Wang, Wang, and Zhang [Strong convergence of viscosity forward-backward algorithm to the sum of two accretive operators in Banach space. Optimization. 2021;70(1):169–190. doi: 10.1080/02331934.2019.1705299]. Also, we provide a new suitable assumption to prove the strong convergence theorem. We apply our main result to the variational inequality problem, the convex minimization problem, and the split feasibility problem. Numerical examples are given to illustrate the performance of the proposed method in the comparison with the related methods.