Revised Algorithm for Finding a Common Solution of Variational Inclusion and Fixed Point Problems

Abstract

Recent research has uncovered an algorithm for locating the common solution to variational inclusion problems with multivalued maximal monotone mapping and $alpha$-inverse strongly monotone mapping, as well as the points that are invariant under non-expansive mapping. In their algorithm Zhang et al. [ZHANG, S., LEE, J. H. W. and CHAN, C. K. Algorithms of common solutions to quasi-variational inclusion and fixed point problems, \text{Applied Mathematics and Mechanics},  \textbf{29(5)}, 571--581 (2008)], $\lambda$ must satisfy a very strict condition, namely $\lambda\in [0,2\alpha]$; thus, it cannot be used for all Lipschitz continuous mappings, despite the fact that inverse strongly monotone implies Lipschitz continuous. This manuscript aims to define a new algorithm that addresses the flaws of the previously described algorithm. Our algorithm is used to solve minimization problems involving the fixed point set of a non-expansive mapping. In addition, we support all of our claims with numerical examples derived from computer simulation.