### 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.

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