Two maximum-based fixed-point methods for the generalized absolute value equation

By employing the expression |x|=2max(x,0)-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|x|=2\max (x,0)-x$$\end{document}, this paper, for the first time, performs a matrix splitting of A+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A+B$$\end{document} to design two fixed-point methods for solving the generalized absolute value equation Ax-B|x|=b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax- B |x |= b$$\end{document}. Under suitable assumptions, we conduct the convergence analysis of these methods. Additionally, we investigate the parameter's effect on the second method's convergence rate, thereby determining the optimal parameter value. Finally, several numerical examples validate both the proposed methods' theoretical results and practical effectiveness.