Functional inequalities in matrix Banach spaces

Using fixed point method, we prove the Hyers–Ulam stability of the following functional inequalities ‖f(x)+f(y)+f(z)‖≤‖f(x+y+z)‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Vert f(x)+f(y)+f(z)\Vert \le \Vert f(x+y+z)\Vert $$\end{document} and ‖f(x)+f(y)+2f(z)‖≤‖2f(x+y2+z)‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Vert f(x)+f(y)+2f(z)\Vert \le \Vert 2f(\frac{x+y}{2}+z)\Vert $$\end{document} in matrix Banach spaces.