Stability of a generalization of Cauchy’s and the quadratic functional equations

Using the fixed point method, we investigate the stability of the following generalization of Cauchy’s and the quadratic functional equations ∑k=0n-1f(x+bky)=nf(x)+nf(y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=0}^{n-1} f(x+ b_{k}y)=nf(x)+nf(y), \end{aligned}$$\end{document}where n∈N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in \mathbb {N}_{2}$$\end{document}, bk=exp(2iπkn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k}=\exp (\frac{2i\pi k}{n})$$\end{document} for 0≤k≤n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le k \le n-1$$\end{document}, in Banach spaces. Also, we prove the hyperstability results of this equation by the fixed point method.