Hyperstability of a quadratic functional equation with involutions in ultrametric n-Banach spaces via fixed point approach

Abstract

Let X be a normed space and Y be an ulrametric n-Banach space. In this paper, we investigate some hyperstability results for the following quadratic functional equation f(x+sigma(y))+f(x+tau(y))=2f(x)+2f(y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\big (x+\sigma (y)\big )+f\big (x+\tau (y)\big )=2f(x)+2f(y),$$\end{document}where f:X -> Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:X\rightarrow Y$$\end{document} is a mapping and sigma,tau:X -> X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ,\tau :X\rightarrow X$$\end{document} are involutions. We also examine the hyperstability of the given equation in its inhomogeneous version f(x+sigma(y))+f(x+tau(y))=2f(x)+2f(y)+chi(x,y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\big (x+\sigma (y)\big )+f\big (x+\tau (y)\big )=2f(x)+2f(y)+\chi (x,y),$$\end{document}where chi:XxX -> Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi :X\times X\rightarrow Y$$\end{document}. Additionally, we elucidate the hyperstability of various special cases of our main results.

Let X be a normed space and Y be an ulrametric n-Banach space. In this paper, we investigate some hyperstability results for the following quadratic functional equation (Formula presented.) where f:X→Y is a mapping and σ,τ:X→X are involutions. We also examine the hyperstability of the given equation in its inhomogeneous version (Formula presented.) where χ:X×X→Y. Additionally, we elucidate the hyperstability of various special cases of our main results.