Superstability of p-Radical Functional Equations Related to Wilson–Kannappan–Kim Functional Equations

In the present paper, we introduce and solve the following p-radical functional equations related to trigonometric functional equations fxp+ypp+fxp-ypp=2f(x)f(y),fxp+ypp+fxp-ypp=2f(x)g(y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f\left( \root p \of {x^{p}+y^{p}}\right) +f\left( \root p \of {x^{p}-y^{p}}\right) =2f(x)f(y),\\&f\left( \root p \of {x^{p}+y^{p}}\right) +f\left( \root p \of {x^{p}-y^{p}}\right) =2f(x)g(y), \end{aligned}$$\end{document}and fxp+ypp+fxp-ypp=2g(x)f(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f\left( \root p \of {x^{p}+y^{p}}\right) +f\left( \root p \of {x^{p}-y^{p}}\right) =2g(x)f(y) \end{aligned}$$\end{document}for all x,y∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\in {\mathbb {R}}$$\end{document} where f, g are functions from R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document} into C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}$$\end{document} and p is an odd positive integer. Furthermore, we investigate the superstability of those functional equations. As consequence, we extend our results to Banach algebras.