Alienation and stability of Jensen's and other functional equations

Let S be a semigroup and K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}$$\end{document} be the field of real or complex numbers. We deal with the stability and alienation of Cauchy's multiplicative (resp. additive) and Jensen's functional equations, starting from the inequalities f(xy)+f(x sigma y)+g(xy)-2f(x)-g(x)g(y)<=epsilon,x,y is an element of S,f(xy)+f(x sigma y)+g(xy)-2f(x)-g(x)-g(y)<=epsilon,x,y is an element of S,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \\ \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)-g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \end{aligned}$$\end{document}where f,g:S -> K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g:S\rightarrow \mathbb {K}$$\end{document} and sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} is an involutive automorphism on S. We also consider analogous problems for Jensen's and the quadratic (resp. Drygas) functional equations with an involutive automorphism.