Alienation and stability of Jensen’s and other functional equations

Let S be a semigroup and \(\mathbb {K}\) 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

$$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \\ \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)-g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \end{aligned}$$

where \(f,g:S\rightarrow \mathbb {K}\) and \(\sigma \) 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.