Alienation and stability of Jensen’s and other functional equations

Let S be a semigroup and \(\mathbb {K}\) $\mathbb{K}$ 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 \(\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \\ \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)-g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \end{aligned}\) $$\begin{array}{cc}\hfill \left|f,(,x,y,),+,f,(,x,\sigma ,y,),+,g,(,x,y,),-,2,f,(,x,),-,g,(,x,),g,(,y,)\right|\le & \epsilon ,x,y\in S,\hfill \\ \hfill \left|f,(,x,y,),+,f,(,x,\sigma ,y,),+,g,(,x,y,),-,2,f,(,x,),-,g,(,x,),-,g,(,y,)\right|\le & \epsilon ,x,y\in S,\hfill \end{array}$$ where \(f,g:S\rightarrow \mathbb {K}\) $f,g:S\to \mathbb{K}$ and \(\sigma \) $\sigma$ 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.