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Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f, g : S → K of the functional equationf(x+y)+f(x+σ(y))+g(x+y)=2f(x)+g(x)g(y)for  x,y∈S.$$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations

Alienation of the Jensen, Cauchy and d’Alembert Equations