Alienation of the Jensen, Cauchy and d’Alembert Equations

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 equation $$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$ 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.

Some remarks on the stability of the multi-Jensen equation

First a stability result of Prager-Schwaiger [Prager W., Schwaiger J., Stability of the multi-Jensen equation, Bull. Korean Math. Soc., 2008, 45(1), 133–142] is generalized by admitting more general domains of the involved function and by allowing the bound to be not constant. Next a result by Cieplinski [Cieplinski K., On multi-Jensen functions and Jensen difference, Bull. Korean Math. Soc., 2008, 45(4), 729–737] is discussed. Finally a characterization of the completeness of a normed space in terms of stability requirements for multi-Jensen functions is presented.