Stability of Wilson’s functional equations with involutions

Abstract

Let S be a commutative semigroup, $${\mathbb{C}}$$ the set of complex numbers, $${\mathbb{R}^+}$$ the set of nonnegative real numbers, $${f, g : S \to \mathbb{C}\, \, {\rm and} \, \, \sigma : S \to S}$$ an involution. In this article, we consider the stability of the Wilson’s functional equations with involution, namely $${f(x + y) + f(x + \sigma y) = 2f(x)g(y)}$$ and $${f(x + y) + f(x + \sigma y) = 2g(x)f(y)}$$ for all $${x, y \in S}$$ in the spirit of Badora and Ger (Functional equations—results and advances, pp 3–15, 2002). As consequences of our results, we obtain the superstability of functional equations studied by Chung et al. (J Math Anal Appl 138:208–292, 1989), Chavez and Sahoo (Appl Math Lett 24:344–347, 2011) and Houston and Sahoo (Appl Math Lett 21:974–977, 2008).