In this paper we will investigate the solutions and stability of the generalized variant of Wilson's functional equationwhere G is a group, σ is an involutive morphism of G and χ is a character of G. (a) We solve (E) when σ is an involutive automorphism, and we obtain some properties about solutions of (E) when σ is an involutive anti-automorphism. (b) We obtain the Hyers Ulam stability of equation (E). As an application, we prove the superstability of the functional equation f (xy) + χ(y)f (σ(y)x) = 2f (x)f (y), x, y ∈ G.
2, where φ is multiplicative. There has been quite a development of the theory of d'Alembert's functional equation (1.1) during the last years, on non abelian groups, as shown in works by Dilian yang about compact groups [10,11,12], Stetkaer [42] for step 2 nilpotent groups, Friis [17] for results on Lie groups and Davison [8,9]