Solutions and stability of variant of Wilson's functional equation (original) (raw)

In this paper we will investigate the solutions and stability of the generalized variant of Wilson's functional equation (E) : f (xy) + χ(y)f (σ(y)x) = 2f (x)g(y), x, y ∈ G, where 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.