In this paper we will investigate the complex-valued 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.

No abstract

No abstract

This paper is mainly concerned with the following functional equation where 𝐺 is a locally compact group, 𝐾 a compact subgroup of its morphisms, and μ is a generalized Gelfand measure. It is shown that continuous and bounded solutions of this equation can be expressed in terms of μ-spherical functions. This extends the previous results obtained by Badora (Aequationes Math. 43: 72–89, 1992) on locally compact abelian groups. In the case where 𝐺 is a connected Lie group, we characterize solutions of the equation in question as joint eigenfunctions of certain operators associated to the left invariant differential operators.

No abstract

We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

Let S be a semigroup. We determine the complex-valued solutions f, g, h of the functional equationin terms of multiplicative functions, solutions of the special caseof the sine addition law, where χ : S → C is a multiplicative function, and also in terms of solutions of the particular caseof the cosine-sine functional equation where χ : S → C is a multiplicative function and ϕ : S → C such that the pair (ϕ, χ) satisfies the sine addition law.

No abstract