Solutions and stability of a variant of Wilson’s functional equation.
Keywords:
Group, Semigroup-Involution, D’Alembert’s equation, Wilson’s equation, Automorphism, Homomorphism, Multiplicative function, Hyers-Ulam stability, SuperstabilityAbstract
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.
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