The stability of some generalizations of the quadratic and Wilson's functional equations

In this paper we prove a stability result for the functional equation 1/L Sigma(lambda is an element of K<bold>) F</bold>( x + lambda y) = alpha(y)G(x) + H(y), x, y <bold> is an element of S</bold>, for the functions F, G, H: S -> X, alpha : S -> K, where (S, +) is an abelian semigroup, K is a finite subgroup of the automorphism group of S, L := |K| and (X, parallel to . parallel to) is the Banach space over a field K <bold> is an element of </bold>{R, C}.