doi:10.1155/2007/79816 Research Article On Stability of a Functional Equation Connected with the Reynolds Operator

Let (X,◦) be an Abelain semigroup, g: X → X, and let K be either R or C. We prove superstability of the functional equation f (x ◦ g(y)) = f (x) f (y) in the class of functions f: X →K. We also show some stability results of the equation in the class of functions f: X →Kn. Copyright © 2007 Adam Najdecki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Throughout this paper n is a positive integer, (X,◦) is a commutative semigroup, K is either the field of reals R or the field of complex numbers C, and g: X → X is an arbitrary function. We study stability of the functional equation f x ◦ g(y)) = f (x) f (y) for x, y ∈ X, (1) in the class of functions f: X →Kn, where (a1,a2,...,an)(b1,b2,...,bn) = (a1b2,a2b2,..., anbn) for (a1,a2,...,an),(b1,b2,...,bn) ∈Kn. (For details concerning the problem of sta-bility of functional equations we refer to, e.g., [1].) Particular cases of (1) are the well-known multiplicative Cauchy equation f (xy)