Stability of generalized Cauchy equations

We investigate the stability of the functional equation f(xy)=g(x)h(y)+k(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(xy) = g(x)h(y) + k(y)$$\end{document}on amenable semigroups. This equation is a common generalization of two Pexider equations stemming from Cauchy’s additive and multiplicative functional equations, and it is a simple case of the Levi-Civita equation.

(1.1) For real functions this equation has already been treated in J. Aczél's fundamental monograph [1], where the composition xy means addition of real numbers (therefore we have f (x + y) = g(x)h(y) + k(y) on p. 120 of [1]). Then chapter 15 of the book [3] is devoted to equation (1.1). By referring to J. Aczél [2] , it is solved under the assumptions that f, g, h, k : S → F, where S is an abelian groupoid with neutral element and F is a field. More precisely, when solving Exercise 1 on p. 250 of [3], it turns out that f, g, h, k satisfy (1.1) if and only if they have one of the following forms (where b, c, u, v ∈ F): where a : S → F solves the additive Cauchy equation where e : S → F solves the multiplicative Cauchy equation The just given four types of functions f, g, h, k : S → F are also solutions of (1.1) in arbitrary (not necessarily commutative) groupoids S.
In the present paper we investigate the stability in the sense of Hyers-Ulam of equation ( and where f x (s) := f (sx), s ∈ S. It is easily seen that the linear functional M has the properties and M (c) = c, for all complex numbers c. Moreover, for convenience, we will write M s (f (s)) instead of M (f ). In paper [11] the stability of the functional equation is under consideration, where X is a set, L is an amenable group of selfmappings of X, K is the field of real or complex numbers, u : The main result of our paper is contained in the next section.
We start with two lemmas.
Proof. Let (z n ) n∈N be a sequence such that Using (2.3) with x = z n and dividing the obtained inequality by |ϕ(z n )|, side by side, we have Letting n → ∞ we obtain Using (2.3) with z n x instead of x and dividing the obtained inequality by |ϕ(z n )|, side by side, we have Passing with n to infinity and taking (2.4) into account we infer that which completes the proof.
Proof. Let M be a right invariant mean on B(S). By (2.5) we infer that, for every y ∈ S, the mapping is bounded. Hence we can define η : S → C by the formula Moreover, since M ≤ 2 (cf.(1.2)), for y ∈ S we have which jointly with (2.5) gives (2.6) and completes the proof.
Proof of Theorem 2.1. From (2.1) we get By (2.7), applied for xy, and (2.1) we obtain Let us consider the following cases.  7), so is f . Functions F = 0, G = g, H = 0 and K = 0 are as required.

h(1) = 0 and there is an
By (2.8) with y = x 0 we see that g is bounded, and therefore, by (2.7), f is also bounded.
and, by (2.8), • If there are x 1 , x 2 ∈ S with g(x 1 ) =: a = b := g(x 2 ) then, by (2.8) applied for x 1 and x 2 in place of x, we have and respectively. We conclude that hence h is bounded. Now, by (2.9), k is also bounded. With F = G = H = K = 0 we get what is required.

h(1) = 0.
We put After dividing both sides of (2.8) by |h(1)| we get (2.10) 3.1. g and k 1 are bounded. By (2.7) f is also bounded, so it is enough to put F = 0, G = 0, H = h and K = 0 to get the assertion of the Theorem. 3.2. g is bounded and k 1 is not.
By (2.7) f is also bounded (M 1 := sup x∈S |f (x)|), and by (2.10) we infer that h 1 is unbounded. Let M 2 > 0 be such that |g(x)| ≤ M 2 for x ∈ S and let (y n ) n∈N be a sequence such that 0 = |h 1 (y n )| n→∞ −→ ∞. By (2.10), for any x 1 , x 2 ∈ S, we obtain |g(x 1 )h 1 (y n ) + k 1 (y n )| ≤ δ + M 2 , n ∈ N, and |g(x 2 )h 1 (y n ) + k 1 (y n )| ≤ δ + M 2 , n ∈ N, Thereby, g is constant (g(x) =: a for x ∈ S). We put F = 0, G = g, H = h and K(x) = −ah(x), x ∈ S. It is obvious that (2.2) is satisfied. In order to finish the proof in this case it is enough to check that the difference k − K is bounded. By (2.1) we have

g is unbounded.
By Lemma 2.2 we infer that On account of Lemma 2.3 there exists k 2 : S → C such that and We define Vol. 89 (2015) Stability of generalized Cauchy equations 55