Hyperstability and Stability of a Logarithm-type Functional Equation

Abstract In 2001, Maksa and Páles [12] introduced a new type’s stability: hyperstability for a class of linear functional equation f(x) + f(y) = 1 n ∑n i=1 f(xφi(y)). Riedel and Sahoo [14] have generalized a functional equation associated with the distance between the probability distributions, which is f(pr, qs) + f(ps, qr) = 2M(rs)f(p, q) + 2M(pq)f(r, s). Elfen etc. [7] obtained the solution of the functional equation f(pr, qs) + f(ps, qr) = 2f(p, q) + 2f(r, s) on semigroup G. The aim of this paper is to investigate the hyperstability and the Hyers-Ulam stability for the above Logarithm-type functional equation considered by Elfen, etc. Namely, if f is an approximative equation related to the above equation, then it is a solution of this equation which exists within ε− bound of a given approximative function f .


Introduction
The following stability problem is well-known as Ulam's stability problem [16]: Let G 1 be a group and let G 2 be a metric group with a metric d(·, ·). Given > 0, does there exist a δ > 0 such that if a mapping h : G 1 → G 2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G 1 , then there exists a homomorphism H : G 1 → G 2 with d(h(x), H(x)) < for all x ∈ G 1 ? In next year, Hyers [11] proved a first partial answer to Ulam's problem for an additive mapping on a Banach space. D. G. Bourgin obtained many excellent results for the stability ( [3], [4]).Hyers' theorem was generalized by Aoki [1] for the case bounded by variables, and their results are improved by Rassias [13] to the case of the linear mapping and by Ger [9] . Gȃvruta [8] proved a further generalization of the Rassias' theorem by using a general control function.
The superstability phenomenon of the exponential equation f (x + y) = f (x)f (y) was discovered by Baker, Lawrence, and Zorzitto [2] in 1979. The superstability for asymptotic phenomenon of the exponential equation was discovered by Ger [9].
In 2001, Maksa and Páles [12] proved a new type's stability for a class of linear functional equation where f is a real-valued mapping defined on a semigroup S, and the mappings ϕ 1 , ϕ 2 , · · · , ϕ n : S → S are pairwise distinct automorphisms. That is as following: Let ε : S × S → R be a function such that there exists a sequence u k that satisfies lim k→∞ ε(u k s, t) = 0 (s, t ∈ S).
Assume that f : S → X satisfies the stability inequality where X is a real normed space. Then, f is a solution of (1). Such a phenomenon is called the hyperstability of the functional equation. Gselmann [10], Brazdȩk and Ciepliński [5] investigated the hyperstability of functional equations. A similar concept was introduced by Sirouni and Kabbaj [15].
Riedel and Sahoo [14] solved a functional equation associated with the distance between the probability distributions. Let M : (0.1) → C be a given multiplicative function. Then, if f : (0.1) 2 → C satisfies the functional equation where M : (0.1) → C is an arbitrary logarithmic function and l : (0.1) 2 → C is a bilogarithmic function. Thus, we will call it a logarithm-type functional equation In addition, Elfen, Riedel and Sahoo [7] solved a functional equation on semigroup G. Its solution type of f onḠ is given by where A : G → C is a homomorphism and ψ : G → C is a symmetric bi-homomorphism. Now we consider the logarithm-type functional equation given by For example, if f (x, y) = ln xy, then f is a solution of the equation (2). In this paper, we investigate the hyperstability and stability of the functional equation (2). Namely, we prove that if f satisfies a stability inequality for the equation (2), then it is also a solution of this equation and also we can find an another solution of it which has an ε−error bound for f .

Hyperstability of the logarithm-type functional equation
In this section, we investigate the hyperstability of the equation (2). Throughout this section, let (G, ·) denote a noncommutative semigroup, X a real normed space, and R the set of real numbers. And let R + denote the set of positive real numbers. Theorem 1. Let ε : G 2 × G 2 −→ R be a function such that there exists a sequence u k ∈ G that satisfies lim k→∞ ε(u k (p, q), (r, s)) = 0 for all p, q, r, s ∈ G. Assume that f : G × G −→ X satisfies the stability inequality for all p, q, r, s ∈ G. Then, Proof. Define a function F : Then, for all p, q, r, s, v, w ∈ G, we have And also, for all p, q, r, s, v, w ∈ G, we have Thus, F satisfies the following functional equation By (3), we get ||F ((p, q), (r, s))|| ≤ ε((p, q), (r, s)), and with the assumed sequence {u k }, we obtain for all p, q, r, s ∈ G.
By applying of (5) and the triangle inequalities, we obtain Hence, we obtain from the assumed sequence {u k } the required result for any r, s, v, w ∈ G.

Stability of the logarithm-type functional equation
In this section, we investigate the stability of the equation (2). Throughout this section, let (G, ·) denote a commutative semigroup, N the set of natural numbers, and X a Banach space.
Theorem 3. Let ε > 0. Assume that f : G × G −→ X satisfies the stability inequality for all p, q, r, s ∈ G. Then there exists a function F : G × G −→ X such that and ||F (p, q) − f (p, q)|| ≤ 39ε 40 for any p, q ∈ G, where F is defined by for any p, q, r, s ∈ G.
Proof. Letting r = p, s = q in (8) and dividing it by 2, we have And also, letting p = q = r = s in (8) and dividing it by 2, we have Let us show that the following inequality holds for every n ∈ N : Replacing p by p 2 and q by q 2 in (9) respectively, and dividing 2 2 , we have Thus by (9),(10), and (12), we have In addition, by letting p by p 2 n−1 and q by q 2 n−1 in (9), and dividing 2 2 n−1 , the following inequality holds for every n ∈ N : By (10), (12), and (14), we have Suppose that the following inequality holds for n ≥ 4 and for any p, q ∈ G: Note that for all n ∈ N . Then, for any p, q ∈ G, based on (14) and (18), we obtain Thus, by induction, inequality (17) holds for all n ≥ 4 and for any p, q ∈ G. Now for n ≥ 4, we have Then, due to (16), (17), and (21), we have ||F (p, q) − f (p, q)|| ≤ 39ε 40 ∀p, q ∈ G.
Finally, the function F defined in (21) holds the required equation (2) as follows: