Local Stability of the Additive Functional Equation and Its Applications

The main purpose of this paper is to prove the Hyers-Ulam stability of the additive functional equation for a large class of unbounded domains. Furthermore, by using the theorem, we prove the stability of Jensen's functional equation for a large class of restricted domains. 1. Introduction. The starting point of studying the stability of functional equations seems to be the famous talk of Ulam [14] in 1940, in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms: let G 1 be a group and let G 2 be a metric group with a metric d(·, ·). Given ε > 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)) < ε for all x ∈ G 1 ? The case of approximately additive mappings was solved by Hyers [3] under the assumption that G 1 and G 2 are Banach spaces. Later, the result of Hyers was significantly generalized by Rassias [11]. It should be remarked that we can find in [4] a lot of references concerning the stability of functional equations (see also [2, 5, 6]). In [12, 13], Skof investigated the Hyers-Ulam stability of the additive functional equation for many cases of restricted domains in R. Later, Losonczi [9] proved the local stability of the additive equation for more general cases and applied the result to the proof of stability of the Hosszú's functional equation. In Section 2, the Hyers-Ulam stability of the additive equation will be investigated for a large class of unbounded domains. Moreover, in Section 3, we will apply the previous result to the proof of the local stability of the Jensen's functional equation on unbounded domains. Throughout this paper, let E 1 and E 2 be a real (or complex) normed space and a Banach space, respectively.