Superstability of Generalized Multiplicative Functionals
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Research Article Superstability of Generalized Multiplicative Functionals Takeshi Miura,1 Hiroyuki Takagi,2 Makoto Tsukada,3 and Sin-Ei Takahasi1 1
Department of Applied Mathematics and Physics, Graduate School of Science and Engineering, Yamagata University, Yonezawa 992-8510, Japan 2 Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan 3 Department of Information Sciences, Toho University, Funabashi, Chiba 274-8510, Japan Correspondence should be addressed to Takeshi Miura, [email protected] Received 2 March 2009; Accepted 20 May 2009 Recommended by Radu Precup Let X be a set with a binary operation ◦ such that, for each x, y, z ∈ X, either x ◦ y ◦ z x ◦ z ◦ y, or z ◦ x ◦ y x ◦ z ◦ y. We show the superstability of the functional equation gx ◦ y gxgy. More explicitly, if ε ≥ 0 and f : X → C satisfies |fx √ ◦ y − fxfy| ≤ ε for each x, y ∈ X, then fx ◦ y fxfy for all x, y ∈ X, or |fx| ≤ 1 1 4ε/2 for all x ∈ X. In the latter case, the √ constant 1 1 4ε/2 is the best possible. Copyright q 2009 Takeshi Miura et al. 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.
1. Introduction It seems that the stability problem of functional equations had been first raised by S. M. Ulam cf. 1, Chapter VI. “For what metric groups G is it true that an ε-automorphism of G is necessarily near to a strict automorphism? An ε-automorphism of G means a transformation f of G into itself such that ρfx · y, fx · fy < ε for all x, y ∈ G.” D. H. Hyers 2 gave an affirmative answer to the problem: if ε ≥ 0 and f : E1 → E2 is a mapping between two real Banach spaces E1 and E2 satisfying fx y − fx − fy ≤ ε for all x, y ∈ E1 , then there exists a unique additive mapping T : E1 → E2 such that fx − T x ≤ ε for all x ∈ E1 . If, in addition, the mapping R t → ftx is continuous for each fixed x ∈ E1 , then T is linear. This result is called Hyers-Ulam stability of the additive Cauchy equation gx y gx gy. J. A. Baker 3, Theorem 1 considered stability of the multiplicative Cauchy equation gxy gxgy: if ε ≥ 0 and f is a complex valued function on a semigroup √ S such that |fxy − fxfy| ≤ ε for all x, y ∈ S, then f is multiplicative, or |fx| ≤ 1 1 4ε /2
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Journal of Inequalities and Applications
for all x ∈ S. This result is called superstability of the functional equation gxy gxgy. Recently, A. Najdecki 4, Theorem 1 proved the superstability of the functional equation gxφy gxgy: if ε ≥ 0, f is a real or complex valued functional from a commutative semigroup X, ◦, and φ is a mapping from X into itself such that |fx ◦ φy − fxfy| ≤ ε for all x, y ∈ X, then fx ◦ φy fxfy holds for all x, y ∈ X, or f is bounded. In this paper, we show that superstability of the functional equation gx
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