Hyers-Ulam stability of Butler-Rassias functional equation
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We will prove the Hyers-Ulam stability of the Butler-Rassias functional equation following an idea by M. T. Rassias. 1. Introduction In 1940, Ulam [9] gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms. Let G1 be a group and let G2 be a metric group with a metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy),h(x)h(y)) < δ for all x, y ∈ G1 , then a homomorphism H : G1 → G2 exists with d(h(x),H(x)) < ε for all x ∈ G1 ? The case of approximately additive functions was solved by Hyers [5] under the assumption that G1 and G2 are Banach spaces. Taking this fact into account, the additive Cauchy functional equation f (x + y) = f (x) + f (y) is said to have the Hyers-Ulam stability. This terminology is also applied to the case of other functional equations. For a more detailed definition of such terminology, one can refer to [4, 6, 7]. In 2003, Butler [3] posed the following problem. Problem 1.1 (Butler [3]). Show that for d < −1, there are exactly two solutions f : R → R of the functional equation f (x + y) − f (x) f (y) = d sinx sin y. Recently, Rassias excellently answered this problem by proving the following theorem (see [8]). Theorem 1.2 (Rassias [8]). Let d < −1 be a constant. The functional equation f (x + y) − f (x) f (y) = d sin x sin y Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:1 (2005) 41–47 DOI: 10.1155/JIA.2005.41
(1.1)
42
Hyers-Ulam stability of Butler-Rassias functional equation
has exactly two solutions in the class of functions f : R → R. More precisely, if a function f : R → R satisfies the Butler-Rassias functional equation for all x, y ∈ R, then f has one of the forms f (x) = c sin x + cosx,
f (x) = −c sin x + cosx,
(1.2)
√
where c = −d − 1 is set. In this paper, we will prove the Hyers-Ulam stability of the Butler-Rassias functional equation (1.1). 2. Preliminaries We follow an idea of Rassias [8] to prove the following lemma. In Section 3, we apply this lemma to the proof of the Hyers-Ulam stability of the Butler-Rassias functional equation (1.1). Lemma 2.1. Let d be a nonzero real number and 0 < ε < |d|. If a function f : R → R satisfies the functional inequality f (x + y) − f (x) f (y) − d sinx sin y ≤ ε
(2.1)
for all x, y ∈ R, then M f := supx∈R | f (x)| is finite and f (x) − f π sinx − cosx ≤ 2 1 + M f ε 2 |d |
(2.2)
for all x ∈ R. Proof. If we replace x by x + z in (2.1), then we have f (x + y + z) − f (x + z) f (y) − d sin(x + z)sin y ≤ ε
(2.3)
for any x, y,z ∈ R. Similarly, if we replace y by y + z in (2.1), then we get f (x + y + z) − f (x) f (y + z) − d sinx sin(y + z) ≤ ε
(2.4)
for x, y,z ∈ R. Using (2.3) and (2.4), we obtain f (x) f (y + z) − f (x + z) f (y) + d sinx sin(y + z) − d sin(x + z)sin y = f (x + y + z) − f (x
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