Journal of Applied MathematicsVolume 2012 (2012), Article ID 610714, 14 pagesdoi:10.1155/2012/610714

Research Article

On a Generalized Hyers-Ulam Stability of Trigonometric Functional Equations

Jaeyoung Chung¹ and Jeongwook Chang²

¹Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea²Department of Mathematics Education, Dankook University, Yongin 448-701, Republic of Korea

Received 4 July 2011; Revised 3 January 2012; Accepted 3 January 2012

Academic Editor: Yongkun Li

Copyright © 2012 Jaeyoung Chung and Jeongwook Chang. 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.

Abstract.

Let be an Abelian group, let be the field of complex numbers, and let . We consider the generalized Hyers-Ulam stability for a class of trigonometric functional inequalities, , where is an arbitrary nonnegative function.

1. Introduction

The Hyers-Ulam stability problems of functional equations go back to 1940 when S. M. Ulam proposed a question concerning the approximate homomorphisms from a group to a metric group (see [1]). A partial answer was given by Hyers et al. [2, 3] under the assumption that the target space of the involved mappings is a Banach space. After the result of Hyers, Aoki [4], and Bourgin [5, 6] dealt with this problem, however, there were no other results on this problem until 1978 when Rassias [7] dealt again with the inequality of Aoki [4]. Following the Rassias' result, a great number of papers on the subject have been published concerning numerous functional equations in various directions [2, 7–21]. The following four functional equations are called trigonometric functional equations.The four functional equations have been investigated separately. The general solutions and regular solutions of the above equations are introduced [22, 23]. In particular, the last equation (1.4) is most interesting in the sense that (1.4) alone characterizes the two trigonometric functions , under some regularities of , which none of the remaining equations are able to do.

In [19], Székelyhidi developed his idea of using invariant subspaces of functions defined on a group or semigroup to obtain the Hyers-Ulam stability of the trigonometric functional equations (1.1) and (1.2). As results, he obtained the Hyers-Ulam stability when for each fixed the difference is a bounded function of and the Hyers-Ulam stability when for each fixed the difference is a bounded function of , where are mappings from an Abelian (amenable) group to the field of complex numbers.

In this paper, we complete the parallel Hyers-Ulam stability to that of [19] for the functional equations (1.3) and (1.4). As results, we obtained the Hyers-Ulam stability when for each fixed the difference is a bounded function of and the Hyers-Ulam stability when for each fixed the difference is a bounded function of .

In fact, the authors [10] obtained weaker versions of the Hyers-Ulam stability for the functional equations (1.3) and (1.4), that is, we proved the Hyers-Ulam stability of (1.3) when the difference is uniformly bounded for all and , and we proved the Hyers-Ulam stability of (1.4) when the difference is uniformly bounded for all and .

So, the results in this paper would be generalizations of those in [10]. We refer the reader to [9, 15, 16, 20, 21] for some related Hyers-Ulam stability of functional equations of trigonometric type.

2. Main Theorems

A function from a semigroup to the field of complex numbers is said to be an additive function provided that and is said to be an exponential function provided that . Throughout this paper, we denote by an Abelian group, the set of complex numbers, and a fixed nonnegative function. For the proof of stabilities of (1.3) and (1.4), we need the following.

Lemma 2.1 (see [2]). Let be a semigroup. Assume that satisfy the inequality; for each , there exists a positive constant such that for all , then either is a bounded function or is an exponential function.

Proof. Suppose that is not exponential, then there are such that . Now we have and hence, In view of (2.1), the right hand side of (2.3) is bounded as a function of . Consequently, is bounded.

We discuss the general solutions of the corresponding trigonometric functional equations

Lemma 2.2 (see [22, 23]). The general solutions of the functional equation (2.4) are given by one of the following: (i) and is arbitrary,(ii) and for some , where is an exponential function,(iii) for some , where is an additive function and is an exponential function satisfying . Also, the general solutions of the functional equation (2.5) are given by one of the following: (i) and for some ,(ii) and , where is an exponential function.

Proof. The solutions of the functional equation (2.4) are given in [23, p. 217, Theorem  11]. For the functional equation (2.5), combining the result of L. Vietoris [22, p. 177] and that of J. A. Baker [23, p. 220], we obtain that every nonconstant function satisfying (2.5) has the form for some exponential function . Thus, using (2.5), we have This completes the proof.

For the proof of the stability of (1.1), we need the following. Throughout this paper, we denote by an arbitrary nonnegative function on .

Lemma 2.3. Let satisfy the inequality for all , then either there exist , not both zero, and such that or else for all .

Proof. Suppose that the inequality (2.9) holds only when . Let and choose satisfying . Now it can be easily calculated that where and . By (2.11), we have Also by (2.11) and (2.12), we have From (2.13) and (2.14), we have Since is bounded by , if we fix , , the right hand side of (2.15) is bounded by a constant , where So by our assumption, the left hand side of (2.15) vanishes, so does the right hand side. Thus, we have Now by the definition of , we have Hence, the right hand side of (2.17) is bounded by . So if we fix , in (2.17), the left hand side of (2.17) is a bounded function of . Thus, by our assumption, we conclude that . This completes the proof.

In the following theorem, we assume that or

For the proof, we discuss the following property.

Lemma 2.4. Let be a bounded exponential function satisfying for some , then there exists such that Furthermore, the constant is the best one.

Proof. Since is a bounded exponential, there exists such that for all and , which implies for all . Assume that , then we have , and we may assume that . If , we have . If , there exists a positive integer such that , and we have . If , then , and we have . Finally, if , there exists a positive integer such that , and we have . Now define by . Then we have for all . Thus, is the biggest one. This completes the proof.

Theorem 2.5. Let satisfy the inequality for all , then satisfies one of the following: (i), is arbitrary,(ii) and are bounded functions,(iii) and for some , where is an exponential function,(iv)there exist and a bounded exponential function such that for all , and satisfies the condition; there exists satisfying for all ,(v)there exist and a bounded exponential function satisfying such that for all , and satisfies one of the following conditions; there exists an additive function such that for all , or there exists an additive function such that for all , where and are the functions given in (2.19) and (2.20).

Proof. In view of Lemma 2.3, we first consider the case when satisfy (2.9). If , is arbitrary which is the case (i). If is a nontrivial bounded function, in view of (2.22), is also bounded which gives the case (ii). If is unbounded, it follows from (2.9) that and for some and a bounded function . Putting (2.28) in (2.22), we have for all . Replacing by and using the triangle inequality, we have, for some , for all . By Lemma 2.1, is an exponential function. If , putting in (2.29), we have Thus, we have since is unbounded. Since is a nonzero bounded exponential function, it follows from the equalities that and , for all . Putting in (2.29) and replacing by multiplying in the result, we have for all . Replacing by in (2.29) and using (2.33), we have First we consider the case for some . Replacing by and by in (2.34), we have for all . From (2.34) and (2.35), using the triangle inequality, putting such that and dividing , we have for all , where , which gives (iv). Now we consider the case , for all . Dividing both the sides of (2.34) by , we have for all , where . By the well-known results in [4], there exists a unique additive function given by such that if , and there exists a unique additive function given by such that if . Multiplying in both sides of (2.39) and (2.41), we get (v). Now we consider the case when satisfy (2.10). In view of Lemma 2.2, the solutions of (2.10) are given by (i), (iii), or contained in the case (v). This completes the proof.

Let be a real normed space, and let be given by , then satisfies the conditions assumed in Theorem 2.5. In view of (2.19) and (2.20), we have if , if . Thus, as a direct consequence of Theorem 2.5, we have the following.

Corollary 2.6. Let satisfy the inequality for all , then satisfies one of the following: (i), is arbitrary,(ii) and are bounded functions,(iii) and for some , where is an exponential function,(iv)there exist and a bounded exponential function such that for all , and satisfies the condition; there exists satisfying for all ,(v)there exist and a bounded exponential function satisfying such that for all , and satisfies one of the following conditions; there exists an additive function such that for all .

Now we prove the stability of (1.2). For the proof, we need the following.

Lemma 2.7. Let satisfy the inequality for all , then either there exist , not both zero, and such that or else for all .

Proof. Suppose that is bounded only when , and let Since we may assume that is nonconstant, we can choose satisfying . Now it can be easily get that where and . From the definition of and the use of (2.53), we have the following two equations: Equating (2.54) and (2.55), we have In (2.56), when are fixed, the right hand side is bounded, so by our assumption, we have Also we can write Thus, if we fix in (2.57), the right hand side of (2.57) is bounded. By our assumption, we have . This completes the proof.

Theorem 2.8. Let satisfy the inequality for all , then satisfies one of the following: (i) and are bounded functions,(ii) and , where is an exponential function,(iii) for a bounded exponential function , and satisfies for all . In particular if , one has .

Proof. In view of Lemma 2.7, we first consider the case when satisfy (2.51). If is bounded, then in view of the inequality (2.59), for each , , is also bounded. It follows from Lemma 2.1 that is bounded or a nonzero exponential function. If is bounded, the case (i) follows. If is a nonzero exponential function, from (2.59), using the triangle inequality, we have for some , for all . Thus, it follows that for all , or else is bounded, and equality (2.62) implies , which gives the case (i).If is unbounded, then in view of (2.59), is also unbounded, and hence, and for some and a bounded function . Putting (2.63) in (2.59), replacing by , and using the triangle inequality, we have From Lemma 2.1, we have for some exponential function . If , we have Putting (2.66) in (2.59), multiplying in the result, and using the triangle inequality, we have for some , for all . Since is unbounded, we have for all , which implies , contradicting to the fact that is unbounded. Thus, it follows that , and we have where is a bounded exponential function. Putting (2.69) in (2.59), we have for all . Replacing by in (2.70) and dividing the result by , we have for all . From (2.69), (2.71), we get (iii). Now we consider the case when satisfy (2.51). In view of Lemma 2.2, the solutions of (2.51) are contained in (i) or given by (ii). Furthermore, if , then putting in (2.70), we have , and from (2.69), we also have . This completes the proof.

In particular, if is a continuous function and , then Theorem 2.8 is reduced as follows.

Corollary 2.9. Let be a continuous function satisfying (2.59) for , then satisfies one of the following: (i) and are bounded functions,(ii) and for some ,(iii)there exists such that for all .

Acknowledgment

The second author was supported by the research fund of Dankook University in 2010.

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