Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

Abstract

Let and be a commutative semigroup and a commutative group, respectively, and the sets of complex numbers and nonnegative real numbers, respectively, and or an involution. In this paper, we first investigate general solutions of the functional equation for all , where . We then prove the Hyers-Ulam stability of the functional equation; that is, we study the functional inequality for all , where and .

1. Introduction

The following four functional equationsare called trigonometric functional equations. The above functional equations have been investigated together or individually. For early historical developments of the above functional equations we refer the reader to [1, p. 177] and [2, pp. 209–217]. In particular, general solutions of (3) are described in [2, pp. 216-217] and general solutions of (4) and regular solutions of (1) and (2) are described in [1, pp. 177–180]. Several authors [3–5] have studied some functional equations with involutions, particular cases of which are the usual d’Alembert’s functional equation [6], Wilson’s functional equation [7], or some related classical functional equations with usual operations. In this paper, generalizing the functional equation (3) we first consider the functional equationfor all , where . As a result we first find all general solutions of functional equation (5). Secondly, we investigate the Hyers-Ulam stability of functional equation (5). The Hyers-Ulam stability problems of functional equations go back to 1940 when Ulam proposed a question concerning the approximate homomorphisms from a group to a metric group (see [8]). Székelyhidi [9] investigated the Hyers-Ulam stability of trigonometric functional equations (1) and (2). As a particular case of the result in [9] he obtained the stability of functional inequalityfor all , where and . On the other hand, we obtained a Székelyhidi type stability of functional equations (3) and (4) (see [10, 11] and also [12–14] for related results). In this paper we obtain a parallel Hyers-Ulam stability to that of [9] for functional equation (3); that is, we obtain the Hyers-Ulam stability of the functional inequalityfor all , where and . Unfortunately, we have no idea yet how to obtain all general solutions of equationfor all , where and its stability, which are the parallel generalizations of the functional equation and stability of (4).

2. General Solutions of (5)

Throughout this section we denote by a 2-divisible commutative semigroup with an identity element. A function is said to be an involution if for all and for all . For simplicity we write instead of . A function is called an exponential function provided that for all and is called an additive function provided that for all . In this section, we present the general solutions of functional equation (5).

Lemma 1 (see [15, Theorem ]). Let satisfy the functional equationfor all . Then either has the formfor all , where is an exponential function satisfying and , orwhere is an exponential function satisfying , and is an additive function on satisfying for all and taking arbitrary values on , and .

In the following, we exclude the trivial cases when for all or for all .

Theorem 2. Let satisfy the functional equationfor all . Then either has the formfor all , where is an exponential function satisfying and , orfor all , where is an exponential function satisfying , and is an additive function on satisfying for all and taking arbitrary values on , and .

Proof. Let and be the even part and odd part of with respect to ; that is,for all . Then we havefor all . Replacing by in (12), we havefor all . From (12) and (17) we havefor all . Putting in (18) we havefor all . Substituting (16) into (12) we havefor all . Replacing by in (20) and using (18) and (19) we havefor all . Adding (21) and (20) we havefor all . Choosing such that , putting in (22), and dividing the result by we havefor all , where . Thus, we havefor all . Thus, it remains to find and . Replacing by in (20) and using (19) we havefor all . Adding (25) and (20) we havefor all . By Lemma 1, has the form (10) or (11). First, we consider the case when has the form (10); that is, , for all and . From (19) we have for all , which implies for all . If , we have for all , and hence for all , which leads to the contradiction for all . Thus, we have andfor all . From (24) and (27) we havefor all , where . Thus, we get (13). Secondly, we consider the case when has the form (11); that is, , , , and for all . From (19) we have for all , which implies . Thus, we have for all . From (23) we have for all . Thus, we get (14). This completes the proof.

In particular if a commutative group, then since , we have the following.

Corollary 3. Let satisfy the functional equationfor all . Then either has the formfor all , where is an exponential function satisfying for some and , or elsefor all , where is an exponential function satisfying for all and is an additive function, and .

In particular, if is a 2-divisible commutative group, then for all if and only if for all . Thus, we have the following.

Corollary 4. Let satisfy the functional equationfor all . Then either has the formfor all , where is an exponential function with , or elsefor all , where is an additive function, and .

Let be a commutative semigroup, and . As an application of Theorem 2, we determine all general solutions of the functional equationfor all .

In the following we exclude the trivial cases when or .

Theorem 5. Let satisfy the functional equation (35). Then either has the formfor all , where are exponential functions satisfying and , or elsefor all , where is an exponential function and is an additive function on taking arbitrary values on , and .

Proof. Let for all . Then by Theorem 2, all general solutions of functional equation (35) are given byfor all , where is exponential functions satisfying for some and , or elsefor all , where is exponential functions satisfying for all , is an additive function on , and . Since can be written in the form for some exponential functions , from (40) and (41) we get (36) and (37), respectively. Now, we consider the case when . Since for all we havefor all . It is easy to see that equality (44) implies . Thus, if and only if . On the other hand, the additive function can be written in the form for all , where are additive functions. Now, the equality implies for all . Thus we have for all and for all . Thus, by letting , we get (40) and (39) from (42) and (43), respectively, since if or , then . This completes the proof.

3. Stability of (5)

Throughout this section we denote by a 2-divisible commutative group and . In this section we consider the stability of functional equation (5); that is, we deal with the functional inequalityfor all . For the proof of the stability of (45) we need the following.

Lemma 6. Let satisfy the functional inequality (45) for all . Then there exist (not both zero) and such thatfor all , or elsefor all .

Proof. Suppose that is bounded only when . Letfor all . Choose satisfying . Then we havefor all , where and . From (48) and (49) we havefor all . Also, from (48) we havefor all . Equating (50) and (51) we havefor all . Fixing in (52) we havefor all . So by our assumption, the left hand side of (52) vanishes, so does its right hand side; that is,for all . From (48) we can writefor all . Using (55) and the triangle inequality we havefor all . Thus, if we fix in (54) the left hand side of (54) is a bounded function of . Hence by our assumption, we have . This completes the proof.

Lemma 7 (see [16, p. 104]). Assume that satisfy the functional inequality; for each , there exists a positive constant such thatfor all ; then either is a bounded function or is an exponential function.

In the following, we assume thatfor all , orfor all .

Lemma 8 (see [17]). Assume that satisfies the functional inequalityfor all . Then there exists a unique additive function given bysuch thatfor all provided that (58) holds, and there exists a unique additive function given bysuch thatfor all provided that (59) holds.

Theorem 9. Let satisfy the functional inequalityfor all ; then satisfies one of the following:(i), is arbitrary,(ii)both and are bounded functions,(iii), for all , where is an exponential function satisfying and ,(iv), for all , where is an exponential function satisfying , and is an additive function satisfying for all , and .(v)there exist and a bounded exponential function satisfying such that for all , and satisfies the condition that there exist , satisfying for all ,(vi)there exist and a bounded exponential function satisfying such that for all , and satisfies one of the following; 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 (58) and (59).

Proof. In view of Lemma 6, if satisfy (47), then by Theorem 2 we get (iii) or (iv). Now, we consider the case when satisfy (46). If , is arbitrary which is the case (i). If is a nontrivial bounded function, in view of (65), is also bounded which gives the case (ii). If is unbounded, it follows from (46) that and can be written asfor all , where and is a bounded function. Putting (71) in (65) we havefor all . Replacing by in (72) we havefor all . Using the triangle inequality with (73) we havefor all , where . By Lemma 7,