Abstract

Let be a commutative semigroup if not otherwise specified and . In this paper we consider the stability of exponential functional equations or , or for all and where is an involution. As main results we investigate bounded and unbounded functions satisfying the above inequalities. As consequences of our results we obtain the Ulam-Hyers stability of functional equations (Chung and Chang (in press); Chávez and Sahoo (2011); Houston and Sahoo (2008); Jung and Bae (2003)) and a generalized result of Albert and Baker (1982).

1. Introduction

Throughout this paper we denote by , , , the set of real numbers, nonnegative real numbers, complex numbers, and the -dimensional Euclidean space, respectively, and , . A function is called exponential provided that for all and is called an involution provided that and for all . In [1], Baker proved the stability of the exponential functional equation: let satisfy the exponential functional inequality for all . Then, either is a bounded function satisfying for all , or is an unbounded exponential function (see also Baker et al. [2]). In particular, if , where is a vector space over the field of rational numbers and is a bounded function satisfying (1) for , then satisfies either for all , or else for all (see Albert and Baker [3]). In [4–6], some functional equations arising from number theory or product of matrices are introduced. The equations therein can be reduced to the functional equations with involution : for all (see Section 4). As we see in Section 4, the equations in [4–6] can be reduced to the equation of the form (5) when with operation of multiplication and for all or for all and the equations in [6] are reduced to those forms (5) in the complex numbers with and in the set of quaternions with , where denotes the conjugate of the quaternion .

As main results of the paper, we consider, in Section 2, the Ulam-Hyers stability of exponential functional equations with involution : for all . In Section 3, as a consequence of our main results we obtain the behaviors of bounded functions satisfying each of the inequalities for all . Also, as a direct consequence of the stability of (7) we obtain a generalized version of the result of Albert and Baker [3]. Our method of proof also works for investigation of bounded solutions of some other functional inequalities (see [7–9]). We refer to [7–12] for the exponential functional equations, inequalities, and related results.

In Section 4, as an application of our result, we obtain the stability of the functional equations: for all , and for all , where and . As stated in [6], (8) and (9) arise from a well-known theorem in number theory.

Reducing (8) and (9) to those in the complex numbers and the quaternions, we obtain the stability of (8) and (9); that is, we investigate bounded and unbounded functions satisfying each of the following functional inequalities: for all and for some , and we investigate bounded and unbounded functions satisfying each of the following functional inequalities: for all and for some . We also refer the reader to [5] for another proof of finding general solutions of (8) and refer to [13] for the stability of inequalities (10)~(13).

2. Stability of (6)

Throughout this section we assume that is a commutative semigroup if not otherwise specified. An exponential function is called -exponential if satisfies for all . We denote -exponential functions by . It is easy to see that if is uniquely -divisible (i.e., for each there exists a unique such that ; we write ), then is a -exponential function if and only if for some exponential function . Throughout this section we denote by an involution. In the following we exclude the trivial cases when or .

Theorem 1. Let satisfy for all . If is a group and is bounded, then for all , where . If is unbounded, then there exists a -exponential function such that for all , where . In particular, if for all , then we have for all .

Proof. First, we assume that is bounded (in this case we assume that is a group). Using the triangle inequality with (15) we have for all . Taking the supremum of the left-hand side of (19) with respect to we have for all , which implies for all . Replacing by in (15) and using the triangle inequality with the result we have for all . Taking the supremum of the left-hand side of (22) with respect to we have for all , which implies for all . From (21) and (24) we get (16). Now, we assume that is unbounded. Putting in (15) we have for all . Now, using the triangle inequality, (15), and (25) we have for all . Since is unbounded, it follows from (26) that for all . Dividing (27) by we have for all , where . Thus, is -exponential and for some -exponential . Replacing by in (25) and multiplying in the result we have for all . Using the triangle inequality with (25) and (29) we have for all . Since is unbounded, we have . Putting in (15) we have for all . Thus, we get (17). For the particular case when for all , replacing by and by in (15) and using triangle inequality with the resulting two inequalities, we have for all , which is for all , since for all . Dividing both sides of (33) by we have for all , where . Since is unbounded, we have for all and hence is unbounded. Replacing by and by in (34) we have two inequalities. Using the triangle inequality with these two resulting inequalities we have for all . Since is unbounded, from (35) we have and hence is independent of . From (34) we have for all and . Since is unbounded we have for all . Putting (37) in (15) we have for all . Since , , and is unbounded, we have . Thus, we get (18). This completes the proof.

Remark 2. We have no idea if the case in (17) occurs or not. For example, it can be verified that if there exists a sequence , , such that as , the case in (17) does not occur; in particular satisfies (18).

Remark 3. In general, the inequality (16) cannot be replaced by the weaker inequality: for all . Indeed, let , , , for all . Then we have for all . However, the inequality (39) fails since

As a direct consequence of Theorem 1 we have the following.

Corollary 4. Let satisfy for all . If is a group and is bounded, then for all . If is unbounded, then is -exponential.

Proof. The inequality (16) implies (43). Now, by Theorem 1, we have for some -exponential function . Putting (44) to (42) and letting we have for all . Since is unbounded, from (45) we have . This completes the proof.

Theorem 5. Let satisfy for all . If is a group and is bounded, then for all , where . If is unbounded, then there exists an exponential function such that for all . In particular, if for all , then we have for all .

Proof. Using the same method as in the proof of Theorem 1, we can show that if is bounded and is a group, then satisfies (47). Assume that is unbounded. Using the triangle inequality and (46) we have for all . Since is unbounded, it follows from (50) that for all . Therefore, is an exponential function: say . Putting and replacing by in (46) we get (48). Assume that for all . Replacing by and by in (46) using triangle inequality with the resulting two inequalities we have for all . Now, using the same method as in the proof of Theorem 1 (after the inequality (33)) we can show that for all . Since is an exponential function, we have and hence . This completes the proof.