Ulam’s Type Stability of Involutional-Exponential Functional Equations

Abstract

Let S be a commutative semigroup, and an involution. In this paper we consider the stability of involution-exponential functional equations for all , where satisfies the growth condition: there exists such that for each . We also consider the stability of -version where is a locally integrable function.

1. Introduction

Throughout this paper we denote by , , a commutative semigroup with an identity element, 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 . An exponential function is called -exponential if satisfies for all and denote by a -exponential function.

In [1], the following functional inequalities with involution are investigated: As a result, all unbounded functions satisfying the inequalities (1) and (2) are exactly described only when is a constant function while only one of unbounded functions satisfying each of (1) and (2) is exactly described when is an arbitrary unbounded function.

In this paper we investigate the functional inequalities (1) and (2) by imposing some growth conditions on , , or . First, we introduce the condition on : where will stand for and .

Secondly, we introduce the condition on ; there exists such that for all .

As a result, we completely determine and satisfying each of the inequalities (1) and (2): if satisfies (3) [resp., satisfies (3)] or satisfies (4), then satisfying (1) [resp., (2)] are of the form for all , where is a -exponential function and is an exponential function.

As an application of our result, we determine all unbounded functions satisfying the functional inequalities for all , where satisfy (3) or satisfies (4) (see [2–5] for related equations) and determine all unbounded functions satisfying the functional inequalities for all , , , , , , , and , where satisfy (3) or satisfies (4) (see [2, 4] for related equations). Finally, we consider the stability of  -version where is a locally integrable function. As a result, we prove that every unbounded solution (i.e., ) of (9) satisfies for almost every , where is an unbounded exponential function. Every bounded solution (i.e.,) satisfies If  , then satisfies either or

We refer the reader to [1, 6–16] for related functional equations and their stabilities. We also refer the reader to [17–19] for some recent developments on the issues of stability and superstability for functional equations.

2. Stability of (1) and (2)

In this section we investigate unbounded functions satisfying (1) and (2) when some of and satisfy (3) or satisfies (4). For bounded solutions of (1) and (2) we refer the reader to [1].

Lemma 1. Assume that is an unbounded exponential function and satisfies (4). Then satisfies (3).

Proof. Since is unbounded, we can choose a such that , where is the constant in (4). Since satisfies (4) we have This completes the proof.

Theorem 2. Let be unbounded functions satisfying for all . Then is a -exponential function. In particular if satisfies (3) or satisfies (4), then there exists a -exponential function such that for all .

Proof. Choosing a sequence , such that as , putting , in (15), dividing the result by , and letting we have for all . Putting in (15) we have for all . Multiplying both sides of (17) by and using (15), (17), and (18) we have for all . Dividing (19) by we have for all , where . From (20) we have for some -exponential . If satisfies (3) or satisfies (4), then, by Lemma 1, we can choose a sequence , such that as . Putting , in (15), dividing the result by , and letting we have for all . Multiplying both sides of (22) by and using (15), (18), and (22) we have for all . Putting in (23) and dividing the result by we have for all . Putting in (15) and using (24) we have for all . Since is unbounded, from (25) we have . Now, from (21) and (24) we get (16). This completes the proof.

We denote by the inner product of and which is defined as , and , where are the real parts of . It is easy to see that if is uniquely -divisible (i.e., for each there exists a unique such that ), then is -exponential if and only if for some exponential function .

Corollary 3. Let , be a polynomial. Suppose that are unbounded function satisfying for all . Then there exists a -exponential function such that for all . In particular if is continuous, then there exists such that for all .

Proof. It is easy to see that satisfies (4). Thus, by Theorem 2 we get (28). Assume that is continuous. It is well known that every continuous exponential functional is given by for some . Thus, from (26) we have for all , where denotes matrix multiplication. Thus, we get (29). This completes the proof.

Remark 4. Let be two nonzero vectors that are not parallel; that is, for all . Then, the hyperplane is not parallel to and hence there exists such that and . If for some , then there exists such that and if and only if . Thus, if for all , then there exists such that and .

Corollary 5. Let be fixed. Suppose that are unbounded continuous function satisfying for all . Then there exists such that for all . If for all , then one has for all .

Proof. Recall that every continuous -exponential functional is given by for all , where denotes matrix multiplication. If for all , then by Remark 4 there exists such that From (32) and (33) we have which implies that satisfies the condition (3). Thus, we get (31). This completes the proof.

Theorem 6. Let be unbounded functions satisfying for all . Then there exists an unbounded -exponential function such that for all . In particular if satisfies (3) or satisfies (4), then one has for all .

Proof. Putting in (35) we have for all . Choose a sequence , such that as . Putting , in (35), dividing the result by , letting , and using (37) we have for all . Multiplying both sides of (38) by and using (35) and (38) we have for all . From (39) we have is an exponential function, say . Now, from (37) we can write for all , where for all . Putting (40) in (35) and using the triangle inequality we have for all . Since is unbounded, from (41) we have for all . Assume that satisfies (3) or satisfies (4). Choose a sequence , such that as . Putting , in (35), dividing the result by , letting , and using (37) we have for all . Multiplying both sides of (42) by and using (35) and (42) we have for all . Putting in (43), replacing by , and dividing the result by we have for all . Putting in (35) and using (40) and (44) we get . This completes the proof.

Using Theorem 6 and applying the same method as in the proof of Corollary 3 we have the following.

Corollary 7. Let , be a polynomial. Suppose that are unbounded function satisfying for all . Then there exists a -exponential function such that for all .

Using Theorem 6 and applying the same method as in the proof of Corollary 5 we have the following.

Corollary 8. Let be fixed. Suppose that are unbounded continuous function satisfying for all . Then there exists such that for all . If for all , then one has for all .

Theorem 9. Let be unbounded functions satisfying for all . Then there exists an unbounded exponential such that for all . In particular if satisfies (3) or satisfies (4), then one has for all .

Proof. Putting in (49) we have for all . Choose a sequence , such that as . Putting , in (49), dividing the result by , letting , and using (51) we have for all . Multiplying both sides of (52) by and using (49) and (52) we have for all . From (53) we have is an exponential function, say . Assume that satisfies (3) or satisfies (4). Choose a sequence , such that as . Putting , in (49), dividing the result by , letting , and using (51) we have for all . Multiplying both sides of (54) by and using (49) and (54) we have for all . Putting in (55), replacing by , and dividing the result by we have for all . Putting in (49) and using (56) we get . This completes the proof.

Using Theorem 9 we have the following.

Corollary 10. Let , be a polynomial. Suppose that are unbounded function satisfying for all . Then there exists an exponential function such that for all .

Corollary 11. Let be fixed. Suppose that are unbounded continuous function satisfying for all . Then there exists such that for all . If for all , then we have for all .

Theorem 12. Let be unbounded functions satisfying for all . Then is an exponential function. In particular, if satisfies the condition (3) or satisfies (4), then there exists an unbounded exponential such that for all .

Proof. Choose a sequence , such that as . Putting , in (61), dividing the result by , and letting we have for all . Multiplying both sides of (63) by and using (61) and (63) we have for all . Therefore, is an exponential function, say . Assume that satisfies (3) or satisfies (4). Choose a sequence , such that as . Putting , in (61), dividing the result by , and letting we have for all . Multiplying both sides of (65) by and using (61) and (65) we have for all . Putting and replacing by in (66) we have for all . Replacing by , we get (62). This completes the proof.

Using Theorem 12 we have the following.

Corollary 13. Let , be a polynomial. Suppose that are unbounded function satisfying for all . Then there exists an exponential function such that for all .

Corollary 14. Let be fixed. Suppose that are unbounded continuous function satisfying for all . Then there exists such that for all . If for all , then we have for all .

3. Applications

In this section we consider the stability of (6)(8). A function is called multiplicative function provided that for all . Let , and for all . Then the functional inequalities (6) and (7) are converted to for all .

Viewing as a multiplicative group, letting , and applying Theorems 2 and 6 to the inequalities (71) we have the following.

Theorem 15. Let be unbounded functions satisfying (6). Then are of the form for all , where is a multiplicative function.

Applying Theorems 9 and 12 to the inequalities (72) we have the following.

Theorem 16. Let be unbounded functions satisfying (7). Then are of the form for all , where is a multiplicative function and is an exponential function satisfying .

Let be the set of quaternions. Recall that , , , , , , and and the conjugate of is given by . We denote . Let , , and for all . Then the functional inequalities (8) are converted to for all .

Applying Theorems 2 and 6 to the inequalities (75) we have the following.

Theorem 17. Let be unbounded functions satisfying (8). Then are of the form for all , where is a multiplicative function.

4. Stability in -Version

Let be a locally integrable function and an involution. In this section, we consider an -version of the stability of functional equation for almost every . More precisely, we study the functional inequality As is well known, inequality (78) implies for all .

We first employ defined by where . It is easy to see that is an infinitely differentiable function with support .

Let be a locally integrable function on and . Then for each , is a smooth function and for almost every as .

In the following, we exclude the case when for almost every .

Theorem 18. Let satisfy (78). Then either there exists an unbounded exponential function such that for almost every , or else If  , then either or

Proof. Applying in (79) we have We also have
Thus, the inequality (78) is converted to the classical functional inequality for all , where .
Choosing such that , putting in (88), using the triangle inequality, and dividing the result by we have for all . Since as , it follows that exists for all . Since for almost every , it follows from (90) that for almost every .
Fixing and letting so that in (88), we have for all . We first consider the case when is unbounded. Let , be a sequence such that . Putting in (92), dividing the result by , and letting we have for all . Multiplying in (93) and using (92) and (93) we have for all . Putting in (94) we have for all . From (95) we have for some . Putting (95) in (94) we have for all . From (96) is an exponential function. Now, we prove that for all . In view of (94), replacing by and by in (88) and letting so that we have for all . Using (95) and (98) we have Letting in (99) so that we have for all . Since is an exponential function, it follows from (100) that for all . Since is unbounded, from (101) we have for all . Now, is written in the form for all . Conversely, let , where is an arbitrary exponential function. Then is an exponential function satisfying for all . Thus, we get (82). From now on, we assume that is bounded, say for all . Then, it follows from (91) that . Thus, we have for all . From the inequality (92), using the method in Theorem 10 of [1] we have for all . Fixing and letting in (104) we have for all . From (105), using the method in Theorem 10 of [1] we have for all , and if , then we have either for all or for all . Since almost every , we get (83), (84), and (85). This completes the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by Basic Science Research Program through the National Foundation of Korea (NRF) funded by the Korea Government (no. 2012R1A1A008507). The author expresses his deep thanks to the referee for useful comments on some possible generalization of Theorem 2.