Journal of Function Spaces
Volume 2016 (2016), Article ID 1036094, 7 pages
http://dx.doi.org/10.1155/2016/1036094

Research Article

On the Superstability of Lobačevskiǐ’s Functional Equations with Involution

Jaeyoung Chung,¹ Bogeun Lee,² and Misuk Ha²

¹Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea
²Department of Mathematics, Jeonbuk National University, Jeonju 561-756, Republic of Korea

Received 29 November 2015; Accepted 22 February 2016

Academic Editor: Alberto Fiorenza

Copyright © 2016 Jaeyoung Chung et al. 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 a uniquely -divisible commutative group and let and be an involution. In this paper, generalizing the superstability of Lobačevskiǐ’s functional equation, we consider or for all , where . As a direct consequence, we find a weaker condition for the functions satisfying the Lobačevskiǐ functional inequality to be unbounded, which refines the result of Găvrută and shows the behaviors of bounded functions satisfying the inequality. We also give various examples with explicit involutions on Euclidean space.

1. Main Results

Throughout this paper, we denote by a uniquely -divisible commutative group (i.e., for each , there exists a unique such that ) and by , , , and the set of real numbers, nonnegative real numbers, complex numbers, and the -dimensional Euclidean space, respectively. A function is called an exponential function provided that for all and is called an involution provided that and for all . We denote by . An exponential function is called -exponential function if satisfies for all and is denoted by . In [1] (see also [2, Theorem ]), Găvrută investigated the superstability of Lobačevskiǐ’s functional equation:

for all and for some (see Albert and Baker [3] and Baker [4] for the superstability of the exponential functional equation). As a result he proved the following.

Theorem 1. Assume that satisfies (1). Then, every bounded function satisfies

for all , and if there exists such that , then satisfies

for all .

A careful observation shows that the upper bound in (2) is not optimal and may be replaced by a better one; namely, with . For example, let . Then, and . Using the better upper bound, we investigate the behavior of bounded functions satisfying (1) (see Remark 3).

As main results in this paper, we generalize Găvrută’s result and consider the inequalities:

for all , where is an unbounded function.

Note that the cases and can be reduced to the special case of form (4) or (5) when , the identity involution on (replacing by and by .

As a direct consequence of our results, we obtain the following.

Theorem 2. Assume that satisfies

for all . Then, every bounded function satisfies

for all , and if there exists such that , then there exist -exponential function and such that

for all .

Remark 3. Let be the identity involution in Theorem 2; we obtain a refined version of Theorem 1. Now, from (7), we have

Thus, if , is a sequence of functions satisfying

for all , then there exists a sequence such that converges uniformly to as (i.e., tends to straight lines).

We refer the reader to [5–7] for related functional equations and their stabilities. We also refer the reader to [8, 9] for some important recent developments on the issues of stability and superstability for functional equations.

2. Stability of (4)

In this section, we prove the superstability of (4). For the proof, we need the following.

Lemma 4. Assume that are unbounded exponential functions satisfying

for all and for some . Then, one has and .

Proof. Since is unbounded exponential function, it is easy to see that for all . Replacing by , , in (11) and dividing the result by we have

If , we have

Since , from (13), we have

for all such that and hence . Let . Choose such that . Then, we have and . Thus, it follows from (14) that and and hence

for all such that . This completes the proof.

From now on, we denote and .

Theorem 5. Assume that satisfy (4). Then, every bounded function satisfies

If is unbounded, then there exists an exponential function such that

for all , and, in particular, if

then there exist -exponential function and such that

for all .

Proof. Assume that is bounded. Replacing by in (4) and using the triangle inequality with the result, we have

for all . Taking the supremum of the left hand side of (20), we have

for all . Fixing and solving the quadratic inequality (21), we obtain

for all . Taking the infimum of the right hand side of (22), we get (16). Now, assume that is unbounded. Putting in (4), we have

for all . Choosing a sequence , , such that as , putting , , in (4), dividing the result by , letting , and using (23), we have

for all . Multiplying both sides of (24) by and using (4) and (24), we have

for all . Thus, is an exponential function and let ; then, we get

for all . Putting in (4), replacing by , and using (26) and the triangle inequality, we have for all , which gives (17). Assume that (18) holds. Then, we can choose a sequence , , such that

Putting , , in (4), dividing the result by , letting , and using (23), we have

for all . Multiplying both sides of (28) by and using (4) and (28), we have

for all . Putting in (29), replacing by , using (26), and dividing the result by , we have

for all . Putting (30) in (23), we have

for all . Applying Lemma 4 in (31), we have for all and . Letting , we get (19). This completes the proof.

In particular, if in Theorem 5, we obtain the following.

Corollary 6. Assume that satisfies

for all . Then, every bounded function satisfies

and if is unbounded, then there exists -exponential function such that

for all .

Proof. By Theorem 5, we get (33) and every unbounded function has the form for some exponential function . Putting and in (32), we have

for all . Using Lemma 4, we have for all . Thus, we get (34). This completes the proof.

Remark 7. In particular, if in Corollary 6, we obtain Theorem 2.

3. Stability of (5)

In this section, we prove the stability of (5). We exclude the trivial case when or .

Theorem 8. Assume that satisfy (5). Then, every bounded function satisfies

If is unbounded, then there exists an exponential function such that

for all . In particular, if

then there exist -exponential function and such that

for all .

Proof. Choosing such that and putting in (5), we get

for all , and it follows that is bounded if and only if is bounded. Assume that and are bounded. Replacing by in (5) and using the triangle inequality with the result, we have

for all . Taking the supremum of the left hand side of (41), we have

for all . Taking the infimum of the right hand side of (42), we have

Now, using the triangle inequality with (5) again, we have

for all . Taking the supremum of in the left hand side of (44) and subtracting from both sides of the result, we have

for all . Taking the supremum of the left hand side of (45), we have

From (43) and (46), we get (36). Now, we assume that and are unbounded. Putting in (5), we have

for all . Choosing a sequence , , such that as , putting , , in (5), dividing the result by , letting , and using (47), we have

for all . Multiplying both sides of (48) by and using (5) and (48), we have

for all . Thus, is an exponential function, say ; then, we get

for all . Assume that (38) holds. Then, we can choose a sequence , , such that

Putting , , in (5), dividing the result by , letting , and using (47), we have

for all . Multiplying both sides of (52) by and using (5) and (52), we have

for all . Putting in (53), replacing by , and using (50), we have

for all . Putting (50) and (54) in (47), we have

for all . Applying Lemma 4, we have for all and . Letting , we get (39). This completes the proof.

In particular, if in Theorem 8, we obtain the following.

Corollary 9. Assume that satisfies

for all . Then, every bounded function satisfies

and if is unbounded, then there exists -exponential function such that

for all .

Proof. By Theorem 8, every bounded function satisfies

for all . Solving (59), we have

for all , which gives (57). If is unbounded, then by Theorem 8 we have

for all . Putting (61) in (56), we have

for all . Since is unbounded, we have for all , which gives (58). This completes the proof.

In particular, if for all in Theorems 5 and 8, we obtain the following.

Corollary 10. Assume that satisfy

for all . If is bounded, then satisfies

If is unbounded, then there exist -exponential function and such that

for all .

Proof. By Theorem 5, every bounded function satisfies

and, by Theorem 8, satisfies

From (66) and (67), we get (64). If is unbounded, then condition (38) holds. Thus, by Theorem 8, we get (65). This completes the proof.

4. Examples

In this section, employing some involutions on Euclidean space, we give various examples satisfying inequalities (4) and (5). We denote by the inner product of and defined as , , and , where are the real parts of . It is easy to see that if is uniquely -divisible, then is -exponential if and only if

for some exponential function .

Example 1. Let in Theorem 5 and let

be a matrix, where with . Then, defines an involution on . Let satisfy

for all and for some , . Then, by Theorem 5, every bounded function satisfies

for all , and if is unbounded, then

for all , where is an exponential function. In particular, if is unbounded, then we can choose such that , and hence

which implies that condition (18) holds. Thus, by Theorem 5, have the form

for all , where and is an exponential function satisfying

for all . In particular, if is Lebesgue measurable, is Lebesgue measurable and has the form for some . From condition (75), we have . Thus, has the form

for all and for some with .

Example 2. Let be an matrix such that , the unit matrix. Suppose that satisfy

for all and for some . Then, by Theorem 5, every bounded function satisfies

for all , and if is unbounded, then

for all . If is unbounded, it is easy to see that condition (18) holds. Thus, by Theorem 5, there exist -exponential function and such that

for all . In particular, if is continuous, then by (68) there exist , and such that

for all .

Example 3. Suppose that are unbounded function satisfying

for all and for some . Then, by Theorem 8, there exists an exponential function such that

for all . Choose such that . Then, we have

which implies that condition (38) holds. Thus, again, by Theorem 8, there exist an exponential function satisfying and such that for all . Now, from , is given by . Thus, we have

for all .

Remark 11. 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 .

Example 4. Let be fixed. Suppose that are unbounded continuous functions satisfying

for all . By Theorem 8, we have

for all and for some , . If for all , then, by Remark 11, there exists such that

From (87) and (88), we have

Thus, condition (38) holds. By Theorem 8, we get

for all .

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2015R1D1A3A01019573).

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