On the Superstability of Lobačevskiǐ’s Functional Equations with Involution

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 satisfiesfor 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 satisfiesfor all . Then, every bounded function satisfiesfor 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 satisfiesIf is unbounded, then there exists an exponential function such thatfor all , and, in particular, ifthen there exist -exponential function and such thatfor all .

Proof. Assume that is bounded. Replacing by in (4) and using the triangle inequality with the result, we havefor all . Taking the supremum of the left hand side of (20), we havefor all . Fixing and solving the quadratic inequality (21), we obtainfor all . Taking the infimum of the right hand side of (22), we get (16). Now, assume that is unbounded. Putting in (4), we havefor 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 getfor 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 thatPutting , , 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 havefor all . Putting (30) in (23), we havefor 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 satisfiesand if is unbounded, then there exists -exponential function such thatfor 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 satisfiesIf is unbounded, then there exists an exponential function such thatfor all . In particular, ifthen there exist -exponential function and such thatfor all .

Proof. Choosing such that and putting in (5), we getfor 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 havefor all . Taking the supremum of the left hand side of (41), we havefor all . Taking the infimum of the right hand side of (42), we haveNow, using the triangle inequality with (5) again, we havefor all . Taking the supremum of in the left hand side of (44) and subtracting from both sides of the result, we havefor all . Taking the supremum of the left hand side of (45), we haveFrom (43) and (46), we get (36). Now, we assume that and are unbounded. Putting in (5), we havefor 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 getfor 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 havefor all . Putting (50) and (54) in (47), we havefor 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 satisfiesfor all . Then, every bounded function satisfiesand if is unbounded, then there exists -exponential function such thatfor all .

Proof. By Theorem 8, every bounded function satisfiesfor all . Solving (59), we havefor all , which gives (57). If is unbounded, then by Theorem 8 we havefor 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 satisfiesIf is unbounded, then there exist -exponential function and such thatfor all .

Proof. By Theorem 5, every bounded function satisfiesand, by Theorem 8, satisfiesFrom (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 iffor some exponential function .

Example 1. Let in Theorem 5 and letbe a matrix, where with . Then, defines an involution on . Let satisfyfor all and for some , . Then, by Theorem 5, every bounded function satisfiesfor 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 satisfyingfor all . In particular, if is Lebesgue measurable, is Lebesgue measurable and has the form for some . From condition (75), we have . Thus, has the formfor 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 satisfiesfor all , and if is unbounded, thenfor all . If is unbounded, it is easy to see that condition (18) holds. Thus, by Theorem 5, there exist -exponential function and such thatfor all . In particular, if is continuous, then by (68) there exist , and such thatfor 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 thatFrom (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).