Abstract
We investigate the generalized Hyers-Ulam stability of homomorphisms and derivations on normed Lie triple systems for the following generalized Cauchy-Jensen additive equation , where are nonzero real numbers. As a results, we generalize some stability results concerning this equation.
1. Introduction
The stability problem of functional equations originated from a question of Ulam [1], concerning the stability of group homomorphisms.
Let be a group and let be a metric group with the metric . Given , there exist a scalar such that if a mapping satisfies the inequality for all , then a homomorphism exists with for all ?
Hyers [2] gave the first affirmative partial answer to the question of Ulam for additive mappings on Banach spaces. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. In 1990, Rassias [5] during 27th international symposium on functional equations asked the question whether such a theorem can also be proved for . In 1991, Gajda [6] following the same approach as in Rassias [5] gave an affirmative solution to this question for . It was proved by Gajda [6], as well as by Rassias and Šemrl [7] that one cannot prove Rassias’ type theorem when .
During the last three decades, several stability problems of functional equations have been investigated by a number of mathematicians; compare or confer [8–12] and references therein.
Now, we introduce the concept of normed Lie triple systems. First of all, ternary algebraic operations were considered in 19th century by several mathematicians such as Cayley [13] who introduced the notion of cubic matrix which in turn was generalized by Kapranov et al. [14] in 1990. There are some applications, although they are still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation. The comments on physical applications of ternary structures can be found in [15–22].
A normed (Banach) Lie triple system is a normed (Banach) space with a trilinear mapping from to satisfying the following axioms:(i),(ii),(iii),(iv),
for all . The concept of Lie triple system was first introduced by Lister [23](see also [24, 25]). Let and be normed Lie triple systems. A -linear mapping is said to be a Lie triple homomorphism if for all . A -linear mapping is called a Lie triple derivation if for all . The third identity asserts that the mapping is a (inner) derivation on [25].
Clearly, every Lie algebra is at the same time a Lie triple system via , and our definition of a homomorphism (derivation) coincides with that of prehomomorphism (prederivation) on a Lie algebra. Also, if is an involutive automorphism of a Lie algebra , then the eigenspace is a Lie triple system; see [25]. We remark that Lie triple systems are important since they give the structure of the tangent space of a symmetric space. Confer [25, 26] for reference. Also some applications of Lie triple systems can be found in Nambuòs approach by modifying the Heisenberg equation of motion [27].
Now, one of the interesting functional equations is the following general Cauchy-Jensen additive equation: which is a generalization of Cauchy or Jensen additive equations, where , and are nonzero real numbers and is a mapping between linear spaces. It is easy to see that a mapping satisfies the above Cauchy-Jensen additive equation if and only if is additive, where if . In this paper, we refine the generalized Hyers-Ulam stability results of [25] for Lie triple homomorphisms and Lie triple derivations on Lie triple systems associated with the general Cauchy-Jensen additive equation (1) and then we apply our results to study stability theorems of Lie triple homomorphisms and Lie triple derivations associated with Cauchy-Jensen additive equation (1) on normed Lie triple systems, which can be regarded as ternary structures. The reader may be referred to [28–30] for some other related stability results on derivations.
Throughout this paper, suppose that is a normed Lie triple system with norm and that is a Banach Lie triple system with norm .
Let be the set of all mappings from to , let be the set of all -linear mappings from to , let be the set of all Lie triple homomorphisms from to , let be the set of all Lie triple derivations on , and let be the set of all nonnegative reals.
2. Stability of Homomorphisms on Normed Lie Triple Systems
In the section, we prove the stability of homomorphisms on normed Lie triple systems associated with the Cauchy-Jensen additive equation. For , we define an operator by for all and all , where is a nontrivial connected subset of with and .
Lemma 1. Let with . Then for all and all if and only if .
Proof. Let for all . Then we note that and for all . Thus, for all . So for all , and then is an additive mapping and for all and all . Hence the assumptions of Lemma 1 in [31] are fulfilled, for instance, in view of the comments in the papers [32, 33], and so we can deduce the lemma. The converse is trivial.
We first consider stability theorem for approximate Lie triple homomorphisms of the functional equation with action of .
Theorem 2. Suppose that with satisfies for all and all . If and are functions such that for all , where , then there exists a unique such that for all .
Proof. Letting , , and in (4) and dividing by , we get
for all . Replacing by in (10) and dividing both sides of (10) by , we get
for all and all nonnegative integers . Therefore, one can use induction to show that
for all nonnegative integers and all . It follows from the convergence of the series (6) that the sequence is a Cauchy sequence in . Thus, we may define a mapping by
for all . Letting in (12) and passing to infinity, we lead to the approximation (9). It follows from (4) that
for all and . Hence
for all and . So, by Lemma 1.
It follows from (5) that
for all . So, .
To prove the uniqueness of such homomorphism subject to (9), we assume that there exists another Lie triple homomorphism satisfying (9). Obviously, we have and for all . By the triangle inequality, (9), and above equality, we have
for all . By letting , we get for any . This completes the proof.
Corollary 3. Let be a positive real number, let , and let be a real number such that either or . Suppose that with satisfies for all and all . Then there exists a unique such thatfor all .
Proof. The proof follows from Theorem 2 by taking for all .
Recently, the investigation on the hyperstability of the Cauchy and Jensen functional equations has been established [34, 35] and some information on it can be found in [36, 37]. Now, we present the following hyperstability result associated with Lie triple homomorphisms between normed Lie triple systems.
From Theorems 2–5 in the paper [34], we can easily derive the following.
Theorem 4. Assume that satisfies the inequality for some , and , . Then
Corollary 5. Let with . Assume that there exist and with , such that for all and all , , and that a function satisfies the functional inequality (5) and (8) for all and some . Then .
Proof. By (23) with , condition (21) holds. Hence Theorem 4 implies that condition (22) is fulfilled. Next, with in (23), we get
which implies that
Hence, by (22),
and next
which yields for . This and (26) show that is additive. Consequently, (26) and Lemma 1 in [31] jointly with the remarks in the papers [32, 33] imply the -linearity of .
Further, it follows from (5) and (8) that, for all and all ,
which tends to zero as . So, .
Corollary 6. Let with and . Assume that there exist and such that for all and all and that a function satisfies the functional inequality (5) and (8) for all . Then .
Proof. It follows from Theorem 2 in [38] (see also, Theorem 1.2 in [35]) and (29) that for all and all . Thus, which implies for all . Now, using the equality (30), we get for any with . Thus, there is defined as for any , which yields and for any . Consequently, and so for any . Therefore, we have for all . Since , we conclude that is additive on , and so we lead to the -linearity of by applying Lemma 1. The remaining proof is the same as the corresponding proof of Corollary 5.
Corollary 7. Let with and . Assume that there exist with such that for all and all and that a function satisfies the functional inequality (5) and (8) for all . Then .
Proof. It follows from Theorem 20 in [36] and (34) that for all and all . Thus, for all and all , one has , which implies that is additive on , and so we lead to the -linearity of by applying Lemma 1. The remaining proof is the same as the corresponding proof of Corollary 5.
Next, we consider stability theorem for approximate Lie triple homomorphisms of the functional equation with action of two scalars and .
Theorem 8. Suppose that with satisfies for and and for all . If and are functions such that for all , where , and the mapping is continuous on for each fixed , then there exists a unique such that for all .
Proof. Applying the same argument as in the proof of Theorem 2, one can deduce the existence of a unique additive mapping given by
satisfying the required inequality (39). By the same reasoning as in the proof of the theorem of [4], the additive mapping is -linear.
Letting , , and in (36), we have
for all . Then, it follows from the last inequality that
for all and all . The right hand side tends to zero as , and so we obtain
for all . Therefore, for each ,
for all , so that . It follows from (37) that
for all . So, .
The remainder of proof is similarly verified by the corresponding proof of Theorem 2.
Corollary 9. Let be a positive real number, let , and let be a real number such that either or . Suppose that with satisfies for all and and for all . If the mapping is continuous on for each fixed , then there exists a unique such thatfor all .
3. Stability of Derivations on Normed Lie Triple Systems
In this section, we prove the stability of derivations on normed Lie triple systems associated with the Cauchy-Jensen additive equation. Now, we consider stability theorem for approximate Lie triple derivations of the functional equation with action of .
Theorem 10. Suppose that with satisfies for all and all . If and are functions such that for all , where , then there exists a unique , defined as , such that for all .
Proof. Putting and in (48), we obtain
for all . Let ; therefore
for all . Replacing by in (55) and dividing both sides of (55) by , we get
for all and all nonnegative integers . Thus, one has
for all nonnegative integers and . It follows from the convergence of the series (50) that the sequence is a Cauchy sequence. From the completeness of , this sequence converges in . So we can define a mapping by for all .
Now, by considering and taking the limit as in (57), we obtain the estimation (53). It follows from (49) thatfor all . So, .
The remaining proof is similar to that of Theorem 2.
Corollary 11. Assume that there exist real numbers , and that , , and are positive real numbers such that . Let and . Suppose with satisfies for all and all . Then there exists a unique such thatfor all , where , .
Now, applying the above main theorem, we present the following hyperstability result associated with Lie triple derivations on Banach Lie triple systems.
Corollary 12. Let with . Assume that there exist and with , such that for all and all and that a function satisfies the functional inequality (49) and (52) for all and some . Then .
Proof. By the same reason as in the proof of Corollary 5, we lead to the -linearity of . Further, it follows from (49) and (52) that for all and all which tends to zero as . So, .
Corollary 13. Let with and . Assume that there exist and such that for all and all and that a function satisfies the functional inequality (49) and (52) for all . Then .
Corollary 14. Let with and . Assume that there exist with such that for all and all and that a function satisfies the functional inequality (49) and (52) for all . Then .
Next, we consider stability theorem for approximate Lie triple derivation of the functional equation with action of and .
Theorem 15. Suppose that with satisfies for and and for all . If and are functions such that for all , where , and the mapping is continuous on for each fixed , then there exists a unique , defined as , such that for all .
Proof. The proof is similar to that of Theorem 10.
Corollary 16. Assume that there exist real numbers , and that , , and are positive real numbers such that . Let and . Suppose that with satisfies for and and for all . If the mapping is continuous on for each fixed , then there exists a unique such thatfor all , where , .
Conflict of Interests
The authors declare that they have no competing interests.
Authors’ Contribution
All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final version of the paper.
Acknowledgments
The authors would like to thank the reviewers and editors for their valuable comments and suggestions on the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2012R1A1A2008139).