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Homomorphisms and derivations in -ternary algebras via fixed point method
Advances in Difference Equations volume 2012, Article number: 137 (2012)
Abstract
Park (J. Math. Phys. 47:103512, 2006) proved the Hyers-Ulam stability of homomorphisms in -ternary algebras and of derivations on -ternary algebras for the following generalized Cauchy-Jensen additive mapping:
In this paper, we improve and generalize some results concerning this functional equation via the fixed-point method.
MSC:39B52, 17A40, 46B03, 47Jxx.
1 Introduction and preliminaries
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Let X and Y be 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.
Theorem 1.1 (Th.M. Rassias)
Let be a mapping from a normed vector space E into a Banach space subject to the inequality
for all , where ϵ and p are constants with and . Then the limit
exists for all and is the unique additive mapping which satisfies
for all . If then inequality (1.1) holds for and (1.2) for . Also, if for each the mapping is continuous in , then L is linear.
Rassias [5] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for . Gajda [6] following the same approach as in Rassias [4], gave an affirmative solution to this question for . It was shown by Gajda [6], as well as by Rassias and Šemrl [7] that one cannot prove a Rassias’ type theorem when . The counterexamples of Gajda [6], as well as of Rassias and Šemrl [7] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings, cf. Găvruta [8], Jung [9], who among others studied the Hyers-Ulam stability of functional equations. The inequality (1.1) that was introduced for the first time by Rassias [4] provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept (cf. the books of Czerwik [10], Hyers, Isac, and Rassias [11]).
Following the terminology of [12], a nonempty set G with a ternary operation is called a ternary groupoid and is denoted by . The ternary groupoid is called commutative if for all and all permutations σ of .
If a binary operation ∘ is defined on G such that for all , then we say that is derived from ∘. We say that is a ternary semigroup if the operation is associative, i.e., if holds for all (see [13]).
A -ternary algebra is a complex Banach space A, equipped with a ternary product of into A, which are -linear in the outer variables, conjugate -linear in the middle variable, and associative in the sense that , and satisfies and (see [12, 14]). Every left Hilbert -module is a -ternary algebra via the ternary product .
If a -ternary algebra has an identity, i.e., an element such that for all , then it is routine to verify that A, endowed with and , is a unital -algebra. Conversely, if is a unital -algebra, then makes A into a -ternary algebra.
A -linear mapping is called a -ternary algebra homomorphism if
for all . If, in addition, the mapping H is bijective, then the mapping is called a -ternary algebra isomorphism. A -linear mapping is called a -ternary derivation if
There are some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation (cf. [16–18]).
Throughout this paper, assume that p, d are nonnegative integers with , and that A and B are -ternary algebras.
The aim of the present paper is to establish the stability problem of homomorphisms and derivations in -ternary algebras by using the fixed-point method.
Let E be a set. A function is called a generalized metric on E if d satisfies
-
(i)
if and only if ;
-
(ii)
for all ;
-
(iii)
for all .
Theorem 1.2 Let be a complete generalized metric space and let be a strictly contractive mapping with constant . Then for each given element , either
for all nonnegative integers n or there exists a nonnegative integer such that
-
(1)
for all ;
-
(2)
the sequence converges to a fixed point of J;
-
(3)
is the unique fixed point of J in the set ;
-
(4)
for all .
2 Stability of homomorphisms in -ternary algebras
Throughout this section, assume that A is a unital -ternary algebra with norm and unit e, and that B is a unital -ternary algebra with norm and unit .
The stability of homomorphisms in -ternary algebras has been investigated in [19]via direct method. In this note, we improve some results in [19]via the fixed-point method. For a given mapping , we define
for all and all .
One can easily show that a mapping satisfies
for all and all if and only if
for all and all .
We will use the following lemma in this paper.
Lemma 2.1 ([20])
Let be an additive mapping such that for all and all . Then the mapping f is -linear.
Lemma 2.2 Let , and be convergent sequences in A. Then the sequence is convergent in A.
Proof Let such that
Since
for all n, we get
for all n. So
This completes the proof. □
Theorem 2.3 Let be a mapping for which there exist functions and such that


for all and all , where . If there exists constant such that
for all , then there exists a unique -ternary algebras homomorphism satisfying
for all .
Proof Let us assume and in (2.1). Then we get
for all . Let . We introduce a generalized metric on E as follows:
It is easy to show that is a generalized complete metric space.
Now, we consider the mapping defined by
Let and let be an arbitrary constant with . From the definition of d, we have
for all . By the assumption and the last inequality, we have
for all . So for any . It follows from (2.4) that . Therefore according to Theorem 1.2, the sequence converges to a fixed point H of Λ, i.e.,
and for all . Also H is the unique fixed point of Λ in the set and
i.e., the inequality (2.3) holds true for all . It follows from the definition of H that

for all and all . Hence
for all and all . So for all and all .
Therefore, by Lemma 2.1, the mapping is -linear.
It follows from (2.2) and (2.5) that

for all . Thus
for all . Therefore, the mapping H is a -ternary algebras homomorphism.
Now, let be another -ternary algebras homomorphism satisfying (2.3). Since and T is -linear, we get and for all , i.e., T is a fixed point of Λ. Since H is the unique fixed point of , we get . □
Theorem 2.4 Let be a mapping for which there exist functions and satisfying (2.1), (2.2),

for all , where . If there exists constant such that
for all , then there exists a unique -ternary algebras homomorphism satisfying
for all .
Proof If we replace x in (2.4) by , then we get
for all . Let . We introduce a generalized metric on E as follows:
It is easy to show that is a generalized complete metric space.
Now, we consider the mapping defined by
Let and let be an arbitrary constant with . From the definition of d, we have
for all . By the assumption and the last inequality, we have
for all , and so for any . It follows from (2.6) that . Thus, according to Theorem 1.2, the sequence converges to a fixed point H of Λ, i.e.,
for all .
The rest of the proof is similar to the proof of Theorem 2.3, and we omit it. □
Corollary 2.5 ([19])
Let r and θ be nonnegative real numbers such that , and let be a mapping such that
and
for all and all . Then there exists a unique -ternary algebra homomorphism such that
for all .
Proof The proof follows from Theorems 2.3 and 2.4 by taking

for all and all . Then we can choose , when and , when and we get the desired results. □
3 Superstability of homomorphisms in -ternary algebras
Throughout this section, assume that A is a unital -ternary algebra with norm and unit e, and that B is a unital -ternary algebra with norm and unit .
We investigate homomorphisms in -ternary algebras associated with the functional equation .
Theorem 3.1 ([19])
Let (resp., ) and θ be nonnegative real numbers, and let be a bijective mapping satisfying (2.1) and
for all . If (resp., ), then the mapping is a -ternary algebra isomorphism.
In the following theorems we have alternative results of Theorem 3.1.
Theorem 3.2 Let and θ be nonnegative real numbers, and let be a mapping satisfying (2.7) and (2.8). If there exist a real number (resp., ) and an element such that (resp., ), then the mapping is a -ternary algebra homomorphism.
Proof By using the proof of Corollary 2.5, there exists a unique -ternary algebra homomorphism satisfying (2.9). It follows from (2.9) that
for all and all real numbers (). Therefore, by the assumption, we get that .
Let and . It follows from (2.8) that

for all . So for all . Letting in the last equality, we get for all . Similarly, one can show that for all when and .
Similarly, one can show the theorem for the case .
Therefore, the mapping is a -ternary algebra homomorphism. □
Theorem 3.3 Let and θ be nonnegative real numbers, and let be a mapping satisfying (2.7) and (2.8). If there exist a real number (resp., ) and an element such that (resp., ), then the mapping is a -ternary algebra homomorphism.
Proof The proof is similar to the proof of Theorem 3.2 and we omit it. □
4 Stability of derivations on -ternary algebras
Throughout this section, assume that A is a -ternary algebra with norm .
Park [19] proved the Hyers-Ulam stability of derivations on -ternary algebras for the functional equation .
For a given mapping , let
for all .
Theorem 4.1 ([19])
Let r and θ be nonnegative real numbers such that , and let be a mapping satisfying (2.7) and
for all . Then there exists a unique -ternary derivation such that
for all .
In the following theorem, we generalize and improve the result in Theorems 4.1.
Theorem 4.2 Let and be functions such that


for all , where . Suppose that is a mapping satisfying


for all and all . If there exists a constant such that
then the mapping is a -ternary derivation.
Proof Let us assume and in (4.3). Then we get
for all . Let . We introduce a generalized metric on E as follows:
It is easy to show that is a generalized complete metric space.
Now, we consider the mapping defined by
Let and let be an arbitrary constant with . From the definition of d, we have
for all . By the assumption and the last inequality, we have
for all . Then for any . It follows from (2.4) that . Thus according to Theorem 1.2, the sequence converges to a fixed point δ of Λ, i.e.,
and for all . Also δ is the unique fixed point of Λ in the set and
i.e., the inequality (2.3) holds true for all . It follows from the definition of δ, (4.1), (4.3), and (4.6) that

for all and all . Hence,
for all and all . So for all and all .
Therefore, by Lemma 2.1 the mapping is -linear.
It follows from (4.2) and (4.4) that
for all . Hence
for all . So the mapping is a -ternary derivation.
It follows from (4.2) and (4.4)

for all . Thus
for all . Hence, we get from (4.7) and (4.8) that
for all . Letting in (4.9), we get
for all . Hence, for all . So the mapping is a -ternary derivation, as desired. □
Corollary 4.3 Let , and θ be nonnegative real numbers, and let be a mapping satisfying (2.7) and
for all . Then the mapping is a -ternary derivation.
Proof Defining
and
for all , and applying Theorem 4.2, we get the desired result. □
Theorem 4.4 Let and be functions such that

for all where . Suppose that is a mapping satisfying (4.3) and (4.4). If there exists a constant such that
then the mapping is a -ternary derivation.
Proof If we replace x in (4.5) by , then we get
for all . Let . We introduce a generalized metric on E as follows:
It is easy to show that is a generalized complete metric space.
Now, we consider the mapping defined by
Let and let be an arbitrary constant with . From the definition of d, we have
for all . By the assumption and last inequality, we have
for all . Then for any . It follows from (4.5) that . Therefore according to Theorem 1.2, the sequence converges to a fixed point δ of Λ, i.e.,
and for all .
The rest of the proof is similar to the proof of Theorem 4.2, and we omit it. □
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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 manuscript.
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Kenari, H., Saadati, R. & Park, C. Homomorphisms and derivations in -ternary algebras via fixed point method. Adv Differ Equ 2012, 137 (2012). https://doi.org/10.1186/1687-1847-2012-137
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DOI: https://doi.org/10.1186/1687-1847-2012-137
Keywords
- -ternary algebra isomorphism
- generalized Cauchy-Jensen functional equation
- Hyers-Ulam stability
- -ternary derivation
- fixed-point method