Orthogonally -Ternary Jordan Homomorphisms and Jordan Derivations: Solution and Stability

Abstract

In this work, by using some orthogonally fixed point theorem, we prove the stability and hyperstability of orthogonally -ternary Jordan homomorphisms between -ternary Banach algebras and orthogonally -ternary Jordan derivations of some functional equation on -ternary Banach algebras.

1. Introduction and Preliminaries

A classical question in the sense of a functional equation says that “when is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?” Ulam [1] raised the question of stability of functional equations and Hyers [2] was the first to give an affirmative answer to the question of Ulam for additive mapping between Banach spaces. In 1987, Rassias [3] proved a generalized version of the Hyers’ theorem for approximately additive maps. The study of stability problem of functional equations have been done by several authors on different spaces such as Banach, -Banach algebras and modular spaces (for example see [4–13]). One of the stimulating aspects is to examine the stability of those functional equations whose general solutions exist and are useful in characterizing entropies [14].

Recently, Eshaghi Gordji et al. [15] introduced the notion of the orthogonal set, which contains the notion of orthogonality in normed space. The study on orthogonal sets has been done by several authors (for example, see [16–18])

Definition 1. (see [15]). Let and be a binary relation. If there exists such that for all ,Then is called an orthogonally set (briefly O-set). We denote this O-set by .
Let be an O-set and be a generalized metric space, then is called orthogonally generalized metric space.

Let be an orthogonally metric space.(i)A sequence is called orthogonally sequence (briefly O-sequence) if for any ,(ii)Mapping is called continuous in if for each O-sequence in with , then . Clearly, every continuous map is continuous at any .(iii) is called orthogonally complete (briefly O-complete) if every Cauchy O-sequence is convergent to a point in X.(iv)Mapping is called -preserving if for all with , then .(v)A mapping is said to be orthogonally contraction (or -contraction) with Lipschitz constant if

By using the concept of orthogonally sets, Bahraini et al. [19], proved the generalization of the Diaz and Margolis [20] fixed point theorem on these sets.

Theorem 1 (see [19]). Let be an O-complete generalized metric space. Let be a -preserving, -continuous, and --contraction. Let be such that for all , or for all , , and consider the “O-sequence of successive approximations with initial element  ”: , , , …, , …. Then, either for all , or there exists a positive integer such that for all . If the second alternative holds, then(i)the O-sequence of is convergent to a fixed point of .(ii) is the unique fixed point of in .(iii)If , then

A -ternary Banach algebra , endowed with a ternary product of into , is a complex Banach space in which the product is -linear in the outer variables, conjugate -linear in the middle variable, and associative in the sense that , for all in and satisfies (see [21]). If is a usual -algebra, then an induced ternary multiplication can defined by . If a -ternary Banach algebra has a unital “e” such that for all , then with binary product and , is a unital -algebra (see [22]).

Definition 2. A -linear mapping between -ternary Banach algebras ; i.e. , is called(1)-ternary homomorphism if(2)-ternary Jordan homomorphism if For all .

Definition 3. A -linear mapping is called(1)-ternary derivation if(2)-ternary Jordan derivation if For all .To prove main results we use the following equivalent assertions.

Lemma 1 (see [23]). Let be a mapping such thatfor all . Then is additive.

Lemma 2 (see [24]). Let and be two ternary Banach algebras. Let be an additive mapping. Then the following assertions are equivalent:(a), for all .(b).

Lemma 3 (see[25]). Let be a ternary Banach algebras. Let be an additive mapping from into . Then the following assertions are equivalent:(a)(b)

In this paper, motivated by the works of [15, 23, 26], we prove the stability of orthogonally -ternary Jordan homomorphism and orthogonally -ternary Jordan derivation of the functional equation

On orthogonally -ternary Banach algebras, where belongs to the set of all complex numbers with for some fixed positive integer number .

2. Main Results

Throughout the paper, let be the set of all complex numbers , where and is a fixed positive integer number and let be two -ternary Banach algebras.

For simplicity, denotewhere and .

Suppose that and are two mappings from into such that for all with for some constant .

Now, we are ready to prove the stability of orthogonally -ternary Jordan homomorphism in -ternary Banach algebras.

Theorem 2. Let be a mapping for whichandand

For all , and with , whose and are defined as (13) and (14). Then there exists a unique orthogonally -ternary Jordan homomorphism such thatfor all .

Proof. Let be the set of all mappings such that or , for all . Define on byand suppose that, for all , if and only ifFor all .
Clearly is an O-complete generalized metric space. Define by , . ThenSo, by definition of on , for every with or and we have . This shows that , i.e. is -contraction. The function is -continuous. In fact, if is an O-sequence in which converges to , then for given , there exists with and such thatFor all and . Therefore by the similar argument, for all and , we haveClearly, is -preserving.
We show that for any , we haveIn (11), put , , and . By induction we haveThen for ,and then, all conditions of Theorem 1 hold.
So, the O-sequence converges to the unique fixed point in the set of , i.e.,Also, for ,and by (26), . Therefore, H satisfies in (18), i.e.,We claim that is the unique desired orthogonally -ternary Jordan homomorphism which satisfies in (18).
First of all, is a additive. In fact, for all , with and using (13), we haveSo by Lemma 1, is additive. By the same proof of Theorem 3 of [27], the mapping is -linear. We show that is unique. Let be another additive mapping satisfying (18). Then, we haveFor all . Letting shows that is unique.
Now, by using (16)and then (14) implies that for all with . On the other hand, by (13) and (17) we haveFor all . Therefore is an orthogonally -ternary Jordan homomorphism satisfying (18).

In the next theorem, we prove that the self-mapping with the same appropriate conditions which satisfied in the functional (11), can be approximated by an orthogonally -ternary Jordan derivation.

Denote

Theorem 3. Let be a mapping satisfying (17) such thatand

For all and with where mappings and are satisfied in (13) and (14). Then, there exists a unique orthogonally -ternary Jordan derivation such thatfor all .

Proof. Similar to proof of Theorem 2, there exists a self-mapping on defined by satisfies (37). By using Lemma 3 and definition of we havefor all with . So for all with . Thus, the mapping is a unique orthogonally -ternary Jordan derivation satisfies (37). Also, by the same argument in the proof of Theorem 2,

Theorems 1 and 2 generalized the result of Rassias [3], whenever we define

For all and , in the sense of orthogonal sets.

As a consequence of Theorem 1, we have hyperstability of orthogonally -ternary Jordan homomorphism between -ternary Banach algebras.

Theorem 4. Let and be nonnegative real numbers, and let be a mapping such that

For all and all with . Then, the mapping is a orthogonally -ternary Jordan homomorphism.

From Theorem 3, we obtain hyperstability of orthogonally -ternary Jordan derivation.

Theorem 5. Let and be nonnegative real numbers. Let be a mapping satisfies (41) such that

For all with . Then the mapping is an orthogonally -ternary Jordan derivation.

3. Conclusions

In this paper, we introduced orthogonally -ternary Jordan homomorphism and -ternary Jordan derivation. Using an orthogonally fixed point theorem, we proved that orthogonally -ternary Jordan homomorphism and orthogonally -ternary Jordan derivation of the functional (11) can be stable and hyperstable in the orthogonally -ternary Banach algebras. The Hyers–Ulam stability theory has many attractions and applications in the field of fractional calculus. For farther research in this field we suggest to see the paper [28, 29].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.