Orthogonally Fixed Points and (m, m)-Hom-Derivation Equations

Abstract

In this paper, we introduce the concept of m-Hom-m-derivation (briefly (m, m)-Hom-derivation) equations in orthogonally Banach algebras. We use the orthogonally fixed point to investigate the hyperstability of (m, m)-Hom-derivation equations.

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] answered to the question of Ulam between Banach spaces. In 1978, Rassias [3] generalized for additive mappings. See [4–9] for more information.

In addition, Hyers–Ulam stability of functional equations on the normed space, Banach space, and Banach algebras was given. Later, other research studies on the Hyers–Ulam stability of fractional-order systems were presented. Obloza, in 1997, was the first author who introduced the Hyers–Ulam stability for linear differential equations [10, 11]. In recent years, Hyers–Ulam stability theory of fractional differential equations has attracted great attention, for example, with the extension of the theory of fractional calculus (integral and derivative of arbitrary order) [12–16]; some researchers tried to extend the above concept for the fractional differential equations [17–20].

Recently, Eshaghi Gordji et al. [21] introduced the notion of the orthogonal set which contains the notion of the orthogonality in normed space. By using the concept of orthogonal sets, Bahraini et al. [22] proved the generalization of the Diaz–Margolis [23] fixed point theorem on these sets. The study on orthogonal sets has been done by several authors (for example, see [24–26]).

Definition 1 (see [21]). Let and be a binary relation. If satisfiesit is called an orthogonal set (or O-set), and we denote this O-set by .
Let be an O-set and be a generalized metric space; then, is called an orthogonally generalized metric space.

Definition 2 (see [21]). Let be an orthogonally metric space.(i)A sequence is called an orthogonal sequence (briefly O-sequence) if(ii)A mapping from into is continuous in if for all O-sequences in with , . Clearly, every continuous map is continuous at any .(iii) is orthogonally complete (briefly O-complete) if every Cauchy O-sequence is convergent to a point in X.(iv)A mapping from into is -preserving if for all with , .(v)A mapping from into is called an orthogonality contraction with Lipschitz constant if

Example 3. Let , and let the metric on be Euclidean metric. For all , is defined by . Let be a mapping defined bywhere . Since and , we have the following cases:(1)If (2)If (3)If (4)If Hence, it is clear that is not complete, but it is easy to see that O-complete, is not continuous and contraction on but it is easy to see that -continuous and --contraction for , and in the end, is -preserving on .

Theorem 1 (see [22]). Suppose that is an O-complete generalized metric space. Let be a -preserving, -continuous, and --contraction. Let satisfy for all , or for all , , and consider the “O-sequence of successive approximations with initial element  :” , , , …, , …. Then, eitheror there exists a positive integer such thatfor all . If the second alternative holds, then(1)The O-sequence of is convergent to a fixed point of .(2) is the unique fixed point of in .(3)If , then

Consider the generalized additive, quadratic, cubic, and quartic functional equationwhere is an integer number and is a real number.

Definition 4. Let be an orthogonally Banach algebra, and satisfying (8) andfor , is called an additive homomorphism, quadratic homomorphism, cubic homomorphism, and quartic homomorphism, respectively, or, briefly, m-homomorphisms.

In 2018, Park et al. [27] defined Hom-derivation, and in the following, we defined (m, m)-Hom-derivations on orthogonally Banach algebra.

Definition 5. Let be m-homomorphisms, . An m-mapping is called (m, m)-Hom-derivations if satisfies

In the present paper, we investigate the stability of orthogonally (m, m)-Hom-derivations for functional equation (8) on orthogonally Banach algebra by using the orthogonally fixed point.

2. Main Results

Throughout the section, let be an orthogonally Banach algebra. Let be mappings between , for all m positive integers less then 5 and , for all with , and ; for simplicity, denote

Suppose that and are mappings from into such that for all with and and for some constant . If we take , and .

It follows (12) and (13): for all with and and .

Theorem 2. Let be two mappings satisfying   where fulfills (12) and (13). Then, there exist unique orthogonally m-homomorphisms and unique orthogonally (m, m)-Hom-derivations such that for all and .

Proof. Let be the set of all mappings such that or , for all . Define on byand suppose that, for all ,for all .
Clearly, is an O-complete generalized metric space. Define by , and . Then,So, by the definition of on , for 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 exist with and such thatTherefore, by the similar argument, for all , , and , we haveClearly, is -preserving.
We show that, for any , we haveIn (16), put . By induction, we haveThen, for ,and then, all conditions of Theorem 3 hold. Furthermore, similarly, mapping holds.
So, the O-sequence converges to the unique fixed point in the set of , i.e.,Also, for ,and by (29), . Therefore, satisfies in (21), i.e.,Similarly,and H satisfies in (20), i.e.,We claim that(1) is the unique desired orthogonally m-homomorphism which satisfies in (20)(2) is the unique desired orthogonally (m, m)-Hom-derivation which satisfies in (21)First of all, and H are two m-mappings. In fact, for all with and and using (12), we haveSo, . Similarly, . By the result in [28], and are orthogonally m-mappings.

Proof. of Claim (1). By using (14) and (18), we havefor all with and .
Therefore, is an orthogonally m-homomorphism.

Proof. of Claim (2). By using (14) and (19), we havefor all with and . So, for all with and . Therefore, is an orthogonally (m, m)-Hom-derivation.

Corollary 1. Let with , and are mappings such that   for all with and ; then, there exist unique orthogonally m-homomorphisms and unique orthogonally (m, m)-Hom-derivations such that 

Proof. By takingand using Theorem 2, we get the desired results.

In the next theorem, we prove the hyperstability of orthogonally (m, m)-Hom-derivations for functional equation (8).

Theorem 3. Suppose there exist functions such that for all with and . Moreover, assume that are mappings such that   for all with and ; then, is an orthogonally (m, m)-Hom-derivation.

Proof. Letting in (47), we get , and by using induction on , we getSimilarly, we haveHence, we havefor all with and . So, by in (53) and using (45), we haveIt follows from (49) and (52) thatfor all with and . Hence, by in (55) and using (46), we have for all with , and it follows from (50) and (51) thatfor all with and . Hence, by in (56) and using (46), we have . Therefore, is an orthogonally (m, m)-Hom-derivation.

Corollary 2. Let with , and are mappings such thatfor all with and ; then, is an (m, m)-Hom-derivation.

3. Conclusions

In this paper, the author introduced orthogonally (m, m)-Hom-derivation equations. Using an orthogonally fixed point theorem, it is proved that orthogonally (m, m)-Hom-derivation equations for orthogonally m-mappings on orthogonally Banach algebras can be stable and hyperstable.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.