Abstract

In this article, a generalized form of -quartic mappings is introduced. The structure of such mappings is studied, and in fact, it is shown that every multiquartic mapping can be described as an equation, namely, the (generalized) multiquartic functional equation. Moreover, by applying two fixed point techniques, some results corresponding to known stability outcomes including Hyers, Rassias, and Găvruţa stabilities are presented. Using a characterization result, an appropriate counterexample is supplied to invalidate the results in the case of singularity.

1. Introduction

When we are speaking about the stability of one variable mappings or an equation, we remember the name of Ulam who proposed the first stability problem regarding of group homomorphisms [1] whether such mappings are stable. Roughly speaking, it means that (for instance) any approximate solution of an additive functional equation , namely, a mapping with the property is near to an exact solution of . In other words, does there exists a unique additive mapping for example on commutative groups such that ? Note that is depended on . This fact is called the stability of additive or Cauchy functional equation. In the case that is additive, we say the functional equation is hyperstable. The Ulam stability problem was partially answered by Hyers [2] for linear mappings on normed spaces and then extended by Aoki [3] (for additive mappings), T. M. Rassias [4] (for linear mappings by considering an unbounded Cauchy difference), and Găvruţa [5] (by a general control function instead of the sum of powers of norms).

In recent decades, the Ulam stability problem has received attention from the authors to study it for multiple variable mappings. Let us state the definition of some of them as follows.

Let be a commutative group, be a linear space over rationals, and be a natural number greater than 1. Let , where . A mapping is called(i)Multiadditive if it is additive in each of components [6], that is,(ii)Multiquadratic if it is quadratic in all variables [7, 8], namely(iii)Multicubic if it is cubic in each of its arguments [9], that is,

Here, this is a question in which equations can describe the mappings as mentioned earlier. Ciepliński [6], Zhao et al. [8], and Bodaghi et al. [9] showed that each of the mappings above can be shown and unified as an equation. Moreover, the stability of multiadditive, multiquadratic, and multicubic mappings is available for instance in [6–10].

Let us recall from [11, 12] two quartic functional equations as follows:

Kang [13] generalized equations (4) and (5) as follows:for fixed integers with , where and for the rest of the paper as well.

Motivated by equations (4) and (5), in [14, 15], the authors introduced two multiquartic mappings and characterized their structures. In fact, for a commutative group , a linear space , and an integer with , a mapping is called multiquartic if it is quartic (fulfills one of quartic functional equations (4) and (5)) in each variable. In [14, 15], the authors proved that every multiquartic mapping can be described as a single equation, and moreover, such equations define the mutiquadratic mappings.

In this study, we introduce an extended class of multiquartic mappings which fulfil quartic equation (6) in each component. We also characterize the structure of such mappings and show that every multiquartic mapping can be unified as a single equation (namely, the multiquartic functional equation) and vice versa (under the quartic condition in each variable). Finally, we prove some Hyers-Ulam and Găvruţa stability results for multiquartic functional equations in quasi- normed spaces by applying a fixed point method. Using a characterization result, we indicate an example for a nonstable multiquartic functional equation.

2. The Structure of Generalized Multiquartic Mappings

Throughout, the set of positive integers and the set of rational numbers are denoted by and , respectively, and also . For any , in which and , we write and , where denotes for the th power of an element of the commutative group .

Definition 1. Let and be vector spaces over . A mapping is called -quartic or generalized multiquartic if is quartic in each of its arguments; that is,Suppose that such that and in which . We wish to denote by when no confusion can arise. Let and , put , where . Put as a subset of including the elements of such that the cardinal of is when .
For the mappings defined in Definition 1, put some conventions as follows:It was shown in [15], Theorem 2.2 that a mapping is multiquartic (in sense of (4)) if and only if it satisfies the following equation:Moreover, it is has been proven in Proposition 1 from [14] that if is a multiquartic mapping (in sense of (5)), then it fulfills the following equation:

Proposition 1. Every generalized multi-quartic mapping fulfills the following equation:in which .

Proof. The proof is based on induction. For, it is clear that f satisfies equation (6). Suppose that (12) holds for some with . We haveThis shows that (12) holds for .
By means of Proposition 1, we find that the mapping defined through satisfies (12) and so one can call the mentioned equation as multiquartic functional equation.

Definition 2. Let and . Consider a mapping and . Then,(i) has the -power condition if for any arbitrary variable ,(ii) is even if for any arbitrary variable ,(iii) has zero condition if for any for which at least one component is zeroGiven with . We remember that is the binomial coefficient defined by .
Let be integers as in (6). We consider the following hypotheses for a mapping and use them from now on. (H1) has the -power condition (H2) has the -power condition and is evenThe proof of the next lemma is similar to the proof of [14], Lemma 2.5, and so we bring some parts of the proof for the sake of completeness.

Lemma 1. Under one of the hypotheses (H1) and (H2), every mapping fulfilling equation (12) has zero condition.

Proof. Assume that (H1) holds. Putting in (12), we haveRelation (16) implies that . Similar to the proof of Proposition 1 from [14], one can continue in this fashion showing that has zero condition. The second part can be obtained similarly.

Theorem 1. Under the assumptions Lemma 1, every mapping satisfying equation (12) can be generalized multiquartic.

Proof. Suppose that (H1) is true. Fix . SetPutting for all in (12) and applying our assumption, we getIt follows from (18) thatThe last equality shows that is quartic in the th component.
(ii) Consider . Interchanging into for all in (12) and applying hypothesis (H2), we obtain the desired result in a similar method of part (i). The proof now is finished.

Corollary 1. For a mapping , the following assertions are equivalent:(i) is (generalized) multiquartic(ii) satisfies (10)(iii) fulfills (12) with either the assumption (H1) or (H2)(iv) fulfills (11) with either the assumption (H1) or (H2)

Proof. It is easily verified that (4) and (5) are equivalent for a one-variable mapping . Moreover, it is shown in [13] that satisfies (4) if and only if (6) is valid for it. (i) (ii) It is proved in [15], Theorem 2.2 (i) (iii) This implication is a direct consequence of Proposition 1 and Theorem 1 (i) (iv) Proposition 1 and Theorem 2.6 from [14] are our tools to obtain the desired result

3. Some Results for the Stability of (12)

This section is devoted to prove the generalized Hyers–Ulam stability of equation (12) in the setting of quasi--normed spaces. The next theorem of fixed point theory was proved in [16]. This result can be useful to reach the desired aims.

Theorem 2. Suppose that is a complete generalized metric space, and is a mapping with the Lipschitz constant . For each element , one of the following cases can happen.(i) for all , or.(ii)there exists such that.(iii) for all ;(iv)there is a fixed point of such that the sequence converges to in which it belongs to the set(v) for all .Let be a vector space and be a commutative group. In the rest of this section, for a mapping , we define the operator throughfor , where is defined in (8).

From now on, we assume that all mappings have zero conditions. With this assumption, we have Hyers’ stability for equation (12) as follows.

Theorem 3. Let . Suppose that is a complete normed space and is a commutative group. If a mapping fulfilling the inequalityfor all , then there exists a unique solution of (12) such thatfor all . Moreover,for all .

Proof. Considering and in (22) and applying the hypotheses, we havefor all , and hence,Thus,Set , where is the set of all mappings from to . Define the generalized metric on .Similar to the proof of [17], Theorem 2.2, it is easily verified that is a complete generalized metric space. Consider the operator defined viafor all . Take and with . Then, , and hence,Therefore, . This shows that . This means that is a strictly contractive with the Lipschitz constant . Relation (27) necessitates thatThus,Applying Theorem 3 for the space , the operator and , we can deduce that the sequence converges to in which is a fixed point of . Indeed, , andOn the other hand, by induction on , it can be shown for all thatand moreover, (24) follows. Obviously, . In addition, it follows from (32) and part (iii) of Theorem 2 thatwhich proves (23). Furthermore,for all . The last relation shows that for all , and therefore, satisfies equation (12). For the uniqueness part, assume that is another solution of equation (12) with zero condition, and moreover, inequality (23) holds for it. Then, satisfies (33), and hence, it is a fixed point of the operator . Besides, by (23), we obtainand consequently, . Using part (ii) of Theorem 2, we find . This finishes the proof.
Here, we recall some basic facts regarding the setting of quasi--normed space.

Definition 3. Let be a linear space over , be a fixed real number with . A quasi--norm is a real-valued function on fulfilling the statements(i) for all (ii) if and only if (iii) for all and (iv)There is a constant such that for all If is a quasi--norm on , then the pair is said to be a quasi--normed space. The smallest possible is called the modulus of concavity of the norm . Moreover, a quasi--normed space is a complete quasi--normed space. Iffor all , then a quasi--norm is called a -norm, and indeed, a quasi--Banach space is said to be a -Banach space. On the other hand, for a -norm, the metric leads us to a translation-invariant metric on . Furthermore, it follows from the Aoki-Rolewicz Theorem [18] that every quasinorm is equivalent to some -norm.
The upcoming fixed-point lemma is presented in [19], Lemma 3.1.

Lemma 2. Let be fixed, with . Let also be a linear space, be a -Banach space with -norm . Suppose that for the function , there exists an with for all . If is a mapping fulfillingfor all , then there exists a unique mapping with andfor all . Moreover, for all .

Next, we prove the Găvruţa stability of (12) in quasi- normed spaces by applying Lemma 2.

Theorem 4. Let be fixed, be a linear space and be a -Banach space and be a function for which there exists an with for all . Suppose that a mapping fulfilling the inequalityfor all . Then, there exists a unique solution of (12) such thatfor all . In particular, if has either the property (H1) or (H2), then it is a generalized multiquartic mapping.

Proof. Similar to the proof of Theorem 3, by putting in (41) and using our hypotheses, we getfor all and sofor all . Lemma 2 implies that there exists a unique mapping such that andfor all . We claim that is a multiquartic map. Here, we note from Lemma 2 that for all , . Now, by (41), we havefor all and . Letting in the above inequality, we find for all , and hence, satisfies (12). The last assertion follows from Theorem 1.
In upcoming corollaries, it is assumed that is a quasi--normed space with quasi--norm , and is a -Banach space with -norm . The following result is a direct consequence of Theorem 3 concerning the stability of (12) when the norm of is controlled by the sum of variables norms of and with positive powers.

Corollary 2. Let and be positive numbers with . If a mapping fulfilling the inequalityfor all , then there exists a unique solution of (12) such thatfor all . Moreover, if has either the property (H1) or (H2), then it is generalized multi-quartic.

The hyperstability of equation (12) is presented as follows.

Corollary 3. Suppose for and with . If a mapping fulfilling the inequalityfor all , then it satisfies (12). In particular, if has one of the hypotheses (H1) and (H2), then it is a generalized multiquartic mapping.

The next proposition was proved in [20], Proposition 4.8.

Proposition 2. Let be a continuous -quartic function. Then, there exists a constant such thatfor all .

We end the paper with the following counterexample for the multiquartic mappings on that its idea is taken from [21]. In fact, we show the assumption cannot be removed in Corollary 2 when in the case that .

Example 1. Let and . Put , where andNote that . Define the function throughHence, for all and is continuous as well. Using the function , define the function byIt is obvious that is an even function in each variable and nonnegative. The series given in the last equality is uniformly convergent of continuous functions, and thus, is continuous and bounded. In other words, for each , we have . Put , where . We claim thatfor all . It is clear that (54) is valid for . Assume that with . Thus, there exists a positive integer such thatHence, and so for all and . Therefore, . If , thenSince is multiquartic function on , for all . It follows from the last equality and (55) thatfor all . If , thenTherefore, fulfills (54) for all . On the contrary, suppose that there is a number and a multiquartic function such that for all . By Proposition 2, there exists such that , and thus,for all . Here, we choose such that . If belongs to such that for all , then for all . Thus, we getThe relation above leads us to a contradiction with (59).

4. Conclusion

In this paper, we have introduced the multiquartic mappings and have investigated the structure of such mappings. In other words, we have shown that the system of functional equations defining a multiquartic mapping can be unified as an equation. We have applied some fixed point theorems to establish the generalized Hyers–Ulam and Găvruţa stability of multiquartic functional equations in Banach and quasi- normed spaces. Finally, by using a characterization result, we have presented a nonstable example to invalidate the results in the special case [21].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The author wrote the first draft, read and approved the final manuscript.

Acknowledgments

The author is highly grateful to the referees for their valuable comments and suggestions enriching the contents of the paper.