Characterization and Stability of Multimixed Additive-Quartic Mappings: A Fixed Point Application

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(i) f is multiadditive [1] if and only if it satisfies (ii) f is multiquadratic [8] if and only if it satisfies where x j = ðx 1j , x 2j , ⋯, x nj Þ ∈ V n with j ∈ f1, 2g. More information about the structure of multiadditive and multiquadratic mappings, we refer for instance to [9,10].
Lee et al. [11] introduced and obtained the general solution of the quartic functional equation which somewhat different from (1) as follows: For the generalized forms of the quartic functional, equations (1) and (4) refer to [12,13]. Recently, in [14] and motivated by (4), a new form of multiquartic mappings was introduced, and the structure of such mappings was described.
Speaking of the stability of a functional equation, we follow the question raised in 1940 by Ulam [15] for group homomorphisms. Hyers [16] presented a partial solution to the problem of Ulam. Later, Hyers' theorem was extended and generalized in various forms by many mathematicians such as Aoki [17] and Rassias [18]. Recall that a functional equation F is said to be stable if any mapping ϕ fulfilling F approximately; then, it is near to an exact solution of F. Next, several stability problems of various functional equations and mappings have been investigated by many mathematicians which can be found in literatures.
In [27], Eshaghi Gordji introduced and obtained the general solution of the following mixed type additive and quartic functional equation He also established the Hyers-Ulam Rassias stability of the above functional equation in real normed spaces. The stability of (5) in non-Archimedean orthogonality spaces is studied in [28]. A different and equivalent form of mixed type additive and quartic functional equation from (5) was introduced by the first author in [29] as follows: It is easily verified that the function f ðxÞ = αx 4 + βx is a solution of equations (5) and (6); the generalized version of equation (6) can be found in [30].
This paper is organized as follows: In the second section, we firstly define multi-additive-quartic mappings and include a characterization of such mappings. In fact, we prove that every multi-additive-quartic mapping can be shown a single functional equation and vice versa (under some extra conditions). Section 3 is devoted to the study of stricture of multimixed additive-quartic mappings. In other words, motivated by equation (6), we introduce the multimixed additive-quartic mappings and reduce the system of n equations defining the multimixed additive-quartic mappings to a single equation, namely, the multimixed additive-quartic functional equation. In Section 4, we prove the Hyers-Ulam stability for the multi-additive-quartic and the multimixed additive-quartic mappings in the setting of Banach spaces by applying a fixed point method [31]. As an application of this result, we establish the stability of multi-additive-quartic mappings. Finally, we show that under some mild conditions every multiadditive and multiquartic functional equations are δ-stable for a small positive number δ.

Characterization of Multi-Additive-Quartic Mappings
Throughout this paper, ℕ and ℚ stand for the set of all positive integers and the rational numbers, respectively, ℕ 0 ≔ ℕ ∪ f0g, ℝ + ≔ ½0,∞Þ. For any l ∈ ℕ 0 , m ∈ ℕ, t = ðt 1 , ⋯, t m Þ ∈ f−1, 1g m , and x = ðx 1 , ⋯, x m Þ ∈ V m , we write lx ≔ ðlx 1 , ⋯, lx m Þ and tx ≔ ðt 1 x 1 , ⋯, t m x m Þ, where ra stands, as usual, for the rth power of an element a of the commutative group V. Let V and W be linear spaces, n ∈ ℕ and k ∈ f0, ⋯, ng. A mapping f : V n ⟶ W is called k-additive and n − k-quartic (briefly, multi-additive-quartic) if f is additive in each of some k variables and satisfies (4) in each of the other variables. In what follows, for simplicity, it is assume that f is additive in each of the first k variables. Moreover, for k = n (k = 0), the above definition leads to the so-called multiadditive (multiquartic) mappings.
In the sequel, we assume that V and W are vector spaces over ℚ. Moreover, we identify To achieve our aims, for the multi-additive-quartic mappings, we use the oncoming notations: For each x 1 , x 2 ∈ V n , we consider the equation for all It is shown in Proposition 2.2 in [14] that if a mapping f : V n ⟶ W is multiquartic, then it satisfies the equation The next proposition shows that the system of n equations defining a multi-additive-quartic mapping can be reduced to (10).
Proof. For k ∈ f0, ng, the result follows from Proposition 2.2 in [14] and Theorem 2 in [1], and so we prove the assertion for the case that k ∈ f1, ⋯, n − 1g. For any x n−k ∈ V n−k , consider the mapping g x n−k : The assumption shows that g x n−k is k-additive, and thus, we can obtain from Theorem 2 in [1] that The above equality implies that for all x k 1 , x k 2 ∈ V k and x n−k ∈ V n−k . Repeat the above method, and for any This mapping is n − k-quartic, and hence, by Proposition 2.2 from [14], we have for all On the other hand, by the definition of h x k , relation (14) converts to for all x n−k 1 , x n−k 2 ∈ V n−k and x k ∈ V k . It now follows between (13) and (15) that for all This finishes the proof.
By Proposition 6, it is easily verified that the mapping f ðz 1 , ⋯, z n Þ = c Q k i=1 z i Q n j=k+1 z 4 j satisfies (10), and so this equation is said to be multi-additive-quartic functional equation.
Definition 2. Let r ∈ ℕ. Consider a mapping f : V n ⟶ W. We say f 3 Journal of Function Spaces (i) Satisfies (has) the r -power condition in the jth variable if for all z 1 , ⋯, z n ∈ V n . Sometimes 4-power condition is called quartic condition.
(ii) Has zero condition if f ðxÞ = 0 for any x ∈ V n with at least one component which is equal to zero We remember that the binomial coefficient for all n, r ∈ ℕ 0 with n ≥ r is defined and denoted by n We wish to show that if a mapping satisfies equation (10), then it is multi-additive-quartic. For doing it, we need the upcoming lemma. The method of the proof of Lemma 3 is similar to the proof of ( [14], Lemma 2.5) and so we include lemma without the proof.

Lemma 3.
Suppose that a mapping f : V n ⟶ W satisfies equation (10). Under one of the following assumptions, f satisfying zero condition.
(i) f satisfies the quartic condition in the last n − k variables (ii) f is even in the last n − k variables Theorem 4. Suppose that a mapping f : V n ⟶ W fulfilling equation (10). Under one of the hypothesis of Lemma 3, f is multi-additive-quartic.
Proof. It follows from Lemma 3; f satisfies zero condition. Putting x n−k 2 = ð0, ⋯, 0Þ in the left side of (10) and applying the hypothesis, we obtain On the other hand, by using Lemma 3, the right side of (10) converts to Now, relations (18) and (19) necessitate that for all x k 1 , x k 2 ∈ V n and x n−k 1 ∈ V n−k . In light of Theorem 2 in [1], we see that f is additive in each of the k first variables. In addition, by considering x k 2 = ð0,⋯,0Þ in (10) and applying again Lemma 3, we have for all x k 1 ∈ V k and x n−k 1 , x n−k 2 ∈ V n−k , and thus, by Theorem 2.6 in [14], f is quartic in each of the last n − k variables. The proof of second part is similar.

Characterization of Multimixed Additive-Quartic Mappings
In this section, we introduce the multimixed additive-quartic mappings and then characterize them as an equation. We start this section with the definition of such mappings.
Definition 5. Let V and W be vector spaces over ℚ, n ∈ ℕ. A mapping f : V n ⟶ W is called n -multimixed additivequartic or briefly multimixed additive-quartic if f satisfies mixed additive-quartic equation (6) in each variable.
Let n ∈ ℕ with n ≥ 2 and x n i = ðx i1 , x i2 , ⋯, x in Þ ∈ V n , where i ∈ f1, 2g. For x 1 , x 2 ∈ V n and q ∈ ℕ 0 with 0 ≤ q ≤ n, put Consider the subset M n q of M as follows: Hereafter, for the multimixed additive-quartic mappings, we use the following notations: Next, we reduce the system of n equations defining the multimixed additive-quartic mapping to obtain a single functional equation. 4 Journal of Function Spaces Proposition 6. If a mapping f : V n ⟶ W is multimixed additive-quartic, it satisfies the equation where f ðM n q Þ and f ðN n ðp 1 ,p 2 Þ Þ are defined in (24) and (8), respectively.
Proof. The proof is based on induction for n. For n = 1, it is obvious that f satisfies (6). Assume that (26) holds for some positive integer n > 1. Then The assertion is now proved.
Since the mapping f ðz 1 , ⋯, z n Þ = Q n j=1 ða j z 4 j + b j z j Þ is multimixed additive-quartic, it satisfies (26) by proposition above, and so this equation is called multimixed additivequartic functional equation.
Here, we bring an elementary lemma without proof.

Lemma 7.
Let n, k, p l ∈ ℕ 0 , such that k + ∑ m l=1 p l ≤ n, where l ∈ f1, ⋯, mg. Then Similar to Lemma 2.1 from [6], we need the following lemma in obtaining our goal in this section. The proof is similar, but we include some parts for the sake of completeness.

Lemma 8.
If a mapping f : V n ⟶ W satisfies equation (26), then it has zero condition.
Proof. Putting x 1 = x 2 = ð0,⋯,0Þ in (26), we have Using Lemma 7 for k = 0 and p 1 , p 2 , the right side of (29) will be as follows: On the other hand, by a simple computation, the left side of (29) is 2 7 It follows from relations (29), (30), and (31) that f ð0, ⋯, 0Þ = 0. One can continue this method to show that f has zero condition.

Journal of Function Spaces
(iv) Even in the jth variable if Proposition 10. Suppose that a mapping f : V n ⟶ W satisfies equation (26). Then, it is multimixed additive-quartic. Moreover, (i) If f is odd in a variable, then it is additive in the same variable (ii) If f is even in a variable, then it is quartic in the same variable Proof. Let j ∈ f1, ⋯, ng be arbitrary and fixed. Set Putting x 2k = 0 for all k ∈ f1,⋯,ng \ fjg in (26) and using Lemma 8, we get The above equalities show that In other words, (6) is true for f * j . Since j is arbitrary, f is a multimixed additive-quartic mapping.
(i) Repeating the proof of Lemma 2.1 (i) from [29] for f * j , we see that f * j ðz + wÞ = f * j ðzÞ + f * j ðwÞ. This means that f is additive in the jth variable (ii) Similar to the previous part, it follows from the proof of part (ii) of Lemma 2.1 in [29] that Therefore, f is quartic in the jth variable.
(i) If f is odd in each variable, then it is multiadditive. Moreover, it satisfies (2) (ii) If f is even in each variable, then it is multiquartic. In particular, it fulfills (11) (iii) If f is odd in each of some k variables and is even in each of the other variables, then it is multi-additivequartic. In addition, (10) is valid for f

Various Stability Results
In this section, we prove some Hyers-Ulam stability results by a fixed point method in the setting of Banach spaces. In what follows, we denote the set of all mappings from E to F by F E . We remember the following theorem which is an essential result in fixed point theory ( [23], Theorem 1). This achievement is a key tool in obtaining our aim in this section.
Theorem 12. Let the hypotheses (A1) Y is a Banach space, E is a nonempty set, j ∈ ℕ, g 1 , ⋯, g j : E ⟶ E, and L 1 , ⋯, L j : E ⟶ ℝ + (A2) T : Y E ⟶ Y E is an operator satisfying the inequality Journal of Function Spaces (A3) Λ : ℝ E + ⟶ ℝ E + is an operator defined through hold, and a function θ : E ⟶ ℝ + and a mapping ϕ : E ⟶ Y fulfill the following two conditions: Then, there exists a unique fixed point ψ of T such that Moreover, ψðxÞ = lim l⟶∞ T l ϕðxÞ for all x ∈ E.
For the rest of this paper and for each mapping f : V n ⟶ W, we consider the difference operator Γ AQ f : V n × V n ⟶ W defined via where f ðM n q Þ and f ðN n ðp 1 ,p 2 Þ Þ are defined in (24) and (8), respectively. In the sequel, all mappings f : V n ⟶ W are assumed that satisfy zero condition. With this assumption, we have the next stability result for functional equation (26) in the odd case.