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

Authors: Abasalt Bodaghi; Idham Arif Alias; Lida Mousavi; Sedigheh Hosseini
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we introduce the multi-additive-quartic and the multimixed additive-quartic mappings. We also describe and characterize the structure of such mappings. In other words, we unify the system of functional equations defining a multi-additive-quartic or a multimixed additive-quartic mapping to a single equation. We also show that under what conditions, a multimixed additive-quartic mapping can be multiadditive, multiquartic, and multi-additive-quartic. Moreover, by using a fixed point technique, we prove the Hyers-Ulam stability of multimixed additive-quartic functional equations thus generalizing some known results.


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## Body

## 1. Introduction

LetV be a commutative group, W be a linear space over rational numbers, and n be an integer with n≥2. A mapping f:Vn⟶W is called
(i)
Multiadditive if it satisfies the Cauchy’s functional equation Ax+y=Ax+Ay in each variable [1](ii)
Multiquadratic if it fulfills quadratic functional equation Qx+y+Qx−y=2Qx+2Qy in each variable [2, 3](iii)
Multicubic if it satisfies the cubic equation C2x+y+C2x−y=2Cx+y+2Cx−y+12Cx in each variable [4, 5](iv)
Multiquartic if it satisfies the quartic equation(1)Qx+2y+Qx−2y=4Qx+y+4Qx−y−6Qx+24Qy,in each variable [6, 7].We have the following observations about a several variables mappingf:Vn⟶W.
(i)
f is multiadditive [1] if and only if it satisfies(2)fx1+x2=∑j1,⋯,jn∈1,2fx1j1,⋯,xnjn.(ii)
f is multiquadratic [8] if and only if it satisfies(3)∑s∈−1,1nfx1+sx2=2n∑j1,⋯,jn∈1,2fx1j1,⋯,xnjn,wherexj=x1j,x2j,⋯,xnj∈Vn with j∈1,2. More information about the structure of multiadditive and multiquadratic mappings, we refer for instance to [9, 10].Bodaghi et al. [4] (resp., [6]) provided a characterization of multicubic (resp., multiquartic) mappings, and they showed that every multicubic (resp., multiquartic) mapping can be shown a single functional equation and vice versa.Lee et al. [11] introduced and obtained the general solution of the quartic functional equation which somewhat different from (1) as follows:
(4)Q2x+y+Q2x−y=4Qx+y+4Qx−y+24Qx−6Qy.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 the last two decades, the stability problem for several variable mappings such as multiadditive, multi-Jensen, multiquadratic, multicubic, and multiquartic mappings by applying direct and fixed point methods has been studied by a number of authors which are available for example in [1, 2, 4, 8, 9, 19–26].In [27], Eshaghi Gordji introduced and obtained the general solution of the following mixed type additive and quartic functional equation
(5)f2x+y+f2x−y=4fx+y+fx−y−37f2y−2fy+2f2x−8fx.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:
(6)fx+2y−4fx+y−4fx−y+fx−2y=127f2y−2fy−6fx.It is easily verified that the functionfx=αx4+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.


## 2. 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, ℕ0≔ℕ∪0,ℝ+≔0,∞. For any l∈ℕ0,m∈ℕ, t=t1,⋯,tm∈−1,1m, and x=x1,⋯,xm∈Vm, we write lx≔lx1,⋯,lxm and tx≔t1x1,⋯,tmxm, where ra stands, as usual, for the rth power of an element a of the commutative group V.LetV and W be linear spaces, n∈ℕ and k∈0,⋯,n. A mapping f:Vn⟶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 thatV and W are vector spaces over ℚ. Moreover, we identify x=x1,⋯,xn∈Vn with xk,xn−k∈Vk×Vn−k, where xk≔x1,⋯,xk and xn−k≔xk+1,⋯,xn. Let n∈ℕ with n≥2 and xin=xi1,xi2,⋯,xin∈Vn, where i∈1,2. Throughout, we shall denote xin by xi if there is no risk of mistake. Put also xik=xi1,⋯,xik∈Vk and xin−k=xi,k+1⋯,xin∈Vn−k. For x1,x2∈Vn and pi∈ℕ0 with 0≤pi≤n and 0≤k≤n−1, set Nn−k=Nn−k=Nk+1,⋯,Nn∣Nj∈x1j±x2j,x1j,x2j, where j∈k+1,⋯,n. Consider the subset Np1,p2n−k of N as follows:
(7)Np1,p2n−k≔Nn−k∈Nn−k∣CardNj:Nj=xij=pii∈1,2.To achieve our aims, for the multi-additive-quartic mappings, we use the oncoming notations:(8)fNp1,p2n−k≔∑Nn−k∈Np1,p2n−kfNn,(9)fz,Np1,p2n−k≔∑Nn−k∈Np1,p2n−kfz,Nn−kz∈V.For eachx1,x2∈Vn, we consider the equation
(10)∑t∈−1,1n−kfx1k+x2k,2x1n−k+tx2n−k=∑p1=0n−k∑p2=0n−k−p1∑i∈1,24n−k−p1−p224p1−6p2fxik,Np1,p2n−k,for allxik=xi1,⋯,xik∈Vk and xin−k=xi,k+1⋯,xin∈Vn−k where i∈1,2.It is shown in Proposition 2.2 in [14] that if a mapping f:Vn⟶W is multiquartic, then it satisfies the equation
(11)∑t∈−1,1nf2x1+tx2=∑p1=0n∑p2=0n−p14n−p1−p224p1−6p2fNp1,p2n−k.The next proposition shows that the system ofn equations defining a multi-additive-quartic mapping can be reduced to (10).Proposition 1.
Letn∈ℕ and k∈0,⋯,n. Suppose that a mapping f:Vn⟶W is k-additive and n−k-quartic (multi-additive-quartic) mapping. Then, f fulfills equation (10).Proof.
Fork∈0,n, 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∈1,⋯,n−1. For any xn−k∈Vn−k, consider the mapping gxn−k:Vk⟶W defined by gxn−kxk≔fxk,xn−k for xk∈Vk. The assumption shows that gxn−k is k-additive, and thus, we can obtain from Theorem 2 in [1] that
(12)gxn−kx1k+x2k=∑j1,j2,⋯,jk∈1,2gxn−kxj11,xj22,⋯,xjkk,x1k,x2k∈Vk.
The above equality implies that(13)fx1k+x2k,xn−k=∑j1,⋯,jk∈1,2fxj11,⋯,xjkk,xn−k,
for allx1k,x2k∈Vk and xn−k∈Vn−k. Repeat the above method, and for any xk∈Vk, define the mapping hxk:Vn−k⟶W via hxkxn−k≔fxk,xn−k,xn−k∈Vn−k. This mapping is n−k-quartic, and hence, by Proposition 2.2 from [14], we have
(14)∑t∈−1,1n−khxk2x1n−k+tx2n−k=∑p1=0n−k∑p2=0n−k−p14n−k−p1−p224p1−6p2hxkNp1,p2n−k,
for allx1n−k,x2n−k∈Vn−k. On the other hand, by the definition of hxk, relation (14) converts to
(15)∑t∈−1,1n−kfxk,2x1n−k+tx2n−k=∑p1=0n−k∑p2=0n−k−p14n−k−p1−p224p1−6p2fxk,Np1,p2n−k,
for allx1n−k,x2n−k∈Vn−k and xk∈Vk. It now follows between (13) and (15) that
(16)∑t∈−1,1n−kfx1k+x2k,2x1n−k+tx2n−k=∑p1=0n−k∑p2=0n−k−p14n−k−p1−p224p1−6p2fx1k+x2k,Np1,p2n−k=∑p1=0n−k∑p2=0n−k−p1∑j1,j2,⋯,jn∈1,24n−k−p1−p224p1−6p2fxj11,xj22,⋯,xjnn,Np1,p2n−k=∑p1=0n−k∑p2=0n−k−p1∑i∈1,24n−k−p1−p224p1−6p2fxik,Np1,p2n−k,
for allxik=xi1,⋯,xik∈Vk and xin−k=xi,k+1⋯,xin∈Vn−k. This finishes the proof.By Proposition6, it is easily verified that the mapping fz1,⋯,zn=c∏i=1kzi∏j=k+1nzj4 satisfies (10), and so this equation is said to be multi-additive-quartic functional equation.Definition 2.
Letr∈ℕ. Consider a mapping f:Vn⟶W. We say f(i)
Satisfies (has) ther-power condition in the jth variable if(17)fz1,⋯,zj−1,2zj,zj+1,⋯,zn=2rfz1,⋯,zj−1,zj,zj+1,⋯,zn,for all z1,⋯,zn∈Vn. Sometimes 4-power condition is called quartic condition.
(ii)
Haszero condition if fx=0 for any x∈Vn with at least one component which is equal to zeroWe remember that the binomial coefficient for alln,r∈ℕ0 with n≥r is defined and denoted by nr≔n!/r!n−r!.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 mappingf:Vn⟶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 variablesTheorem 4.
Suppose that a mappingf:Vn⟶W fulfilling equation (10). Under one of the hypothesis of Lemma 3, f is multi-additive-quartic.Proof.
It follows from Lemma3; f satisfies zero condition. Putting x2n−k=0,⋯,0 in the left side of (10) and applying the hypothesis, we obtain
(18)2n−k×24n−kfx1k+x2k,x1n−k=25n−kfx1k+x2k,2x1n−k.
On the other hand, by using Lemma3, the right side of (10) converts to
(19)∑p1=0n−kn−kp14n−k−p124p12n−k−p1fxj11,xj22,⋯,xjkk,x1n−k=∑p1=0n−kn−kp18n−k−p124p1fxj11,xj22,⋯,xjkk,x1n−k=25n−k∑j1,j2,⋯,jk∈1,2fxj11,xj22,⋯,xjkk,x1n−k.
Now, relations (18) and (19) necessitate that
(20)fx1k+x2k,x1n−k=∑j1,j2,⋯,jn∈1,2fxj11,xj22,⋯,xjnn,x1n−k,
for allx1k,x2k∈Vn and x1n−k∈Vn−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 x2k=0,⋯,0 in (10) and applying again Lemma 3, we have
(21)∑t∈−1,1n−kfx1k,2x1n−k+tx2n−k=∑p1=0n−k∑p2=0n−k−p14n−k−p1−p224p1−6p2fx1k,Np1,p2n−k,
for allx1k∈Vk and x1n−k,x2n−k∈Vn−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.


## 3. 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.
LetV and W be vector spaces over ℚ, n∈ℕ. A mapping f:Vn⟶W is called n-multimixed additive-quartic or briefly multimixed additive-quartic if f satisfies mixed additive-quartic equation (6) in each variable.Letn∈ℕ with n≥2 and xin=xi1,xi2,⋯,xin∈Vn, where i∈1,2. For x1,x2∈Vn and q∈ℕ0 with 0≤q≤n, put
(22)M=Mn=M1,⋯,Mn∣Mj∈x1j±2x2j,2x2j,j∈1,⋯,n.Consider the subsetMqn of M as follows:
(23)Mqn≔Mn∈M∣CardMj:Mj=2x2j=q.Hereafter, for the multimixed additive-quartic mappings, we use the following notations:(24)fMqn≔∑Mn∈MqnfMn,(25)fMqn,z≔∑Mn∈MqnfMn,zz∈V.Next, we reduce the system ofn equations defining the multimixed additive-quartic mapping to obtain a single functional equation.Proposition 6.
If a mappingf:Vn⟶W is multimixed additive-quartic, it satisfies the equation
(26)∑q=0n−127qfMqn=∑p1=0n∑p2=0n−p14n−p1−p2−6p1−247p2fNp1,p2n,
wherefMqn and fNp1,p2n are defined in (24) and (8), respectively.Proof.
The proof is based on induction forn. For n=1, it is obvious that f satisfies (6). Assume that (26) holds for some positive integer n>1. Then
(27)∑q=0n+1−127q+1fMqn+1=∑q=0n∑t∈1,−1−127qfMqn,x1n+1+2tx2,n+1−127∑q=0n−127qfMqn,2x2,n+1=∑p1=0n∑p2=0n−p1∑t∈1,−14n−p1−p2−6p1−247p2fNp1,p2n,x1,n+1+2tx2,n+1−127∑p1=0n∑p2=0n−p14n−p1−p2−6p1−247p2fNp1,p2n,2x2,n+1=4∑p1=0n∑p2=0n−p1∑s∈1,−14n−p1−p2−6p1−247p2fNp1,p2n,x1,n+1+sx1,n+1−6∑p1=0n∑p2=0n−p14n−p1−p2−6p1−247p2fNp1,p2n,x1,n+1−247∑p1=0n∑p2=0n−p14n−p1−p2−6p1−247p2fNp1,p2n,x2,n+1=∑p1=0n+1∑p2=0n+1−p14n+1−p1−p2−6p1−247p2fNp1,p2n+1.
The assertion is now proved.Since the mappingfz1,⋯,zn=∏j=1najzj4+bjzj is multimixed additive-quartic, it satisfies (26) by proposition above, and so this equation is called multimixed additive-quartic functional equation.Here, we bring an elementary lemma without proof.Lemma 7.
Letn,k,pl∈ℕ0, such that k+∑l=1mpl≤n, where l∈1,⋯,m. Then
(28)n−kn−k−∑l=1mpl∑l=1mpl∑l=1m−1pl⋯p1+p2p1=n−kp1n−k−p1p2⋯n−k−∑l=1m−1plpm.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 mappingf:Vn⟶W satisfies equation (26), then it has zero condition.Proof.
Puttingx1=x2=0,⋯,0 in (26), we have
(29)∑q=0nnn−q−127q2n−qf0,⋯,0=∑p2=0n∑p1=0n−p2nn−p1−p2p1+p2p12n−p1−p24n−p1−p2−6p1−247p2f0,⋯,0.
Using Lemma7 for k=0 and p1,p2, the right side of (29) will be as follows:
(30)∑p2=0n∑p1=0n−p2nn−p1−p2p1+p2p12n−p1−p24n−p1−p2−6p1−247p2f0,⋯,0=2n∑p2=0nnp2−127p2∑p1=0n−p2n−p2p14n−p1−p2−3p1f0,⋯,0=2n∑p2=0nnp2−127p24−3n−p2f0,⋯,0=2n−127+1nf0,⋯,0=−107nf0,⋯,0.
On the other hand, by a simple computation, the left side of (29) is
(31)27nf0,⋯,0.
It follows from relations (29), (30), and (31) that f0,⋯,0=0. One can continue this method to show that f has zero condition.Definition 9.
A mappingf:Vn⟶W is
(iii)
Odd in thejth variable if(32)fz1,⋯,zj−1,−zj,zj+1,⋯,zn=−fz1,⋯,zj−1,zj,zj+1,⋯,zn.(iv)
Even in thejth variable if(33)fz1,⋯,zj−1,−zj,zj+1,⋯,zn=fz1,⋯,zj−1,zj,zj+1,⋯,zn.Proposition 10.
Suppose that a mappingf:Vn⟶W satisfies equation (26). Then, it is multimixed additive-quartic. Moreover,
(i)
Iff is odd in a variable, then it is additive in the same variable(ii)
Iff is even in a variable, then it is quartic in the same variableProof.
Letj∈1,⋯,n be arbitrary and fixed. Set
(34)fj∗z≔fz1,⋯,zj−1,z,zj+1,⋯,zn.
Puttingx2k=0 for all k∈1,⋯,n\j in (26) and using Lemma 8, we get
(35)2n−1fj∗z+2w+fj∗z−2w−127fj∗2w=∑p1=0n−1n−1p14n−p1−6p12n−p1−1fj∗z+w+fj∗z−w+∑p1=1nn−1p1−14n−p1−6p12n−p1fj∗z+∑p1=0nn−1p14n−p1−1−6p1−2472n−p1−1fj∗w=42n−1∑p1=0n−1n−1p14n−1−p1−3p1fj∗z+w+fj∗z−w−62n−1∑p1=0n−1n−1p14n−1−p1−3p1fj∗z−2472n−1∑p1=0n−1n−1p14n−1−p1−3p1fj∗w=4×2n−1fj∗z+w+fj∗z−w−6×2n−1fj∗z−247×2n−1fj∗w.
The above equalities show that(36)fj∗z+2w+fj∗z−2w−127fj∗2w=4fj∗z+w+fj∗z−w−6fj∗z−247fj∗w.
In other words, (6) is true for fj∗. Since j is arbitrary, f is a multimixed additive-quartic mapping.
(i)
Repeating the proof of Lemma 2.1 (i) from [29] for fj∗, we see that fj∗z+w=fj∗z+fj∗w. 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(37)fj∗2z+w+fj∗2z−w=4fj∗z+w+fj∗z−w+24fj∗z−6fj∗w.
Therefore,f is quartic in the jth variable.Corollary 11.
Suppose a mappingf:Vn⟶W satisfies equation (26).
(i)
Iff is odd in each variable, then it is multiadditive. Moreover, it satisfies (2)(ii)
Iff is even in each variable, then it is multiquartic. In particular, it fulfills (11)(iii)
Iff is odd in each of some k variables and is even in each of the other variables, then it is multi-additive-quartic. In addition, (10) is valid for f


## 4. 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 fromE to F by FE. 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, g1,⋯,gj:E⟶E, and L1,⋯,Lj:E⟶ℝ+
(A2)T:YE⟶YE is an operator satisfying the inequality
(38)Tλx−Tμx≤∑i=1jLixλgix−μgix,λ,μ∈YE,x∈E,
(A3)Λ:ℝ+E⟶ℝ+E is an operator defined through
(39)Λδx≔∑i=1jLixδgixδ∈ℝ+E,x∈E,hold, and a function θ:E⟶ℝ+ and a mapping ϕ:E⟶Y fulfill the following two conditions:
(40)Tϕx−ϕx≤θx,θ∗x≔∑l=0∞Λlθx<∞x∈E.
Then, there exists a unique fixed pointψ of T such that
(41)ϕx−ψx≤θ∗xx∈E.
Moreover,ψx=liml⟶∞Tlϕx for all x∈E.For the rest of this paper and for each mappingf:Vn⟶W, we consider the difference operator ΓAQf:Vn×Vn⟶W defined via
(42)ΓAQfx1,x2≔∑q=0n−127qfMqn−∑p1=0n∑p2=0n−p14n−p1−p2−6p1−247p2fNp1,p2n,wherefMqn and fNp1,p2n are defined in (24) and (8), respectively. In the sequel, all mappings f:Vn⟶W are assumed that satisfy zero condition. With this assumption, we have the next stability result for functional equation (26) in the odd case.Theorem 13.
Letβ∈−1,1 be fixed, V be a linear space, and W be a Banach space. Suppose that ϕ:Vn×Vn⟶ℝ+ is a mapping satisfying
(43)liml⟶∞12nβlϕ2βlx1,2βlx2=0,
for allx1,x2∈Vn and
(44)Φx≕712n12nβ+1/2∑l=0∞12nβlϕ0,2βl+β−1/2x<∞,
for allx∈Vn. Assume also f:Vn⟶W is a mapping fulfilling the inequality
(45)ΓAQfx1,x2≤ϕx1,x2,
for allx1,x2∈Vn. If f is odd in each variable, then there exists a unique multiadditive mapping A:Vn⟶W such that
(46)fx−Ax≤Φx,
for allx∈Vn.Proof.
Replacingx1,x2 by 0,x1 in (45) and using the assumptions, we have
(47)−127nf2x−−247nfx≤ϕ0,x,
for allx=x1∈Vn (here and the rest of the proof) and so
(48)fx−12nf2x≤724nϕ0,x.
Set(49)θx≔712n12nβ+1/2ϕ0,2β−1/2x,andTθx≔12nβθ2βxθ∈WVn.
Then, relation (48) can be modified as
(50)fx−Tfx≤θxx∈Vn.
DefineΛηx≔1/2nβη2βx for all η∈ℝ+Vn. It is seen that Λ has the form (A3) of Theorem 12 for which E=Vn, g1x=2βx, and L1x=1/2nβ. Furthermore, for each λ,μ∈WVn, we get
(51)Tλx−Tμx=12nβλ2βx−μ2βx≤L1xλg1x−μg1x.
The above relation portrays that the hypothesis (A2) holds. By induction onl, one can check that for any l∈ℕ0, we have
(52)Λlθx≔12nβlθ2βlx=712n12nβ+1/2lϕ0,2βl+β−1/2x.
Now, relations (44) and (52) necessitate that all assumptions of Theorem 12 are satisfied. Hence, there exists a unique mapping A:Vn⟶W such that
(53)Ax=liml⟶∞Tlfx=12nβA2βxx∈Vn,and (46) holds. In continuation, we prove that
(54)ΓAQTlfx1,x2≤12nβlϕ2βlx1,2βlx2,
for allx1,x2∈Vn and l∈ℕ0. We argue by induction on l. Clearly, inequality (54) is valid for l=0 by (45). Assume that (54) is true for an l∈ℕ0. Then
(55)∥ΓAQTl+1fx1,x2∥=∑q=0n−127qTl+1fMqn−∑p1=0n∑p2=0n−p14n−p1−p2−6p1−247p2Tl+1fNp1,p2n=12nβ∑q=0n−127qTl+1f2βMqn−∑p1=0n∑p2=0n−p14n−p1−p2−6p1−247p2Tl+1f2βNp1,p2n=12nβΓAQTlf2βx1,2βx2≤12nβl+1ϕ2βl+1x1,2βl+1x2,
for allx1,x2∈Vn. Letting l⟶∞ in (54) and applying (43), we arrive at ΓAQAx1,x2=0 for all x1,x2∈Vn. This means that the mapping A satisfies (26), and so it is multiadditive by Corollary 11. This finishes the proof.Here, in analogy with Theorem13, we bring the next stability result for functional equation (26) in the even case.Theorem 14.
Letβ∈−1,1 be fixed, V be a linear space, and W be a Banach space. Suppose that ϕ:Vn×Vn⟶ℝ+ is a mapping satisfying
(56)liml⟶∞124nβlϕ2βlx1,2βlx2=0,for all x1,x2∈Vn and
(57)Ψx≕72n124nβ+1/2∑l=0∞124nβlϕ2βl+β+1/2x,0<∞,for all x∈Vn. Assume also f:Vn⟶W is a mapping fulfilling the inequality
(58)ΓAQfx1,x2≤ϕx1,x2,for all x1,x2∈Vn. If f is even in each variable, then there exists a unique solution Q:Vn⟶W of (26) such that
(59)∥fx−Qx∥≤Ψx,for all x∈Vn. In particular, if Q is even mapping in each variable, then it is multiquartic.Proof.
Replacingx1,x2 by 0,x1 in (58) and applying the hypotheses, we obtain
(60)∑q=0nnq−127q2n−qf2x−∑p2=0nnp2−247p24n−p2×2n−p2fx≤ϕ0,x,where x=x1∈Vn (here and the rest of the proof). On the other hand
(61)∑q=0nnq−127q2n−q=−127+2n=27n,∑p2=0nnp2−247p24n−p2×2n−p2=−247+8n=327n.
By the relations above (60) will be
(62)27nf2x−327nfx≤ϕ0,x,and so
(63)fx−124nf2x≤732nϕ0,x.
One can rewrite (63) as
(64)fx−Tfx≤θxx∈Vn,where
(65)θx≔72n124nβ+1/2ϕ0,2β−1/2x,Tθx≔124nβθ2βxθ∈WVn.
Similar to the proof of Theorem13, consider Ληx≔1/24nβη2βx for all η∈ℝ+Vn, and hence, Λ satisfies (A3) of Theorem 12 with E=Vn, g1x=2βx, and L1x=1/24nβ. Moreover, for each λ,μ∈WVn, we obtain
(66)Tλx−Tμx=124nβλ2βx−μ2βx≤L1xλg1x−μg1x.
The last relation implies that the hypothesis (A2) is true. It is easily checked by induction onl that for any l∈ℕ0 and x∈Vn(67)Λlθx≔124nβlθ2βlx=732n124nβ+1/2lψ0,2βl+β−1/2x.
It now follows between (57) and (67) that all assumptions of Theorem 12 hold, and thus, there exists a unique mapping Q:Vn⟶W such that
(68)Qx=liml⟶∞Tlfx=124nβQ2βxx∈Vn,and (46) is valid. Similar to the proof of Theorem 13, one can show that
(69)ΓAQTlfx1,x2≤124nβlψ2βlx1,2βlx2,for all x1,x2∈Vn and l∈ℕ0. Letting l⟶∞ in (69) and applying (56), we arrive at ΓAQQx1,x2=0 for all x1,x2∈Vn, and therefore, the mapping Q satisfies (26). The last part follows from part (ii) of Corollary 11.Here and subsequently, it is assumed thatV is a normed space and W is a Banach space unless otherwise stated explicitly. In the following corollary, we show that the multiadditive and multiquartic mappings are stable. Since the proof is routine, we include it without proof.Corollary 15.
Givenα∈ℝ. Suppose that f:Vn⟶W is a mapping satisfying the inequality
(70)∥ΓAQfx1,x2∥≤∑i=12∑j=1n∥xij∥α,for all x1,x2∈Vn.
(i)
Ifα≠n and f is odd in each variable, then there exists a unique multiadditive mapping A:Vn⟶W such that(71)∥fx−Ax∥≤712n12α−2n∑j=1n∥x1j∥α.(ii)
Ifα≠4n and f is even in each variable, then there exists a unique solution Q:Vn⟶W of (26) such that(72)∥fx−Qx∥≤72n12α−24n∑j=1n∥x1j∥α,for all x=x1∈Vn. Moreover, if Q is even mapping in each variable, then it is multiquartic.The upcoming corollaries are direct consequences of Theorems13 and 14 when the functional equation (26) is controlled by a small positive number δ.Corollary 16.
Letδ>0 and f:Vn⟶W be a mapping satisfying the inequality
(73)∥ΓAQfx1,x2∥≤δ,for all x1,x2∈Vn.
(i)
Iff is odd in each variable, then there exists a unique multiadditive mapping A:Vn⟶W such that(74)∥fx−Ax∥≤712nδ2n−1,for all x∈Vn(ii)
Iff is even in each variable, then there exists a unique solution Q:Vn⟶W of (26) such that(75)∥fx−Qx∥≤72nδ24n−1,for all x∈Vn. Furthemore, if Q is even mapping in each variable, then it is multiquartic.Proof.
Letting the constant functionϕx1,x2=δ for all x1,x2∈Vn and using Theorem 13 and Theorem 14 in the case β=1, one can obtain the desired result.Given the mappingf:Vn⟶W, we define the operator Γf:Vn×Vn⟶W through
(76)Γfx1,x2≔∑t∈−1,1n−kfx1k+x2k,2x1n−k+tx2n−k−∑p1=0n−k∑p2=0n−k−p1∑i∈1,24n−k−p1−p224p1−6p2fxik,Np1,p2n−k,for all xik=xi1,⋯,xik∈Vk and xin−k=xi,k+1⋯,xin∈Vn−k where i∈1,2 and fNp1,p2n−k are defined in (8).In the next result, we show that the functional equation (10) can be stable.Theorem 17.
Letβ∈−1,1 be fixed, V be a linear space, and W be a Banach space. Suppose that ϕ:Vn×Vn⟶ℝ+ is a mapping satisfying the inequality
(77)∑l=0∞124n−3kβlϕ2βl−∣β−1∣/2x1,2βl−∣β−1∣/2x2<∞,for all x1,x2∈Vn. Assume also f:Vn⟶W is a mapping satisfying the inequality
(78)∥Γfx1,x2∥≤ϕx1,x2,for all x1,x2∈Vn. Then, there exists a unique solution F:Vn⟶W of (10) such that
(79)∥fx−Fx∥≤Φx,for all x=xk,xn−k∈Vn, where
(80)Φx=124n−3k∣β+1∣/2+n−k∑l=0∞124n−3kβlϕ2βl−∣β−1∣/2x,2βl−∣β−1∣/2xk,0.Proof.
Puttingx1k=x2k=xk and x1n−k=xn−k,x2n−k=0 in (79), we have
(81)2n−kf2x−∑p1=0nn−kp12k4n−k−p124p12n−k−p1fx≤ϕx,xk,0,in which x=xk,xn−k. A computation shows that (81) can be rewritten as follows:
(82)2n−kf2x−25n−4kfx≤ϕx,xk,0,and so
(83)f2x−24n−3kfx≤12n−kϕx,xk,0.
Set(84)ξx≔124n−3kβ+1/2+n−kϕx2β−1/2,xk2β−1/2,0,and Tξx≔1/24n−3kβξ2βx where ξ∈WVn. Then, relation (83) can be modified as
(85)fx−Tfx≤ξxx∈Vn.
DefineΛηx≔1/24n−3kβη2βx for all η∈ℝ+Vn,x=xk,xn−k∈Vn. The rest of the proof is similar to the proof of Theorem 13.Corollary 18.
Letδ>0. If f:Vn⟶W is a mapping satisfying the inequality
(86)∥Γfx1,x2∥≤δ,for all x1,x2∈Vn, then there exists a unique solution F:Vn⟶W of (10) such that
(87)∥fx−Fx∥≤δ2n−k24n−3k−1,for all x∈Vn.Proof.
Setting the constant functionϕx1,x2=δ for all x1,x2∈Vn and applying Theorem 17 in the case β=1, the result can be found.


## 5. Conclusion

In the present paper, we introduced the multi-additive-quartic and multimixed additive-quartic mappings. Indeed, we characterized the mentioned mappings and then unified the system of functional equations defining a multi-additive-quartic or a multimixed additive-quartic mapping to a single equation. We also showed that under which conditions a multimixed additive-quartic mapping is multiadditive, multiquartic, and multi-additive-quartic. Finally, we applied a fixed point theorem to establish the Hyers-Ulam stability of multi-additive-quartic mappings and multimixed additive-quartic functional equations.



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*Source: 9943199-2021-11-12.xml*