# Generalized Multiquartic Mappings, Stability, and Nonstability

Authors: Abasalt Bodaghi
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, a generalized form ofn-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.


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## Body

## 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 ℱ, namely, a mapping h with the property hx+y−hx−hy≤ε is near to an exact solution of ℱ. In other words, does there exists a unique additive mapping A for example on commutative groups such that hx−Ax≤δε? Note that δε is depended on ε. This fact is called the stability of additive or Cauchy functional equation. In the case that f 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.LetV,+ be a commutative group, W be a linear space over rationals, and n be a natural number greater than 1. Let zi,zi′∈V, where i∈1,…,n. A mapping f:Vn⟶W is called(i)
Multiadditive if it is additive in each of components [6], that is,(1)fz1,…,zi−1,zi+zi′,zi+1,…,zn=fz1,…,zi−1,zi,zi+1,…,zn+fz1,…,zi−1,zi′,zi+1,…,zn.(ii)
Multiquadratic if it is quadratic in all variables [7, 8], namely(2)fz1,…,zi−1,zi+zi′,zi+1,…,zn+fz1,…,zi−1,zi−zi′,zi+1,…,zn=2fz1,…,zi−1,zi,zi+1,…,zn+2fz1,…,zi−1,zi′,zi+1,…,zn.(iii)
Multicubic if it is cubic in each of itsn arguments [9], that is,(3)fz1,…,zi−1,2zi+zi′,zi+1,…,zn+fz1,…,zi−1,2zi−zi′,zi+1,…,zn=2fz1,…,zi−1,zi+zi′,zi+1,…,zn+2fz1,…,zi−1,+zi−zi′,zi+1,…,zn+12fz1,…,zi−1,zi,zi+1,…,zn.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:(4)hx+2y+hx−2y=4hx+y+hx−y−6hx+24hy,(5)h2x+y+h2x−y=4hx+y+hx−y+24hx−6hy.Kang [13] generalized equations (4) and (5) as follows:(6)hax+by+hax−by=a2b2hx+y+hx−y+Ka2hx−Kb2hy,for fixed integers a,b with a,b≠0,±1, where K=2a2−b2 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 G, a linear space V, and an integer n with n≥2, a mapping f:Gn⟶V 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-β 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 byℕ and ℚ, respectively, and also ℕ0≔ℕ∪0. For any p∈ℕ0,k∈ℕ, r=r1,…,rk∈−1,1k in which −1,1k=−1,1×⋯×−1,1︷k−times and x=x1,…,xk∈Vk, we write px≔px1,…,pxk and rx≔r1x1,…,rkxk, where px denotes for the p th power of an element x of the commutative group V.Definition 1.
LetV and W be vector spaces over ℚ. A mapping f:Vn⟶W is called n-quartic or generalized multiquartic if f is quartic in each of its n arguments; that is,(7)fz1,…,zi−1,azi+bzi′,zi+1,…,zn+fz1,…,zi−1,azi−bzi′,zi+1,…,zn=a2b2fz1,…,zi−1,zi+zi′,zi+1,…,zn+fz1,…,zi−1,+zi−zi′,zi+1,…,zn+Ka2fz1,…,zi−1,zi,zi+1,…,zn−Kb2fz1,…,zi−1,zi′,zi+1,…,zn.
Suppose thatn∈ℕ such that n≥2 and xin=xi1,…,xin∈Vn in which i∈1,2. We wish to denote xin by xi when no confusion can arise. Let x1,x2∈Vn and pi∈0,1,…,n, put Nn=Nn=N1,…,Nn|Nj∈x1j±x2j,x1j,x2j, where j∈0,1,…,n. Put Np1,p2n as a subset of N including the elements of N such that the cardinal of Nj:Nj=xij is pi when i∈1,2.
For the mappings defined in Definition1, put some conventions as follows:(8)fNp1,p2n≔∑Nn∈Np1,p2nfNn,(9)fNp1,p2n,w≔∑Nn∈Np1,p2nfNn,ww∈V.
It was shown in [15], Theorem 2.2 that a mapping f:Vn⟶W is multiquartic (in sense of (4)) if and only if it satisfies the following equation:(10)∑t∈−1,1nfx1+2tx2=∑p1=0n∑p2=0n−p14n−p1−p2−6p124p2fNp1,p2n.
Moreover, it is has been proven in Proposition1 from [14] that if f:Vn⟶W is a multiquartic mapping (in sense of (5)), then it fulfills the following equation:(11)∑t∈−1,1nf2x1+tx2=∑p1=0n∑p2=0n−p14n−p1−p224p1−6p2fNp1,p2n.Proposition 1.
Every generalized multi-quartic mappingf:Vn⟶W fulfills the following equation:(12)∑t∈−1,1nfax1+tbx2=∑p1=0n∑p2=0n−p1ab2n−p1−p2Ka2p1−Kb2p2fNp1,p2n,in which x1,x2∈Vn.Proof.
The proof is based on induction. Forn=1, it is clear that f satisfies equation (6). Suppose that (12) holds for some n∈ℕ with n>1. We have(13)∑t∈−1,1n+1fax1n+1+tbx2n+1=a2b2∑t∈−1,1n∑s∈−1,1fax1n+tbx2n,x1,n+1+sx2,n+1+Ka2∑t∈−1,1nfax1n+tbx2n,x1,n+1−Kb2∑t∈−1,1nf2x1n+tbx2n,x2,n+1=a2b2∑p1=0n∑p2=0n−p1∑t∈−1,1ab2n−p1−p2Ka2p1−Kb2p2fNp1,p2n,x1,n+1+tx2,n+1+Ka2∑p1=0n∑p2=0n−p1ab2n−p1−p2Ka2p1−Kb2p2fNp1,p2n,x1,n+1−Kb2∑p1=0n∑p2=0n−p1ab2n−p1−p2Ka2p1−Kb2p2fNp1,p2n,x2,n+1=∑p1=0n+1∑p2=0n+1−p1ab2n+1−p1−p2Ka2p1−Kb2p2fNp1,p2n+1.
This shows that (12) holds for n+1.
By means of Proposition1, we find that the mapping f:ℝn⟶ℝ defined through fr1,…,rn=cr14,…,rn4 satisfies (12) and so one can call the mentioned equation as multiquartic functional equation.Definition 2.
Letm∈ℚ and r∈ℕ. Consider a mapping f:Vn⟶W and v1,…,vn∈Vn. Then,(i)
f has the m,r-power condition if for any arbitrary variable vj,(14)fv1,…,vj−1,mvj,vj+1,…,vn=mrfv1,…,vj−1,vj,vj+1,…,vn.(ii)
f is even if for any arbitrary variable vj,(15)fv1,…,vj−1,−vj,vj+1,…,vn=fv1,…,vj−1,vj,vj+1,…,vn.(iii)
f has zero condition if fv=0 for any v∈Vn for which at least one component is zero
Givenn,k∈ℕ0 with n≥k. We remember that nk is the binomial coefficient defined by n!/k!n−k!.
Leta,b be integers as in (6). We consider the following hypotheses for a mapping f:Vn⟶W and use them from now on.(H1)f has the a,4-power condition(H2)f has the b,4-power condition and is even
The 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 mappingf:Vn⟶W fulfilling equation (12) has zero condition.Proof.
Assume that (H1) holds. Puttingx1=x2=0,…,0 in (12), we have(16)2nf0,…,0=∑p1=0n∑p2=0n−p1np1n−p1p2ab2n−p1−p2Ka2p1−Kb2p22n−p1−p2f0,…,0=∑p1=0nnp12n−p1Ka2p1∑p2=0n−p1n−p1p2ab2n−p1−p2−Kb22p2f0,…,0=∑p1=0nnp12n−p1Ka2p1a2b2−Kb22n−p1f0,…,0=2b4+Ka2nf0,…,0.
Relation (16) implies that f0,…,0=0. Similar to the proof of Proposition 1 from [14], one can continue in this fashion showing that f has zero condition. The second part can be obtained similarly.Theorem 1.
Under the assumptions Lemma1, every mapping f:Vn⟶W satisfying equation (12) can be generalized multiquartic.Proof.
Suppose that (H1) is true. Fixj∈1,…,n. Set(17)f∗x1j,x2j≔fx11,…,x1,j−1,x1j+x2j,x1,j+1,…,x1n+fx11,…,x1,j−1,x1j−x2j,x1,j+1,…,x1n,f∗x1j≔fx1=fx11,…,x1n,f∗x2j≔fx11,…,x1,j−1,x2j,x1,j+1,…,x1n.Putting x2k=0 for all k∈1,…,n/j in (12) and applying our assumption, we get(18)2n−1×a4n−1fx11,…,x1,j−1,ax1j+bx2j,x1,j+1,…,x1n+fx11,…,x1,j−1,ax1j−bx2j,x1,j+1,…,x1n=2n−1fax11,…,ax1,j−1,ax1j+bx2j,ax1,j+1,…,ax1n+fax11,…,ax1,j−1,ax1j−bx2j,ax1,j+1,…,ax1n=∑p1=0n−1n−1p1ab2n−p1Ka2p12n−1−p1f∗x1j,x2j+∑p1=1nn−1p1−1ab2n−p1Ka2p12n−p1f∗x1j−Kb2∑p1=1nn−1p1−1ab2n−p1Ka2p1−12n−p1f∗x2j=ab2∑p1=0n−1n−1p1ab2n−1−p1Ka2p12n−1−p1f∗x1j,x2j+Ka2∑p1=0n−1n−1p12a2b2n−1−p1Ka2p1f∗x1j−Kb2∑p1=0n−1n−1p12a2b2n−1−p1Ka2p1f∗x2j=ab22a2b2+Ka2n−1f∗x1j,x2j+Ka22a2b2+Ka2n−1f∗x1j−Kb22a2b2+Ka2n−1f∗x2j=ab22n−1a4n−1f∗x1j,x2j+Ka22n−1a4n−1f∗x1j−Kb22n−1a4n−1f∗x2j.It follows from (18) that(19)fx11,…,x1,j−1,ax1j+x2j,x1,j+1,…,x1n+fx11,…,x1,j−1,ax1j−x2j,x1,j+1,…,x1n=a2b2f∗x1j,x2j+Ka2f∗x1j−Kb2f∗x2j.
The last equality shows thatf is quartic in the j th component.
(ii) Considerj∈1,…,n. Interchanging x1k,x2k into 0,x1k for all k∈1,…,n/j 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 mappingf:Vn⟶W, the following assertions are equivalent:(i)
f is (generalized) multiquartic(ii)
f satisfies (10)(iii)
f fulfills (12) with either the assumption (H1) or (H2)(iv)
f 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 Q. Moreover, it is shown in [13] that Q 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-β-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 thatX,d is a complete generalized metric space, and T:X⟶X is a mapping with the Lipschitz constant L<1. For each element x∈X, one of the following cases can happen.(i)
dTnx,Tn+1x=∞ for all n∈ℕ0, or.(ii)
there existsn0∈ℕ such that.(iii)
dTnx,Tn+1x<∞ for all n≥n0;(iv)
there is a fixed pointx∗ of T such that the sequence Tnx converges to x∗ in which it belongs to the set(20)X∗=x∈X:dTn0x,x<∞.(v)
dx,x∗≤1/1−Ldx,Tx for all x∈X∗.
LetW be a vector space and G,+ be a commutative group. In the rest of this section, for a mapping f:Gn⟶W, we define the operator Df:Gn×Gn⟶W through(21)Dfx1,x2≔∑t∈−1,1nfax1+tbx2−∑p1=0n∑p2=0n−p1ab2n−p1−p2Ka2p1−Kb2p2fNp1,p2n,for x1,x2∈Gn, where fNp,qn is defined in (8).From now on, we assume that all mappingsf:Gn⟶W have zero conditions. With this assumption, we have Hyers’ stability for equation (12) as follows.Theorem 3.
Letε>0. Suppose that W is a complete normed space and G,+ is a commutative group. If a mapping f:Gn⟶W fulfilling the inequality(22)Dfx1,x2≤ε,for all x1,x2∈Gn, then there exists a unique solution ℱ:Gn⟶W of (12) such that(23)fx−ℱx≤12na4n−1ε,for all x∈Gn. Moreover,(24)ℱx=liml⟶∞1a4nlfalx,for all x∈Gn.Proof.
Consideringx=x1 and x2=0 in (22) and applying the hypotheses, we have(25)2nfax−∑p1=0nnp1ab2n−p1Ka2p12n−p1fx≤ε,for all x∈Gn, and hence,(26)2nfax−2na4nfx≤ε.Thus,(27)fax−a4nfx≤12nε.
SetX≔f∈WGn|fhas zero condition, where WGn is the set of all mappings from Gn to W. Define the generalized metric d on X.(28)dg1,g2≔infC∈0,∞:g1x−g2x≤Cg1,g2ε,x∈Gn.Similar to the proof of [17], Theorem 2.2, it is easily verified that Ω,d is a complete generalized metric space. Consider the operator T:X⟶X defined via(29)Tfx≔1a4nfax,for all x∈Gn. Take g1,g2∈Ω and Cg1,g2∈0,∞ with dg1,g2≤Cg1,g2. Then, g1x−g2x≤Cg1,g2ε, and hence,(30)Tg1x−Tg2x≤1a4ng1ax−g2ax≤1a4nCg1,g2ε.
Therefore,dTg1,Tg2≤1/a4nCg1,g2. This shows that dTg1,Tg2≤1/a4ndg1,g2. This means that T is a strictly contractive with the Lipschitz constant 1/a4n. Relation (27) necessitates that(31)Tfx−fx≤1a4nfax−fx≤12n×a4nε.Thus,(32)dTf,f≤12n×a4n<∞.
Applying Theorem3 for the space X,d, the operator T and n0=0, we can deduce that the sequence Tlfl∈ℕ converges to ℱ in X,d which is a fixed point of T. Indeed, ℱx=liml⟶∞Tlfx, and(33)ℱx=1a4nfx.On the other hand, by induction on l, it can be shown for all x∈Gn that(34)Tlfx≔1a4nlfalx,and moreover, (24) follows. Obviously, f∈X∗. In addition, it follows from (32) and part (iii) of Theorem 2 that(35)df,ℱ≤11−1/a4ndTf,f≤12na4n−1,which proves (23). Furthermore,(36)Dℱx1,x2=liml⟶∞1a4nlDfalx1,alx2≤liml⟶∞1a4nlε=0,for all x1,x2∈Gn. The last relation shows that Dℱx1,x2=0 for all x1,x2∈Gn, and therefore, ℱ satisfies equation (12). For the uniqueness part, assume that F:Gn⟶W is another solution of equation (12) with zero condition, and moreover, inequality (23) holds for it. Then, F satisfies (33), and hence, it is a fixed point of the operator T. Besides, by (23), we obtain(37)df,F≤12na4n−1<∞,and consequently, F∈Λ. Using part (ii) of Theorem 2, we find F=ℱ. This finishes the proof.
Here, we recall some basic facts regarding the setting of quasi-β-normed space.Definition 3.
LetV be a linear space over ℝ, β be a fixed real number with 0<β<1. A quasi-β-norm is a real-valued function on V fulfilling the statements(i)
v≥0 for all v∈V(ii)
v=0 if and only if v=0(iii)
rv=rβ|v for all v∈V and r∈ℝ(iv)
There is a constantK≥1 such that u+v≤Ku+v for all u,v∈V
If⋅ is a quasi-β-norm on V, then the pair V,⋅ is said to be a quasi-β-normed space. The smallest possible K is called the modulus of concavity of the norm ⋅. Moreover, a quasi-β-normed space is a complete quasi-β-normed space. If(38)u+vp≤up+vp,for all u,v∈V, then a quasi-β-norm ⋅ is called a β,p-norm, and indeed, a quasi-β-Banach space is said to be a β,p-Banach space. On the other hand, for a p-norm, the metric du,v≔u−vp leads us to a translation-invariant metric on V. Furthermore, it follows from the Aoki-Rolewicz Theorem [18] that every quasinorm is equivalent to some p-norm.
The upcoming fixed-point lemma is presented in [19], Lemma 3.1.Lemma 2.
Letj∈−1,1 be fixed, a,s∈ℕ with a≥2. Let also V be a linear space, W be a β,p-Banach space with β,p-norm ⋅W. Suppose that for the function ψ:V⟶0,∞, there exists an L<1 with ψajv<Lajsβψv for all v∈V. If f:V⟶W is a mapping fulfilling(39)fav−asfvW≤ψv,for all v∈V, then there exists a unique mapping F:V⟶W with Fav=asFv and(40)fv−FvW≤1asβ1−Ljψv,for all v∈V. Moreover, Fv=liml⟶∞fajlv/ajls for all v∈V.Next, we prove the Găvruţa stability of (12) in quasi-β normed spaces by applying Lemma 2.Theorem 4.
Letj∈−1,1 be fixed, V be a linear space and W be a β,p-Banach space and φ:Vn×Vn⟶ℝ+ be a function for which there exists an 0<L<1 with φajx1,ajx2≤a4njβLφx1,x2 for all x1,x2∈Vn. Suppose that a mapping f:Vn⟶W fulfilling the inequality(41)Dfx1,x2≤φx1,x2,for all x1,x2∈Vn. Then, there exists a unique solution Q:Vn⟶W of (12) such that(42)fx−Qx≤11−Lj12nβ×a4nβφx,0,for all x∈Vn. In particular, if Q has either the property (H1) or (H2), then it is a generalized multiquartic mapping.Proof.
Similar to the proof of Theorem3, by putting x2=0 in (41) and using our hypotheses, we get(43)2nfax−2na4nfx≤φx,0,for all x=x1∈Vn and so(44)fax−a4nfx≤12nβφx,0,for all x∈Vn. Lemma 2 implies that there exists a unique mapping Q:Vn⟶W such that Qax=a4nQx and(45)fx−Qx≤11−Lj12nβ×a4nβφx,0,for all x∈Vn. We claim that Q is a multiquartic map. Here, we note from Lemma 2 that for all x∈Vn, Qx=liml⟶∞fajlx/a4njl. Now, by (41), we have(46)Dfajlx1,ajlx2a4njl≤a−4njlβφajlx1,ajlx2≤a−4njlβa4njβLlφx1,x2=Llφx1,x2,for all x1,x2∈Vn and l∈ℕ. Letting l⟶∞ in the above inequality, we find DQx1,x2=0 for all x1,x2∈Vn, and hence, Q satisfies (12). The last assertion follows from Theorem 1.
In upcoming corollaries, it is assumed thatV is a quasi-α-normed space with quasi-α-norm ⋅V, and W is a β,p-Banach space with β,p-norm ⋅W. The following result is a direct consequence of Theorem 3 concerning the stability of (12) when the norm of Dfx1,x2 is controlled by the sum of variables norms of x1 and x2 with positive powers.Corollary 2.
Letθ and λ be positive numbers with λ≠4nβ/α. If a mapping f:Vn⟶W fulfilling the inequality(47)Dfx1,x2W≤θ∑k=12∑j=1nxkjVλ,for all x1,x2∈Vn, then there exists a unique solution Q:Vn⟶W of (12) such that(48)fx−QxW≤θ2nβa4nβ−aαλ,∑j=1nx1jVλ,λ∈0,4nβα,aαλθ2nβ×a4nβaαλ−a4nβ,∑j=1nx1jVλ,λ∈4nβα,∞,for all x=x1∈Vn. Moreover, if Q has either the property (H1) or (H2), then it is generalized multi-quartic.The hyperstability of equation (12) is presented as follows.Corollary 3.
Supposeλij>0 for i∈1,2 and j∈1,…,n with ∑i=12∑j=1nλij≠4nβ/α. If a mapping f:Vn⟶W fulfilling the inequality(49)Dfx1,x2W≤θ∏i=12∏j=1nxijVλij,for all x1,x2∈Vn, then it satisfies (12). In particular, if f 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.
Letf:ℝn⟶ℝ be a continuous n-quartic function. Then, there exists a constant c∈ℝ such that(50)fr1,…,rn=c∏j=1nrj4,for all r1,…,rn∈ℝ.We end the paper with the following counterexample for the multiquartic mappings onℝn that its idea is taken from [21]. In fact, we show the assumption λ≠4n cannot be removed in Corollary 2 when V=W=ℝ in the case that α=β=1.Example 1.
Letδ>0 and n∈ℕ. Put μ=24n−1/28nSt4nδ, where t=maxa,b and(51)S=2n+∑p1=0n∑p2=0n−p1ab2n−p1−p2Ka2p1Kb2p2.
Note thatt≥2. Define the function ψ:ℝn⟶ℝ through(52)ψr1,…,rn=μ∏j=1nrj4,for allrjwithrj<1,μ,otherwise.Hence, ψr1,…,rn≤μ for all r1,…,rn∈ℝn and ψ is continuous as well. Using the function ψ, define the function f:ℝn⟶ℝ by(53)fr1,…,rn=∑l=0∞ψ2lr1,…,2lrn24nl,rj∈ℝ.It is obvious that f is an even function in each variable and nonnegative. The series given in the last equality is uniformly convergent of continuous functions, and thus, f is continuous and bounded. In other words, for each r1,…,rn∈ℝn, we have fr1,…,rn≤24n/24n−1μ. Put xi=xi1,…,xin, where i∈1,2. We claim that(54)Dfx1,x2≤δ∑i=12∑j=1nxij4n,for all x1,x2∈ℝn. It is clear that (54) is valid for x1=x2=0. Assume that x1,x2∈ℝn with ∑i=12∑j=1nxij4n<1/24nt4n. Thus, there exists a positive integer N such that(55)124nN+1t4n<∑i=12∑j=1nxij4n<124nNt4n.Hence, xij4n<∑i=12∑j=1nxij4n<1/24nt4n and so 2Nxij<1/t for all i∈1,2 and j∈1,…,n. Therefore, 2N−1xij<1. If y1,y2∈xij|i∈1,2,j∈1,…,n, then(56)2N−1y1±y2,2N−1ay1±2by2⊆−1,1.
Sinceψ is multiquartic function on −1,1n, Dψ2lx1,2lx2=0 for all l∈0,1,2,…,N−1. It follows from the last equality and (55) that(57)Df2lx1,2lx2∑i=12∑j=1nxij4n≤∑l=N∞Dψ2lx1,2lx224nl∑i=12∑j=1nxij4n≤∑l=0∞μS24nl+N∑i=12∑j=1nxij4n≤μS24nt4n∑l=0∞124nl=μS28n24n−1t4n=δ,for all x1,x2∈ℝn. If ∑i=12∑j=1nxij4n≥1/24nt4n, then(58)Df2lx1,2lx2∑i=12∑j=1nxij4n≤28n24n−1t4nμS=δ.Therefore, f fulfills (54) for all x1,x2∈ℝn. On the contrary, suppose that there is a number b∈0,∞ and a multiquartic function Q:ℝn⟶ℝ such that fr1,…,rn−Qr1,…,rn<b∏j=1nrj4 for all r1,…,rn∈ℝns. By Proposition 2, there exists c∈ℝ such that Qr1,…,rn=c∏j=1nrj4, and thus,(59)fr1,…,rn≤c+b∏j=1nrj4,for all r1,…,rn∈ℝn. Here, we choose p∈ℕ such that p+1μ>c+b. If r=r1,…,rn belongs to ℝn such that rj∈0,1/2p for all j∈1,…,n, then 2lrj∈0,1 for all l=0,1,…,p. Thus, we get(60)fr1,…,rn=∑l=0∞ψ2lr1,…,2lr224nl=∑l=0pμ24nl∏j=1nrj424nl=p+1μ∏j=1nrj4>c+b∏j=1nrj4.The 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-β normed spaces. Finally, by using a characterization result, we have presented a nonstable example to invalidate the results in the special case [21].



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