# Characterization and Stability of Multi-Euler-Lagrange Quadratic Functional Equations

Authors: Abasalt Bodaghi; Hossein Moshtagh; Amir Mousivand
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The aim of the current article is to characterize and to prove the stability of multi-Euler-Lagrange quadratic mappings. In other words, it reduces a system of equations defining the multi-Euler-Lagrange quadratic mappings to an equation, say, the multi-Euler-Lagrange quadratic functional equation. Moreover, some results corresponding to known stability (Hyers, Rassias, and Gӑvruta) outcomes regarding the multi-Euler-Lagrange quadratic functional equation are presented in quasi-β-normed and Banach spaces by using the fixed point methods. Lastly, an example for the nonstable multi-Euler-Lagrange quadratic functional equation is indicated.


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## Body

## 1. Introduction

The celebrated Ulam challenge [1] arises from this question that how we can find an exact solution near to an approximate solution of an equation. This phenomenon of mathematics is called the stability of functional equations which has many applications in nonlinear analysis. The mentioned question has been partially solved by Hyers [2], Aoki [3], and Rassias [4] for the linear, additive, and linear (unbounded Cauchy difference) mappings, respectively. Next, many Hyers-Ulam stability problems for miscellaneous functional equations were studied by authors in the spirit of Rassias approach (see for instance [5–14] and other resources).During the last two decades, stability problems for multivariable mappings were studied and extended by a number of authors. One of the mappings is the multiquadratic mapping, studied, for example, in [15–17]. Recall that a multivariable mapping f:Vn⟶W is said to be multiquadratic [11] if it fulfills the famous quadratic equation
(1)Qu+v+Qu−v=2Qu+2Qv,in each component. Note that equation (1) is a suitable tool for obtaining some characterizations in the setting of inner product spaces and in fact plays a prominent role. In other words, any square norm on an inner product space fulfills
(2)u+v2+u−v2=2u2+v2,which is called the parallelogram equality. However, some functional equations have been applied to characterize inner product spaces and are available in [18, 19] and references therein. In addition, the quadratic functional equation was used to characterize inner product spaces in [20, 21].A lot of information about equation (1) and some equations which are equivalent to it (in particular, about their solutions and stability) and more applications can be found for instance in [22–24]. Park was the first author who studied the stability of multiquadratic in the setting of Banach algebras [16]. After that, some authors introduced various versions of multiquadratic mappings and investigated the Hyers-Ulam stability of such mappings in Banach spaces and non-Archimedean spaces; these results are available for instance in [15, 25–29]. As for an unification of the multiquadratic mappings, Zhao et al. [17] were the first authors who described the structure of multiquadratic mappings, and in fact, they showed that f:Vn⟶W is a multiquadratic mapping if and only if the equation
(3)∑t∈−1,1nfv1+tv2=2n∑i1,⋯,in∈1,2fv1i1,⋯,vninholds, where vi=x1i,⋯,vni∈Vn and i∈1,2.Rassias [30] introduced the following notion of a generalized Euler-Lagrange-type quadratic mapping and investigated its generalized stability.Definition 1.
Suppose thatV and W are linear spaces. A nonlinear mapping Q:V⟶W satisfying the functional equation
(4)Qau+bv+Qbu−av=a2+b2Qu+Qvis called 2-dimensional quadratic, where u,v∈V and a,b are the fixed reals with a2+b2>1.It is easily seen that the Euler-Lagrange equality(5)au+bv2+bu−av2=a2+b2u2+v2is valid for Q, defined in Definition 1 with any fixed reals a,b, and hence, (4) is also called Euler-Lagrange quadratic functional equation; we refer to [31] for Euler-Lagrange type cubic functional equation and its stability. Note that equation (4) is a general form of (1) in the case that a=b=1, and so the function Qv=v2 satisfies (4). Next, Xu [32] extended the definition above to several variable mappings and presented the next definition.Definition 2.
LetV and W be vector spaces. A mapping f:Vn⟶W is said to be the n-Euler-Lagrange quadratic or multi-Euler-Lagrange quadratic if the mapping
(6)v↦fv1,⋯,vi−1,v,vi+1,⋯,vnsatisfies (4), for all i∈1,⋯,n and all vi∈V.In this article, we include a characterization of multi-Euler-Lagrange quadratic mappings and show that every multi-Euler-Lagrange quadratic mapping can be described as an equation (namely, the multi-Euler-Lagrange quadratic equation). Under the quadratic condition (2-power condition) in each variable, every multivariable mappings satisfying the mentioned earlier equation is multi-Euler-Lagrange quadratic (Theorem5). Furthermore, we bring two Hyers-Ulam stability results for multi-Euler-Lagrange quadratic functional equations in quasi-β-normed and Banach spaces which their proof is based according to some known fixed point methods; see [33, 34] for more stability results in quasi-β-Banach spaces setting. Finally, we indicate an example to show that the multi-Euler-Lagrange quadratic functional equation is nonstable in the case of singularity.


## 2. Characterization of Multi-Euler-Lagrange Quadratic Mappings

Throughout, we consider the following known notations:(i)
ℕ=the set of all natural numbers(ii)
ℤ= the set of all integer numbers(iii)
ℚ= the set of all rational numbers(iv)
ℕ0≔ℕ∪0(v)
ℝ+≔0,∞LetV be a linear space over ℚ. Given n∈ℕ, p∈ℕ0,s=s1,⋯,sn∈ℚn, and v=v1,⋯,vn∈Vn. We write sv≔s1v1,⋯,snvn and pv≔pv1,⋯,pvn which belong to Vn. Here and subsequently, V is linear space over ℚ and vin=vi1,vi2,⋯,vin∈Vn, in which i∈1,2. Furthermore, for given the fixed elements ain=ai1,ai2,⋯,ain∈ℤn such that aij≠0,±1, where i=1,2 and j=1,⋯,n (here and the rest of the paper). We will write ain and vin simply ai and vi, respectively, when no confusion can arise.Forv1,v2∈Vn and a1,a2∈ℤn, set
(7)Aj+1=∑i=12aijvij,Aj−1=∑i=12−1i+1a3−i,jvij,j∈1,⋯,n.In continuation, we show that the equation(8)∑t1,⋯tn∈−1,+1fA1t1,⋯,Antn=∏j=1na1j2+a2j2∑l1,⋯,ln∈1,2fvl11,⋯,vlnnis a general form of (4) for the multivariable case. In other words, we prove that every multi-Euler-Lagrange quadratic mapping fulfills (1) and vice versa. For doing this, we need some definitions and the upcoming lemma.Definition 3.
LetV and W be vector spaces over ℚ and f:Vn⟶W be a multivariable mapping.
(i)
We sayf satisfies (has) the 2-power (quadratic) condition in the jth component if
(9)fx1,⋯,xj−1,a∗xj,xj+1,⋯,xn=a∗2fx1,⋯,xj−1,xj,xj+1,⋯,xn,for all x1,⋯,xn∈V, where a∗∈a1j,a2j for all j∈1,⋯,n(ii)
Iffx1,⋯,xn=0 when the fixed xj is zero, then we say that f has zero functional equation in the jth variable. Moreover, if fx1,⋯,xn=0 for any x1,⋯,xn∈Vn with at least one xj is zero, we say f has zero functional equationWe consider two hypotheses as follows:(H1)f has the quadratic condition in all variables.(H2)f has zero functional equation.Remark 4.
It is clear that if a mappingf:Vn⟶W satisfies the quadratic condition in the jth variable, then it has zero functional equation in the same variable. Therefore, if f fulfills (H1), then it satisfies (H2).Theorem 5.
For a mappingf:Vn⟶W, the following assertions are equivalent:
(i)
f is multi-Euler-Lagrange quadratic(ii)
f fulfills (8) and H1Proof.
i ⇒ ii In view of [30], one can show that f satisfies H1. By induction on n, we now proceed the rest of this implication so that f satisfies equation (8). Obviously, f satisfies equation (4) for n=1. The induction hypothesis is
(10)∑t1,⋯tn∈−1,+1fA1t1,⋯,Antn=∏j=1na1j2+a2j2∑l1,⋯,ln∈1,2fvl11,⋯,vlnn.
Then(11)∑t1,⋯,tn+1∈−1,1fA1t1,⋯,An+1tn+1=∑t1,⋯,tn∈−1,1fA1t1,⋯,An+1+1+∑t1,⋯,tn∈−1,1fA1t1,⋯,An+1−1=a1,n+12+a2,n+12∑t1,⋯,tn∈−1,1fA1t1,⋯,Antn,v1,n+1+∑t1,⋯,tn∈−1,1fA1t1,⋯,Antn,v2,n+1=a1,n+12+a2,n+12∏j=1na1j2+a2j2∑l1,⋯,ln∈1,2fvl11,⋯,vlnn,v1,n+1+∑l1,⋯,ln∈1,2fvl11,⋯,vlnn,v2,n+1=∏j=1n+1a1j2+a2j2∑l1,⋯,ln+1∈1,2fvl11,⋯,vln+1,n+1.
(ii) ⇒ (i) Letj∈1,⋯,n be arbitrary and fixed. Taking v2k=0 for all k∈1,⋯,n\j in (8) and applying Remark 4, the left side will be as follows:
(12)fa11v11,⋯,a1,j−1v1,j−1,Aj+1,a1,j+1v1,j+1,⋯,a1nv1n+fa21v11,⋯,a2,j−1v1,j−1,Aj+1,a2,j+1v1,j+1,⋯,a2nv1n+fa11v11,⋯,a1,j−1v1,j−1,Aj−1,a1,j+1v1,j+1,⋯,a1nv1n+fa21v11,⋯,a2,j−1v1,j−1,Aj−1,a2,j+1v1,j+1,⋯,a2nv1n=a112a212a122a222⋯a1,j−12a2,j−12a1,j+12a2,j+12⋯a1n2a2n2fv11,⋯,v1,j−1,Aj+1,v1,j+1,⋯,v1n+fv11,⋯,v1,j−1,Aj−1,v1,j+1,⋯,v1n.
Once again, the same replacements convert the right side of (8) to
(13)∏k≠jk=1n−1a1k2+a2k2a1j2+a2j2fv11,⋯,v1,j−1,v1j,v1,j+1,⋯,v1n+fv11,⋯,v1,j−1,v2j,v1,j+1,⋯,v1n.
It follows from (12) and (13) that f is Euler-Lagrange a1j,a2j-quadratic in the jth component, and this completes the proof.We should note that Theorem5 necessitates that the mapping f:ℝn⟶ℝ defined through fx1,⋯,xn=C∏j=1nxj2 fulfills equation (8). Hence, this equation can be called the multi-Euler-Lagrange quadratic functional equation.


## 3. Stability and Nonstability Results

The goals of this section are to prove miscellaneous result stability of multi-Euler-Lagrange quadratic equation (14) such as Hyers and Găvruta stability. Here, we mention a special case of equation (8) in which a1=a,⋯,a and a2=b,⋯,b, and so (8) converts to
(14)∑t1,⋯tn∈a,b,b,afA1t1,⋯,Antn=a2+b2n∑l1,⋯,ln∈1,2fvl11,⋯,vlnn,in which
(15)Aja,b=av1j+bv2j,andAjb,a=bv1j−av2j,and m=a2+b2 (used here and from now on) for all j∈1,⋯,n.For a setE, a function d:E×E⟶0,∞ is said to be a generalized metric on E provided that d fulfills the statements below, for all u,v,w∈E.
(i)
du,v=0 if and only if u=v(ii)
du,v=dv,u(iii)
du,w≤du,v+dv,wThe next theorem from [35] is one of fundamental results in fixed point theory and useful to achieve our first purpose in this section.Theorem 6.
Suppose thatΩ,d is a complete generalized metric space and J:Ω⟶Ω is a mapping such that its Lipschitz constant is L<1. Then, for each element x∈Ω, one of following cases can be happen:
(i)
dJnx,Jn+1x=∞foralln≥0or(ii)
There is ann0∈ℕ such that dJnx,Jn+1x<∞ for all n≥n0, and the sequence Jnx is convergent to a fixed point x∗ of J which belongs to the set Λ=x∈Ω:dJn0x,x<∞. Moreover, dx,x∗≤1/1−Ldx,Jx for all x∈Λ
In the sequel, for any mappingf:Vn⟶W, we define the operator Df:Vn×Vn⟶W via
(16)Dfv1,v2≔∑t1,⋯tn∈a,b,b,afA1t1,⋯,Antn−mn∑l1,⋯,ln∈1,2fvl11,⋯,vlnn,for the fixed nonzero integers a,b where Aja,b and Ajb,a are defined in (15) for all j=1,⋯,n.
In the incoming stability result for equation (14), Dfv1,v2 is controlled by a small positive number ε. We recall that for i=1,2, we consider vi=vi1,⋯,vin∈Vn.Theorem 7.
Givenε>0. Let V and W be a linear space and a complete normed space, respectively. Suppose that a mapping f:Vn⟶W fulfilling H2 and
(17)Dfv1,v2≤ε,for all v1,v2∈Vn. Then, there exists a unique solution Q:Vn⟶W of (14) such that
(18)fv−Qv≤mn+1m2n−1ε,for all v∈Vn. In addition,
(19)Qv=liml⟶∞1m2nlfmlv,for all v∈Vn.Proof.
Puttingv2=0 in (17) and using the assumption H2, we have
(20)f~v1−mnfv1≤ε,for all v1∈Vn, where
(21)f~v1=∑al11,⋯,alnn∈a,bfal11v11,⋯,alnnv1n.
Setv1=v for simply and for the rest of the proof, all the equations and inequalities are valid for all v∈Vn. Once more, by replacing v1,v2 instead of av1,bv1=av,bv in (17), we get
(22)fmv−mnf~v≤ε.
Multiplying both sides of (20) by mn and plugging to (22), we obtain
(23)fmv−m2nfv≤fmv−mnf~v+mnf~v−m2nfv≤mn+1ε,and thus
(24)fmv−m2nfv≤mn+1ε.
LetΩ≔f:Vn⟶WfsatisfiesH2. For each f,g∈Ω, we define the function d on Ω as follows:
(25)dg,h≔infC∈0,∞:gv−hv≤Cg,hε,forallv∈Vn.
Similar to the proof of ([36], Theorem 2.2), it is seen that Ω,d is a complete generalized metric space. Define J:Ω⟶Ω through
(26)Jfv≔1m2nfmv,for all v∈Vn. Take g,h∈Ω and Cg,h∈0,∞ with dg,h≤Cg,h. Then, gv−hv≤Cg,hε, and hence
(27)Jgv−Jhv≤1m2ngmv−hmv≤1m2nCg,hε.
Therefore,dJg,Jh≤1/m2nCg,h. This shows that dJg,Jh≤1/m2ndg,h and in fact J is a strictly contractive operator such that its Lipschitz is 1/m2n. It concludes from (24) that
(28)Jfv−fv≤1m2nfmv−fv≤mn+1m2nε.
Hence,(29)dJf,f≤mn+1m2n<∞.
An application of Theorem6 for the space Ω,d, the operator J, n0=0, and x=f, shows that the sequence Jlfl∈ℕ is convergent in Ω,d and its limit; Q is a fixed point of J. Indeed, Qv=liml⟶∞Jlfv, and
(30)Qv=1m2nQmv,v∈Vn.
In other words, by induction onl, it is easily verified that for each v∈Vn, we have
(31)Jlfv≔1m2nlfmlv,and (19) follows. Note that clearly f∈Λ, and hence, part (iii) of Theorem 6 and (29) necessitate that
(32)df,Q≤11−1/m2ndJf,f≤mn+1m2n−1,which proves (18). In addition,
(33)DQv1,v2=liml⟶∞1m2nlDfmlv1,mlv2≤liml⟶∞1m2nlε=0,for all v1,v2∈Vn. The last relation shows that DQv1,v2=0 for all v1,v2∈Vn and means that Q fulfills (14). Let us finally suppose that Q:Vn⟶W is another solution of equation (14) satisfies H2 such that inequality (18) holds. Then, Q satisfies (30), and so it is a fixed point of J. Furthermore, by (18), we get
(34)df,Q≤mn+1m2n−1<∞,and consequently, Q∈Λ. It now follows from part (ii) of Theorem 6 that Q=Q. This finishes the proof.Remark 8.
In the proof of Theorem7, if we put v1=0, we can not reach to (20) unless it is assumed that f is even in each component. Recall from [33] that f:Vn⟶W is even in the kth component if
(35)fx1,⋯,xk−1,−xk,xk+1,⋯,xn=fx1,⋯,xk−1,xk,xk+1,⋯,xn.
In other words, this condition is redundant, and we do not need it.Hereafter, we concentrate our mind on the quasi-β-normed spaces.Definition 9.
Letβ be a fix real number with 0<β<1 and K denote either ℝ or ℂ. Suppose that E is a vector space over K. A quasi-β-norm is a real-valued function on E fulfilling the next conditions for all x,y∈E and t∈K.
(i)
x≥0 and moreover x=0⇔x=0(ii)
tx=tβx(iii)
There exists a real numberM≥1 such that x+y≤Mx+yWhenβ=1, the norm above is a quasinorm. Recall that M is the modulus of concavity of the norm ·. Moreover, if · is a quasi-β-norm on E, the pair E,· is said to be a quasi-β-normed space. Similar to normed spaces, a complete quasi-β-normed space is called a quasi-β-Banach space. For 0<p≤1, if x+yp≤xp+yp, for all x,y∈E, then the quasi-β-norm · is called a β,p-norm. In this case, every quasi-β-Banach space is said to be a β,p-Banach space. A result of the Aoki-Rolewicz theorem [37] shows that every quasinorm can be equivalent to a p-norm, for some p.A main tool of this section is the upcoming fixed point lemma which has been proved in ([38], Lemma 3.1).Lemma 10.
Given the fixedj∈−1,1 and a,t∈ℕ with a≥2. Suppose that V is a linear space and W is a β,p-Banach space with β,p-norm ·W. If ϕ:V⟶0,∞ is a function such that there exists an L<1 with ϕajv<Lajtβϕv for all v∈V and g:V⟶W is a mapping satisfying
(36)gav−atgvW≤ϕv,for all v∈V, then there exists a uniquely determined mapping G:V⟶W such that Gav=atGv and
(37)gv−GvW≤1atβ1−Ljϕv,v∈V.
Furthermore, for eachv∈V, we have Gv=liml⟶∞gajlv/ajlt.In the next theorem, we prove the Găvruta stability of (14) in quasi-β-normed spaces.Theorem 11.
Givenj∈−1,1. Let V be a vector space over ℚ and W be a β,p-Banach space. Assume that φ:Vn×Vn⟶ℝ+ is a function such that φmjv1,mjv2≤m2njβLφv1,v2 for all v1,v2∈Vn, where 0<L<1. If a mapping f:Vn⟶W satisfying H2 and
(38)Dfv1,v2W≤φv1,v2,v1,v2∈Vn,then there is a unique solution Q:Vn⟶W of (14) so that
(39)fv−QvW≤11−Lj1m2nβφ~v,v∈Vn,where
(40)φ~v=Mmnβφv,0+φav,bv,whereas M is the modulus of concavity of the norm ·W.Proof.
Settingv2=0 in (38) and applying H2, we have
(41)f~v−mnfvW≤φv,0,for all v1≔v∈Vn, where f~v=f~v1 is defined in (21). Interchanging v1,v2 into av1,bv1=av,bv in (38), we obtain
(42)fmv−mnf~vW≤φav,bv,for all v∈Vn. Multiplying both sides of (41) by mnβ, we get
(43)mnf~v−m2nfvW≤mnβφv,0,for all v∈Vn. It follows from (42), (43), and part (iii) of Definition 9 that
(44)fmv−m2nfvW≤φ~v,for all v∈Vn, where φ~v is defined in (40). By Lemma 10, there exists a mapping Q:Vn⟶W which is unique such that Qmv=m2nQv and
(45)fv−QvW≤11−Lj1m2nβφ~v,v∈Vn.
Lastly, we show thatQ fulfilling (14). Note that Lemma 10 implies that for each v∈Vn, Qv=liml⟶∞fmjlv/m2njl. For each v1,v2∈Vn and l∈ℕ, by (38), we find
(46)Dfmjlv1,mjlv2m2njlW≤m−2njlβφmjlv1,mjlv2≤m−2njlβm2njβLlφv1,v2=Llφv1,v2.
Takingl⟶∞ in the last relation, we observe that DQv1,v2=0 for all v1,v2∈Vn, and therefore, Q fulfills (14).The following corollary is a consequence of Theorem11 when the norm of Dfv1,v2 is controlled by sum of variable norms of v1 and v2 with positive powers.Corollary 12.
LetV be a quasi-α-normed space with quasi-α-norm ·V, and W be a β,p-Banach space with β,p-norm ·W. Let θ and λ be positive numbers with λ≠2nβ/α. If a mapping f:Vn⟶W satisfying
(47)Dfv1,v2W≤θ∑k=12∑l=1nvklVλ,for all v1,v2∈Vn, then there exists a unique solution Q:Vn⟶W of (14) such that
(48)fv−QvW≤θΛm2nβ−mαλ∑l=1nv1lVλ,λ∈0,2nβα,mαλΛθm2nβmαλ−m2nβ∑l=1nv1lVλ,λ∈2nβα,∞,for all v=v1∈Vn, where Λ=Mmnβ+aαλ+bαλ.Proof.
Takingφv1,v2=θ∑k=12∑l=1nvklVλ, the result concludes from Theorem 11.We bring an elementary lemma without proof as follows.Lemma 13.
If a functiong:ℝ⟶ℝ is a continuous and satisfies (1), then it has the form gx=cx2, for all x∈ℝ, where c=g1.It is easily seen that whena=b=1 in (14), then this equation and (3) are the same. In the upcoming result, we extend Lemma 13 for multivariable functions. In fact, we use it to make a counterexample.Proposition 14.
Suppose thatf:ℝn⟶ℝ is a continuous which satisfies (3). Then, f has the form
(49)fr1,⋯,rn=cr12⋯rn2,r1,⋯,rn∈ℝ,where c is a constant in ℝ.Proof.
We first recall from Theorem 2 in [17] that f is a n-quadratic mapping. By induction on n, we proceed the proof. For n=1, (49) holds by Lemma 13. Assume that (49) is valid for a n∈ℕ, and f:ℝn+1⟶ℝ is a continuous n+1-quadratic function. Fix the variables r1,⋯,rn in ℝ. Then, the function r↦fr1,⋯,rn,r is quadratic and continuous, and hence, by Lemma 13, f has the form
(50)fr1,⋯,rn,r=cr2,r∈ℝ,where c is a constant in ℝ. One should note that c depends on r1,⋯,rn, and hence
(51)c=cr1,⋯,rn.
Lettingr=1 in (50) and applying (51), we have
(52)c=cr1,⋯,rn=fr1,⋯,rn,1.
It is known thatf is n+1-quadratic and c is an n-quadratic function. Therefore, by the induction assumption, there exists a real number c0 so that
(53)c=cr1,⋯,rn=c0r12⋯rn2.
It now follows from (50) and (53) that (49) holds for n+1.
Here, we present a nonstable example for the multiquadratic mappings onℝn (see [8]). Indeed, for the case α=β=a=b=1, we show that the assumption λ≠2n can not be eliminated in Corollary 12.Example 1.
Givenn∈ℕ and δ>0. Set μ≔22n−1/24n2n+4nδ. The function ψ:ℝn⟶ℝ is defined via
(54)ψr1,⋯,rn=μ∏j=1nrj2,forallrjwithrj<1,μ,otherwise.
Considerf:ℝn⟶ℝ as a function defined by
(55)fr1,⋯,rn=∑l=0∞ψ2lr1,⋯,2lrn22nl,rj∈ℝ.
Obviously,f is a nonnegative function and moreover is an even function in all components. Additionally, ψ is bounded by μ and continuous. Since f is a uniformly convergent series of continuous functions, it is continuous and bounded. In other words, we get fr1,⋯,rn≤22n/22n−1μ for all r1,⋯,rn∈ℝn. For i∈1,2, take xi=xi1,⋯,xin. We shall prove that
(56)Dfx1,x2≤δ∑i=12∑j=1nxij2n,for all x1,x2∈ℝn. Clearly, (56) holds for x1=x2=0. Let x1,x2∈ℝn with
(57)∑i=12∑j=1nxij2n<122n.
Inequality (57) necessitates that there is N∈ℕ such that
(58)122nN+1<∑i=12∑j=1nxij2n<122nN,and so xij2n<∑i=12∑j=1nxij2n<1/22nN. It follows the last relation that 2Nxij<1 for all i=1,2 and j=1,⋯,n. Hence, 2N−1xij<1. Let y1,y2∈xiji=1,2,j=1,⋯,n. Then 2N−1y1±y2<1. It is known that ψ is multiquadratic function on −1,1n, and hence, Dψ2lx1,2lx2=0 for all l∈0,1,2,⋯,N−1. Now, the last equality and relation (58) imply that
(59)Df2lx1,2lx2∑i=12∑j=1nxij2n≤∑l=N∞Dψ2lx1,2lx222nl∑i=12∑j=1nxij2n≤∑l=0∞μ2n+4n22nl+N∑i=12∑j=1nxij2n≤μ2n+4n∑l=0∞122nl≤μ2n+4n22n22n22n−1=μ2n+4n24n22n−1=δ,for all x1,x2∈ℝn. Hence, (56) is valid for case (57). If ∑i=12∑j=1nxij2n≥1/22n, then
(60)Df2lx1,2lx2∑i=12∑j=1nxij2n≤22n22n22n−1μ2n+4n=δ.
Therefore,f satisfies in (56) for all x1,x2∈ℝn. Assume that there exists a number b∈0,∞ and a multiquadratic function Q:ℝn⟶ℝ for which the inequality fr1,⋯,rn−Qr1,⋯,rn<b∏j=1nrj2 is valid for all r1,⋯,rn∈ℝn. An application of Proposition 14 shows that there is a constant c∈ℝ such that Qr1,⋯,rn=c∏j=1nrj2, and hence
(61)fr1,⋯,rn≤c+b∏j=1nrj2,r1,⋯,rn∈ℝn.
Furthermore, chooseN∈ℕ such that Nμ>c+b. Take r=r1,⋯,rn∈ℝn in which rj∈0,1/2N−1 for all j∈1,⋯,n, then 2lrj∈0,1 for all l=0,1,⋯,N−1. Therefore
(62)fr1,⋯,rn=∑l=0∞ψ2lr1,⋯,2lr222nl≥∑l=0N−1μ22nl∏j=1nrj222nl=Nμ∏j=1nrj2>c+b∏j=1nrj2,which is a contradiction with (61).We close the paper by an alternative stability result for equation (14) as follows.Corollary 15.
LetV be a quasi-α-normed space with quasi-α-norm ·V and W be a β,p-Banach space with β,p-norm ·W. Suppose λil>0 for i∈1,2 and l∈1,⋯,n with λ=λ∗+λ•≠2nβ/α, where λ∗=∑l=1nλ1l and λ•=∑l=1nλ2l. If a mapping f:Vn⟶W fulfilling the inequality
(63)Dfv1,v2W≤θ∏i=12∏l=1nvilVλil,for all v1,v2∈Vn, then there exists a unique solution Q:Vn⟶W of (14) so that
(64)fv−QvW≤θΩm2nβ−mαλ∏l=1nv1lV2λ1l,λ∈0,2nβα,mαλΩθm2nβmαλ−m2nβ∏l=1nv1lV2λ1l,λ∈2nβα,∞,for all v=v1∈Vn, where Ω=Maαλ∗bαλ•.Proof.
Settingφv1,v2=θ∏i=12∏l=1nvilVλil in Theorem 11, one can obtain the desired results.


## 4. Conclusion

In this paper, by using Euler-Lagrange type quadratic functional equations, we have defined the multi-Euler-Lagrange quadratic mappings and have studied the structure of such mappings. Indeed, we have described the multi-Euler-Lagrange quadratic mapping as an equation. In continuation, we have shown that some fixed point theorems can be applied to prove the Hyers-Ulam stability version of multi-Euler-Lagrange quadratic functional equations in the setting of quasi-β-normed and Banach spaces. In the last part, we have brought an example which shows that such functional equations can be nonstable in the some cases.The current work provides guidelines for further research and proposals for new directions and open problems with relevant discussions. Here, we give some questions and information on the connections between the fixed point theory and the Hyers-Ulam stability.(1)
Which equation can describe the multi-Euler-Lagrange cubic mappings defined in [31]? Are these mappings stable on various Banach spaces? Can the known fixed point methods be useful to prove their Hyers-Ulam stability?(2)
Definition of the multiadditive-quartic mappings by using [14] as a system of n functional equations. The characterization of such mappings and discussion about their stability via a fixed point approach(3)
Applying the functional equations indicated in [5, 12, 13, 34], we can generalize such mappings and equations to multiple variables



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