# Generalized Stability of Euler-Lagrange Quadratic Functional Equation

Authors: Hark-Mahn Kim; Min-Young Kim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The main goal of this paper is the investigation of the general solution and the generalized Hyers-Ulam stability theorem of the following Euler-Lagrange type quadratic functional equationf(ax+by)+af(x-by)=(a+1)b2f(y)+a(a+1)f(x), in (β,p)-Banach space, where a,b are fixed rational numbers such that a≠-1,0 and b≠0.


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## Body

## 1. Introduction

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.LetG be a group and let G′ be a metric group with metric ρ(·,·). Given ϵ>0, does there exist a δ>0 such that if f:G→G′ satisfies ρ(f(xy),f(x)f(y))<δ for all x,y∈G, then a homomorphism h:G→G′ exists with ρ(f(x),h(x))<ϵ for all x∈G?In 1941, the first result concerning the stability of functional equations was presented by Hyers [2]. He has answered the question of Ulam for the case where G1 and G2 are Banach spaces.LetE1 and E2 be real vector spaces. A function f:E1→E2 is called a quadratic function if and only if f is a solution function of the quadratic functional equation
(1.1)f(x+y)+f(x-y)=2f(x)+2f(y).It is well known that a functionf between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function B such that f(x)=B(x,x) for all x, where the mapping B is given by B(x,y)=(1/4)(f(x+y)-f(x-y)). See [3, 4] for the details. The Hyers-Ulam stability of the quadratic functional equation (1.1) was first proved by Skof [5] for functions f:E1→E2, where E1 is a normed space and E2 is a Banach space. Cholewa [6] demonstrated that Skof’s theorem is also valid if E1 is replaced by an Abelian group G. Assume that a function f:G→E satisfies the inequality
(1.2)‖f(x+y)+f(x-y)-2f(x)-2f(y)‖≤δ,
for some δ≥0 and for all x,y∈G. Then there exists a unique quadratic function Q:G→E such that
(1.3)‖f(x)-Q(x)‖≤δ2,
for all x∈G. Czerwik [7] proved the Hyers-Ulam-Rassias stability of quadratic functional equation (1.1). Let E1 and E2 be a real normed space and a real Banach space, respectively, and let p≠2 be a positive constant. If a function f:E1→E2 satisfies the inequality
(1.4)‖f(x+y)+f(x-y)-2f(x)-2f(y)‖≤ϵ(‖x‖p+‖y‖p),
for some ϵ>0 and for all x,y∈E1, then there exists a unique quadratic function q:E1→E2 such that
(1.5)‖f(x)-q(x)‖≤2ϵ|4-2p|‖x‖p,
for all x∈E1. Furthermore, according to the theorem of Borelli and Forti [8], we know the following generalization of stability theorem for quadratic functional equation. Let G be an Abelian group and E a Banach space, and let f:G→E be a mapping with f(0)=0 satisfying the inequality
(1.6)‖f(x+y)+f(x-y)-2f(x)-2f(y)‖≤φ(x,y),
for all x,y∈G. Assume that one of the series
(1.7)Φ(x,y)∶={∑k=0∞122(k+1)φ(2kx,2ky)<∞,∑k=0∞22kφ(x2(k+1),y2(k+1))<∞,
then there exists a unique quadratic function Q:G→E such that
(1.8)‖f(x)-Q(x)‖≤Φ(x,x),
for all x∈G. During the last three decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability of several functional equations, and there are many interesting results concerning this problem [9–13].The notion of quasi-β-normed space was introduced by Rassias and Kim in [14]. This notion is a generalization of that of quasi-normed space. We consider some basic concepts concerning quasi-β-normed space. We fix a real number β with 0<β≤1 and let 𝕂 denote either ℝ or ℂ. Let X be a linear space over 𝕂. A quasi-β-norm ∥·∥ is a real-valued function on X satisfying the following:(1)
∥x∥≥0 for all x∈X and ∥x∥=0 if and only if x=0,(2)
∥λx∥=|λ|β∥x∥ for all λ∈𝕂 and all x∈X,(3)
there is a constantK≥1 such that ∥x+y∥≤K(∥x∥+∥y∥) for all x,y∈X.The pair(X,∥·∥) is called a quasi-β-normed space if ∥·∥ is a quasi-β-norm on X. The smallest possible K is called the modulus of concavity of ∥·∥. A quasi-β-Banach space is a complete quasi-β-normed space. A quasi-β-norm ∥·∥ is called a (β,p)-norm (0<p≤1) if
(1.9)‖x+y‖p≤‖x‖p+‖y‖p,
for all x,y∈X. In this case, the quasi-β-Banach space is called a (β,p)-Banach space. We observe that if x1,x2,…,xn are nonnegative real numbers, then
(1.10)(∑i=1nxi‍)p≤∑i=1nxip‍,
where 0<p≤1 [15].J. M. Rassias investigated the stability of Ulam for the Euler-Lagrange functional equation(1.11)f(ax+by)+f(bx-ay)=(a2+b2)[f(x)+f(y)]
in the paper of [16]. Gordji and Khodaei investigated the generalized Hyers-Ulam stability of other Euler-Lagrange quadratic functional equations [17]. Jun et al. [18] introduced a new quadratic Euler-Lagrange functional equation
(1.12)f(ax+y)+af(x-y)=(a+1)f(y)+a(a+1)f(x),
for any fixed a∈ℤ with a≠0,-1, which was a modified and instrumental equation for [19], and solved the generalized stability of (1.12). Now, we improve the functional equation (1.12) to the following functional equations:
(1.13)f(ax+by)+af(x-by)=(a+1)f(by)+a(a+1)f(x),(1.14)f(ax+by)+af(x-by)=(a+1)b2f(y)+a(a+1)f(x),
for any fixed rational numbers a,b∈ℚ with a≠0,-1 and b≠0, which are generalized versions of (1.12). In this paper, we establish the general solution of (1.13) and (1.14) and then prove the generalized Hyers-Ulam stability of (1.13) and (1.14). We remark that there are some interesting papers concerning the stability of functional equations in quasi-Banach spaces [15, 20–23] and quasi-β-normed spaces [14, 24, 25].


## 2. General Solution of (1.13) and (1.14)

First, we present the general solution of (1.14) in the class of all functions between vector spaces.Lemma 2.1.
LetX and Y be vector spaces over 𝕂. Then a mapping f:X→Y is a solution of the functional equation (1.12) for any fixed rational number a∈ℚ with a≠0,-1 if and only if f is quadratic.Proof.
See the same proof in [18].Lemma 2.2.
LetX and Y be vector spaces over 𝕂. Then a mapping f:X→Y is a solution of the functional equation (1.13) if and only if f is quadratic.Proof.
We assume that a mappingf:X→Y satisfies the functional equation (1.13). Letting by=u in (1.13), then (1.13) is equivalent to (1.12). Then by Lemma 2.1, f is quadratic. Conversely, if f is quadratic, then it is obvious that f satisfies (1.13).Theorem 2.3.
LetX and Y be vector spaces over 𝕂. Then a mapping f:X→Y with f(0)=0 satisfies the functional equation (1.14) if and only if f is quadratic. In this case, f(ax)=a2f(x) and f(bx)=b2f(x) hold for all x∈X.Proof.
We assume that a mappingf:X→Y with f(0)=0 satisfies the functional equation (1.14). Then replacing y in (1.14) by 0, we also get the equality f(ax)=a2f(x) for all x∈X. Now, we decompose f into the even part and the odd part by setting
(2.1)fe(x)=12(f(x)+f(-x)),fo(x)=12(f(x)-f(-x)),
for all x∈X. Then fe and fo satisfy the functional equation (1.14). Therefore, we may assume without loss of generality that f is even and satisfies (1.14) for all x,y∈X. If we replace x in (1.14) by 0, then we get
(2.2)f(by)+af(-by)=(a+1)b2f(y),
for all y∈X. From this equality, we have f(by)=b2f(y) for all y∈X. Therefore, (1.14) implies (1.13) for all x,y∈X. By Lemma 2.2,  f is quadratic.
Now, we assume thatf is odd and satisfies (1.14) for all x,y∈X. For the case a=1, we have
(2.3)f(x+by)+f(x-by)=2b2f(y)+2f(x),
for all x,y∈X. Setting x by 0 in (2.3), one obtains f≡0. Let a≠1. Replacing x by 0 in (1.14), we have
(2.4)(1-a)f(by)=(a+1)b2f(y),
for all y∈X. From (1.14) and (2.4), we get
(2.5)f(ax+by)+af(x-by)=(1-a)f(by)+a(a+1)f(x),
for all x,y∈X. Putting by=u in (2.5), then we obtain
(2.6)f(ax+u)+af(x-u)=(1-a)f(u)+a(a+1)f(x),
for all x,u∈X. Replacing u by au in (2.6), we get
(2.7)f(ax+au)+af(x-au)=(1-a)f(au)+a(a+1)f(x),
for all x,u∈X. Since f(ax)=a2f(x), (2.7) yields
(2.8)af(x+u)+f(x-au)=(1-a)af(u)+(a+1)f(x),
for all x,u∈X. Interchanging x and u in (2.8), we have by oddness of f(2.9)-f(ax-u)+af(x+u)=(1-a)af(x)+(a+1)f(u),
for all x,u∈X. Replacing u by -u in (2.6), we get
(2.10)f(ax-u)+af(x+u)=-(1-a)f(u)+a(a+1)f(x),
for all x,u∈X. Adding (2.9) and (2.10) side by side, this leads to
(2.11)f(x+u)=f(x)+f(u),
for all x,u∈X. Therefore, f is additive and so f(ax)=af(x) for all x∈X and for any odd function satisfying (1.14). Using the equality f(ax)=a2f(x), we obtain f(x)=0 for all x∈X. Therefore, f(x)=fe(x)+fo(x) is a quadratic mapping, as desired.
Conversely, iff is quadratic, then it is obvious that f satisfies (1.14).We note thatf(0)=0 if a+b2≠1 and f satisfies (1.14).


## 3. Generalized Stability of (1.14) for a≠1

For convenience, we use the following abbreviation: for any fixed rational numbersa and b with a≠-1,0,1 and b≠0,
(3.1)Df(x,y)∶=f(ax+by)+af(x-by)-(a+1)b2f(y)-a(a+1)f(x),
for all x,y∈X, which is called the approximate remainder of the functional equation (1.14) and acts as a perturbation of the equation.From now on, letX be a vector space, and let Y be a (β,p)-Banach space unless we give any specific reference. We will investigate the generalized Hyers-Ulam stability problem for the functional equation (1.14). Thus, we find some conditions such that there exists a true quadratic function near an approximate solution of (1.14).Theorem 3.1.
Letφ:X×X→[0,∞) be a function such that
(3.2)Φ(x)∶=∑n=0∞‍1|a|2βnp(φ(anx,0))p<∞,(3.3)lim⁡n→∞1|a|2βnφ(anx,any)=0,
for all x,y∈X. Suppose that a function f:X→Y with f(0)=0 satisfies
(3.4)‖Df(x,y)‖Y≤φ(x,y),
for all x,y∈X. Then there exists a unique quadratic function Q:X→Y satisfying
(3.5)‖f(x)-Q(x)‖Y≤1|a|2β[Φ(x)]1/p,
for all x∈X. The function Q is given by
(3.6)Q(x)=lim⁡k→∞1a2kf(akx),
for all x∈X.Proof.
Lettingy by 0 in (3.4), we get
(3.7)‖f(ax)-a2f(x)‖Y≤φ(x,0),
for all x∈X. Multiplying both sides by 1/|a|2β in (3.7), we have
(3.8)‖1a2f(ax)-f(x)‖Y≤1|a|2βφ(x,0),
for all x∈X. Replacing x by anx and multiplying both sides by 1/|a|2nβ in (3.8), we have
(3.9)‖1a2(n+1)f(an+1x)-1a2nf(anx)‖Y≤1|a|2β(n+1)φ(anx,0),
for all x∈X. Next we show that the sequence {(1/a2n)f(anx)} is a Cauchy sequence. For any m,n∈ℕ, m>n≥0, and x∈X, it follows from (3.9) that
(3.10)‖1a2(m+1)f(am+1x)-1a2nf(anx)‖Yp=‖∑i=nm‍1a2(i+1)f(ai+1x)-1a2if(aix)‖Yp≤∑i=nm‍‖1a2(i+1)f(ai+1x)-1a2if(aix)‖Yp≤∑i=nm‍1|a|2βp(i+1)(φ(aix,0))p=1|a|2βp∑i=nm‍1|a|2βpi(φ(aix,0))p,
for all x∈X. It follows from (3.2) and (3.10) that the sequence {(1/a2n)f(anx)} is a Cauchy sequence in Y for all x∈X. Since Y is a (β,p)-Banach space, the sequence {(1/a2n)f(anx)} converges for all x∈X. Therefore, we can define a mapping Q:X→Y by
(3.11)Q(x)=lim⁡n→∞1a2nf(anx),
for all x∈X. Taking m→∞ and n=0 in (3.10), we have
(3.12)‖Q(x)-f(x)‖Yp≤1|a|2βp∑i=0∞‍1|a|2βpi(φ(aix,0))p=1|a|2βpΦ(x),
for all x∈X. Therefore,
(3.13)‖Q(x)-f(x)‖Y≤1|a|2β[Φ(x)]1/p,
for all x∈X, that is, the mapping Q satisfies (3.5). It follows from (3.3) and (3.4) that
(3.14)‖DQ(x,y)‖Y=lim⁡n→∞‖1a2nDf(anx,any)‖Y=lim⁡n→∞1|a|2βn‖Df(anx,any)‖Y≤lim⁡n→∞1|a|2βnφ(anx,any)=0,
for all x,y∈X. Therefore, Q satisfies (1.14), and so the function Q is quadratic.
To prove the uniqueness of the quadratic functionQ, let us assume that there exists a quadratic function Q′:X→Y satisfying the inequality (3.5). Then we have
(3.15)‖Q(x)-Q′(x)‖Yp=‖1a2nQ(anx)-1a2nQ′(anx)‖Yp=1a2nβp‖Q(anx)-Q′(anx)‖Yp≤1a2nβp(‖Q(anx)-f(anx)‖Yp+‖Q′(anx)-f(anx)‖Yp)≤1|a|2nβp2|a|2βpΦ(anx)=2|a|2βp(n+1)∑i=0∞‍1|a|2βpi(φ(ai+nx,0))p=2|a|2βp∑i=n∞‍1|a|2βpi(φ(aix,0))p,
for all x∈X and n∈ℕ. Therefore, letting n→∞, one has Q(x)-Q′(x)=0 for all x∈X, completing the proof of uniqueness.In the following corollary, we get a stability result of (1.14).Corollary 3.2.
LetX be a quasi-α-normed space for fixed real number α with 0<α≤1. Let θ1,θ2,θ3,α1,α2,γ1,γ2 be positive reals such that either (1)  |a|>1, (α1+α2)α<2β, and γiα<2β or (2)  |a|<1, (α1+α2)α>2β, and γiα>2β, for i=1,2. Assume that a function f:X→Y with f(0)=0 satisfies the inequality
(3.16)‖Df(x,y)‖Y≤θ1‖x‖α1‖y‖α2+θ2‖x‖γ1+θ3‖y‖γ2,
for all x,y∈X. Then there exists a unique quadratic function Q:X→Y which satisfies the inequality
(3.17)‖f(x)-Q(x)‖Y≤θ2‖x‖γ1(|a|2βp-|a|γ1αp)1/p,
for all x∈X. The function Q is given by
(3.18)Q(x)=lim⁡k→∞f(akx)a2k,
for all x∈X.Proof.
Letφ(x,y)=θ1∥x∥α1∥y∥α2+θ2∥x∥γ1+θ3∥y∥γ2. Then
(3.19)Φ(x)=∑n=0∞‍1|a|2βnp(φ(anx,0))p=∑n=0∞‍1|a|2βnpθ2p‖anx‖γ1p=θ2p‖x‖γ1p∑n=0∞‍|a|(γ1α-2β)np<∞,(3.20)lim⁡n→∞1|a|2βnφ(anx,any)=lim⁡n→∞1|a|2βn[θ1(‖anx‖α1‖any‖α2)+θ2‖anx‖γ1+θ3‖any‖γ2]=θ1(‖x‖α1‖y‖α2)lim⁡n→∞|a|((α1+α2)α-2β)n+θ2‖x‖γ1lim⁡n→∞|a|(γ1α-2β)n+θ3‖y‖γ2lim⁡n→∞|a|(γ2α-2β)n=0.
By Theorem 3.1, there exists a unique quadratic mapping Q:X→Y such that
(3.21)‖f(x)-Q(x)‖Y≤1|a|2β[Φ(x)]1/p=θ2‖x‖γ1|a|2β(∑n=0∞|a|(γ1α-2β)np‍)1/p=θ2‖x‖γ1(|a|2βp-|a|γ1αp)1/p,
for all x∈X.Theorem 3.3.
Letφ:X×X→[0,∞) be a function such that
(3.22)Ψ(x)∶=∑n=0∞‍|a|2βnp(φ(xan+1,0))p<∞,(3.23)lim⁡n→∞|a|2βnφ(xan,yan)=0,
for all x,y∈X. Suppose that a function f:X→Y with f(0)=0 satisfies
(3.24)‖Df(x,y)‖Y≤φ(x,y),
for all x,y∈X. Then there exists a unique quadratic function Q:X→Y satisfying
(3.25)‖f(x)-Q(x)‖Y≤[Ψ(x)]1/p,
for all x∈X. The function Q is given by
(3.26)Q(x)=lim⁡k→∞a2kf(xak),
for all x∈X.Proof.
Lettingy by 0 in (3.24), we get
(3.27)‖f(ax)-a2f(x)‖Y≤φ(x,0),
for all x∈X. Replacing x by x/a in (3.27), we have
(3.28)‖f(x)-a2f(xa)‖Y≤φ(xa,0),
for all x∈X. Replacing x by x/an and multiplying both sides by |a|2βn in (3.28), we have
(3.29)‖a2nf(xan)-a2(n+1)f(xan+1)‖Y≤|a|2βnφ(xan+1,0),
for all x∈X. Next we show that the sequence {a2nf(x/an)} is a Cauchy sequence. For any m,n∈ℕ, m>n≥0, and x∈X, it follows from (3.29) that
(3.30)‖a2nf(xan)-a2(m+1)f(xam+1)‖Yp=‖∑i=nm‍a2if(xai)-a2(i+1)f(xai+1)‖Yp≤∑i=nm‍‖a2if(xai)-a2(i+1)f(xai+1)‖Yp≤∑i=nm‍|a|2βpi(φ(xai+1,0))p.
It follows from (3.22) and (3.30) that the sequence {a2nf(x/an)} is a Cauchy sequence in Y for all x∈X. Since Y is a (β,p)-Banach space, the sequence {a2nf(x/an)} converges for all x∈X. Therefore, we can define a mapping Q:X→Y by
(3.31)Q(x)=lim⁡n→∞a2nf(xan),
for all x∈X. The rest of the proof is similar to the corresponding proof of Theorem 3.1.Corollary 3.4.
LetX be a quasi-α-normed space for fixed real number α with 0<α≤1. Let θ1,θ2,θ3,α1,α2,γ1,γ2 be positive reals such that either (1)  |a|>1, (α1+α2)α>2β, and γiα>2β or (2)  |a|<1, (α1+α2)α<2β, and γiα<2β, for i=1,2. Assume that a function f:X→Y with f(0)=0 satisfies the inequality
(3.32)‖Df(x,y)‖Y≤θ1‖x‖α1‖y‖α2+θ2‖x‖γ1+θ3‖y‖γ2,
for all x,y∈X. Then there exists a unique quadratic function Q:X→Y which satisfies the inequality
(3.33)‖f(x)-Q(x)‖Y≤θ2‖x‖γ1(|a|γ1αp-|a|2βp)1/p,
for all x∈X. The function Q is given by
(3.34)Q(x)=lim⁡k→∞a2kf(xak),
for all x∈X.Proof.
Letφ(x,y)=θ1∥x∥α1∥y∥α2+θ2∥x∥γ1+θ3∥y∥γ2. Then φ satisfies the conditions (3.22) and (3.23). Applying Theorem 3.3, we obtain the results, as desired.


## 4. Generalized Stability of (1.13)

For convenience, we use the following abbreviation: for any fixed rational numbersa and b with a≠-1,0 and b≠0,
(4.1)Ef(x,y)∶=f(ax+by)+af(x-by)-(a+1)f(by)-a(a+1)f(x),
for all x,y∈X, which is called the approximate remainder of the functional equation (1.13) and acts as a perturbation of the equation.We will investigate the generalized Hyers-Ulam stability problem for the functional equation (1.13).Theorem 4.1.
Letφ:X×X→[0,∞) be a function such that
(4.2)Φ(x)∶=∑n=0∞‍1|a+1|2βnp(φ((a+1)nx,(a+1)nxb))p<∞,(4.3)lim⁡n→∞1|a+1|2βnφ((a+1)nx,(a+1)ny)=0,
for all x,y∈X. Suppose that a function f:X→Y with f(0)=0 satisfies
(4.4)‖Ef(x,y)‖Y≤φ(x,y),
for all x,y∈X. Then there exists a unique quadratic function Q:X→Y satisfying
(4.5)‖f(x)-Q(x)‖Y≤1|a+1|2β[Φ(x)]1/p,
for all x∈X. The function Q is given by
(4.6)Q(x)=lim⁡k→∞1(a+1)2kf((a+1)kx),
for all x∈X.Proof.
Replacingx by by in (4.4), we get
(4.7)‖f((a+1)by)-(a+1)2f(by)‖Y≤φ(by,y),
for all y∈X. Letting by be x in (4.7), we have
(4.8)‖f((a+1)x)-(a+1)2f(x)‖Y≤φ(x,xb),
for all x∈X. Multiplying both sides by 1/|a+1|2β in (4.8), we have
(4.9)‖1(a+1)2f((a+1)x)-f(x)‖Y≤1|a+1|2βφ(x,xb),
for all x∈X. Replacing x by (a+1)ix and multiplying both sides by 1/|a+1|2iβ in (4.9), we have
(4.10)‖1(a+1)2(i+1)f((a+1)i+1x)-1(a+1)2if((a+1)ix)‖Y≤1|a+1|2β(i+1)φ((a+1)ix,(a+1)ixb),
for all x∈X. Next we show that the sequence {(1/(a+1)2n)f((a+1)nx)} is a Cauchy sequence. For any m,n∈ℕ, m>n≥0, and x∈X, it follows from (4.10) that
(4.11)‖1(a+1)2(m+1)f((a+1)m+1x)-1(a+1)2nf((a+1)nx)‖Yp=‖∑i=nm‍1(a+1)2(i+1)f((a+1)i+1x)-1(a+1)2if((a+1)ix)‖Yp≤∑i=nm‍‖1(a+1)2(i+1)f((a+1)i+1x)-1(a+1)2if((a+1)ix)‖Yp≤∑i=nm‍1|a+1|2βp(i+1)(φ((a+1)ix,(a+1)ixb))p=1|a+1|2βp∑i=nm‍1|a+1|2βpi(φ((a+1)ix,(a+1)ixb))p,
for all x∈X. It follows from (4.2) and (4.11) that the sequence {f((a+1)nx)/(a+1)2n} is a Cauchy sequence in Y for all x∈X. Since Y is a (β,p)-Banach space, the sequence {f((a+1)nx)/(a+1)2n} converges for all x∈X. Therefore, we can define a mapping Q:X→Y by
(4.12)Q(x)=lim⁡n→∞1(a+1)2nf((a+1)nx),
for all x∈X. The rest of the proof is similar to the corresponding proof of Theorem 3.1.In the following corollary, we get a stability result of (1.13).Corollary 4.2.
LetX be a quasi-α-normed space for fixed real number α with 0<α≤1. Let θ1,θ2,θ3,α1,α2,γ1,γ2 be positive reals such that either (1)  |a+1|>1, (α1+α2)α<2β, and γiα<2β or (2)  |a+1|<1, (α1+α2)α>2β, and γiα>2β, for i=1,2. Assume that a function f:X→Y with f(0)=0 satisfies the inequality
(4.13)‖Ef(x,y)‖Y≤θ1‖x‖α1‖y‖α2+θ2‖x‖γ1+θ3‖y‖γ2,
for all x,y∈X. Then there exists a unique quadratic function Q:X→Y which satisfies the inequality
(4.14)‖f(x)-Q(x)‖Y≤{θ1p‖x‖(α1+α2)p|b|αα2p(|a+1|2βp-|a+1|(α1+α2)αp)Q+θ2p‖x‖γ1p|a+1|2βp-|a+1|γ1αp+θ3p‖x‖γ2p|b|γ2αp(|a+1|2βp-|a+1|γ2αp)}1/p,
for all x∈X. The function Q is given by
(4.15)Q(x)=lim⁡k→∞1(a+1)2kf((a+1)kx),
for all x∈X.Proof.
Letφ(x,y)=θ1∥x∥α1∥y∥α2+θ2∥x∥γ1+θ3∥y∥γ2. Then φ satisfies the conditions (4.2) and (4.3). By Theorem 4.1, there exists a unique quadratic mapping Q:X→Y such that
(4.16)‖f(x)-Q(x)‖Y≤1|a+1|2β[∑n=0∞‍1|a+1|2βnp(φ((a+1)nx,(a+1)nxb))p]1/p≤{θ1p‖x‖(α1+α2)p|b|αα2p(|a+1|2βp-|a+1|(α1+α2)αp)(x)+θ2p‖x‖γ1p|a+1|2βp-|a+1|γ1αp+θ3p‖x‖γ2p|b|γ2αp(|a+1|2βp-|a+1|γ2αp)}1/p,
for all x∈X.Theorem 4.3.
Letφ:X×X→[0,∞) be a function such that
(4.17)Ψ(x)∶=∑n=0∞‍|a+1|2βnp(φ(x(a+1)n+1,x(a+1)n+1b))p<∞,lim⁡n→∞|a+1|2βnφ(x(a+1)n,y(a+1)n)=0,
for all x,y∈X. Suppose that a function f:X→Y with f(0)=0 satisfies
(4.18)‖Ef(x,y)‖Y≤φ(x,y),
for all x,y∈X. Then there exists a unique quadratic function Q:X→Y satisfying
(4.19)‖f(x)-Q(x)‖Y≤[Ψ(x)]1/p,
for all x∈X. The function Q is given by
(4.20)Q(x)=lim⁡k→∞(a+1)2kf(x(a+1)k),
for all x∈X.Proof.
Replacingx by x/(a+1) in (4.8), we have
(4.21)‖f(x)-(a+1)2f(xa+1)‖Y≤φ(xa+1,x(a+1)b),
for all x∈X. The rest of the proof is similar to the corresponding proof of Theorem 3.3.Corollary 4.4.
LetX be a quasi-α-normed space for fixed real number α with 0<α≤1. Let θ1,θ2,θ3,α1,α2,γ1,γ2 be positive reals such that either (1)  |a+1|>1 and (α1+α2)α>2β, γiα>2β or (2)  |a+1|<1 and (α1+α2)α<2β, γiα<2β, for i=1,2. Assume that a function f:X→Y with f(0)=0 satisfies the inequality
(4.22)‖Ef(x,y)‖Y≤θ1‖x‖α1‖y‖α2+θ2‖x‖γ1+θ3‖y‖γ2,
for all x,y∈X. Then there exists a unique quadratic function Q:X→Y which satisfies the inequality
(4.23)‖f(x)-Q(x)‖Y≤{θ1p‖x‖(α1+α2)p|b|αα2p(|a+1|(α1+α2)αp-|a+1|2βp)Q+θ2p‖x‖γ1p|a+1|γ1αp-|a+1|2βp+θ3p‖x‖γ2p|b|αγ2p(|a+1|γ2αp-|a+1|2βp)}1/p,
for all x∈X. The function Q is given by
(4.24)Q(x)=lim⁡k→∞(a+1)2kf(x(a+1)k),
for all x∈X.Proof.
Letφ(x,y)=θ1∥x∥α1∥y∥α2+θ2∥x∥γ1+θ3∥y∥γ2. Then φ satisfies the conditions (4.17). Applying Theorem 4.3, we obtain the results, as desired.



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*Source: 219435-2012-08-15.xml*