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 equation f(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.
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.
Let G 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.
Let E1 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 function f 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:
∥x∥≥0 for all x∈X and ∥x∥=0 if and only if x=0,
∥λx∥=|λ|β∥x∥ for all λ∈𝕂 and all x∈X,
there is a constant K≥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.
Let X 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.
Let X 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 mapping f: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.
Let X 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 mapping f: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 that f 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, if f is quadratic, then it is obvious that f satisfies (1.14).
We note that f(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 numbers a 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, let X 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.
Let φ:X×X→[0,∞) be a function such that
(3.2)Φ(x)∶=∑n=0∞1|a|2βnp(φ(anx,0))p<∞,(3.3)limn→∞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)=limk→∞1a2kf(akx),
for all x∈X.
Proof.
Letting y 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)=limn→∞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=limn→∞‖1a2nDf(anx,any)‖Y=limn→∞1|a|2βn‖Df(anx,any)‖Y≤limn→∞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 function Q, 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.
Let X 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)=limk→∞f(akx)a2k,
for all x∈X.
Proof.
Let φ(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)limn→∞1|a|2βnφ(anx,any)=limn→∞1|a|2βn[θ1(‖anx‖α1‖any‖α2)+θ2‖anx‖γ1+θ3‖any‖γ2]=θ1(‖x‖α1‖y‖α2)limn→∞|a|((α1+α2)α-2β)n+θ2‖x‖γ1limn→∞|a|(γ1α-2β)n+θ3‖y‖γ2limn→∞|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.
Let φ:X×X→[0,∞) be a function such that
(3.22)Ψ(x)∶=∑n=0∞|a|2βnp(φ(xan+1,0))p<∞,(3.23)limn→∞|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)=limk→∞a2kf(xak),
for all x∈X.
Proof.
Letting y 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)=limn→∞a2nf(xan),
for all x∈X. The rest of the proof is similar to the corresponding proof of Theorem 3.1.
Corollary 3.4.
Let X 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)=limk→∞a2kf(xak),
for all x∈X.
Proof.
Let φ(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 numbers a 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.
Let φ: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)limn→∞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)=limk→∞1(a+1)2kf((a+1)kx),
for all x∈X.
Proof.
Replacing x 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)=limn→∞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.
Let X 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)=limk→∞1(a+1)2kf((a+1)kx),
for all x∈X.
Proof.
Let φ(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.
Let φ: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<∞,limn→∞|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)=limk→∞(a+1)2kf(x(a+1)k),
for all x∈X.
Proof.
Replacing x 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.
Let X 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)=limk→∞(a+1)2kf(x(a+1)k),
for all x∈X.
Proof.
Let φ(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.
Acknowledgment
This study was supported by the Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (no. 2012R1A1A2008139).
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