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volume 3, issue 3, article 33, 2002.

Received 11 January, 2002;

accepted 19 February, 2002.

Communicated by:Th. M. Rassias

Abstract Contents

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Journal of Inequalities in Pure and Applied Mathematics

ON THE HYERS-ULAM STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS

ICK-SOON CHANG AND HARK-MAHN KIM

Department of Mathematics, Chungnam National University, Taejon 305-764, Korea

EMail:ischang@@math.cnu.ac.kr EMail:hmkim@@math.cnu.ac.kr

c

2000Victoria University ISSN (electronic): 1443-5756 002-02

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Abstract

In this paper, we obtain the general solution and the generalized Hyers-Ulam stability for quadratic functional equationsf(2x+y) +f(2x−y) =f(x+y) + f(x−y) + 6f(x)andf(2x+y) +f(x+ 2y) = 4f(x+y) +f(x) +f(y).

2000 Mathematics Subject Classification:39B22, 39B52, 39B72.

Key words: Hyers-Ulam-Rassias stability; Quadratic function

Contents

1 Introduction. . . 3

2 Solution of (1.3), (1.4) . . . 6

3 Stability of (1.3) . . . 9

4 Stability of (1.4) . . . 19 References

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1. Introduction

In 1940, S.M. Ulam [20] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of homomorphisms:

LetG1be a group and letG2be a metric group with the metricd(·,·). Given > 0, does there exist a δ > 0such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1, then there exists a homomorphismH :G1 →G2 withd(h(x), H(x))< for allx∈G1?

In other words, we are looking for situations when the homomorphisms are stable, i.e., if a mapping is almost a homomorphism, then there exists a true ho- momorphism near it. If we turn our attention to the case of functional equations, we can ask the question: When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation.

The case of approximately additive functions was solved by D. H. Hyers [9]

under the assumption that G1 andG2 are Banach spaces. In 1978, a general- ized version of the theorem of Hyers for approximately linear mappings was given by Th. M. Rassias [17]. During the last decades, the stability problems of several functional equations have been extensively investigated by a number of authors [2,6, 11,15]. The terminology generalized Hyers-Ulam stability orig- inates from these historical backgrounds. These terminologies are also applied to the case of other functional equations. For more detailed definitions of such terminologies, we can refer to [10,12,18].

The functional equation

(1.1) f(x+y) +f(x−y) = 2f(x) + 2f(y)

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is related to a symmetric biadditive function ([1], [16]). It is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a functionf between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive functionBsuch thatf(x) = B(x, x) for allx(see [1], [16]). The biadditive functionB is given by

(1.2) B(x, y) = 1

4(f(x+y)−f(x−y)).

A Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by F. Skof for functionsf :E1 →E2, whereE1 is a normed space andE2 a Banach space (see [19]). P. W. Cholewa [3] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an abelian group.

In the paper [4], S. Czerwik proved the Hyers-Ulam-Rassias stability of the quadratic functional equation (1.1). A. Grabiec [8] has generalized these results mentioned above. K. W. Jun and Y. H. Lee [13] proved the Hyers-Ulam-Rassias stability of the pexiderized quadratic equation (1.1).

Now, we introduce the following functional equations, which are somewhat different from (1.1),

f(2x+y) +f(2x−y) = f(x+y) +f(x−y) + 6f(x), (1.3)

f(2x+y) +f(x+ 2y) = 4f(x+y) +f(x) +f(y).

(1.4)

In this paper, we establish the general solution and the generalized Hyers- Ulam stability problem for the equations (1.3), (1.4), which are equivalent to (1.1). It is significant for us to decrease the possible estimator of the stability

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problem for the functional equations. This work is possible if we consider the stability problem in the sense of Hyers-Ulam-Rassias for the functional equa- tions (1.3), (1.4). As a result, we have much better possible upper bounds for the equations (1.3), (1.4) than those of Czerwik [4] and Skof-Cholewa [3].

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2. Solution of (1.3), (1.4)

Let R+ denote the set of all nonnegative real numbers and let bothE1 andE2 be real vector spaces. We here present the general solution of (1.3), (1.4).

Theorem 2.1. A function f : E1 → E2 satisfies the functional equation (1.1) if and only iff : E1 → E2 satisfies the functional equation (1.4) if and only if f : E1 → E2 satisfies the functional equation (1.3). Therefore, every solution of functional equations (1.3) and (1.4) is also a quadratic function.

Proof. Letf :E1 →E2 satisfy the functional equation (1.1). Puttingx= 0 = y in (1.1), we get f(0) = 0.Setx = 0in (1.1) to getf(y) = f(−y). Letting y = x andy = 2x in (1.1), respectively, we obtain that f(2x) = 4f(x) and f(3x) = 9f(x)for allx∈E1. By induction, we lead tof(kx) =k2f(x)for all positive integerk.Replacingxandyby2x+yandx+ 2yin (1.1), respectively, we have

f(2x+y) +f(x+ 2y) = 1

2[f(3x+ 3y) +f(x−y)]

(2.1)

= 4f(x+y) + 1

2[f(x+y) +f(x−y)]

= 4f(x+y) +f(x) +f(y) for allx, y ∈E1.

Letf :E1 →E2 satisfy the functional equation (1.4). Puttingx= 0 = yin (1.4), we getf(0) = 0.Sety= 0in (1.4) to getf(2x) = 4f(x). Lettingy =x andy=−2xin (1.4), we obtain thatf(3x) = 9f(x)andf(x) =f(−x)for all

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x∈E1. Puttingxandybyx+yandx+yin (1.4), respectively, we obtain f(2x+ 3y) +f(x+ 3y) = 4f(x+ 2y) +f(x+y) +f(y), (2.2)

f(3x+y) +f(3x+ 2y) = 4f(2x+y) +f(x) +f(x+y).

(2.3)

Adding (2.2) to (2.3) and using (1.4), we obtain

(2.4) f(2x+ 3y) +f(3x+ 2y) +f(x+ 3y) +f(3x+y)

= 18f(x+y) + 5f(x) + 5f(y) for allx, y∈E1.Replacingyby2yandxby2xin (1.4), respectively, we have

4f(x+y) +f(x+ 4y) = 4f(x+ 2y) +f(x) + 4f(y), (2.5)

4f(x+y) +f(4x+y) = 4f(2x+y) + 4f(x) +f(y) (2.6)

for allx, y ∈E1.Adding (2.5) to (2.6) and using (1.4), we get (2.7) f(x+ 4y) +f(4x+y) = 8f(x+y) + 9f(x) + 9f(y) for allx, y ∈E1.

On the other hand, using (1.4), we get f(x+ 4y) +f(4x+y)

(2.8)

=f(6x+ 9y) +f(9x+ 6y)−4f(5x+ 5y)

= 9f(2x+ 3y) + 9f(3x+ 2y)−100f(x+y), which yields the relation by virtue of (2.7)

(2.9) f(2x+ 3y) +f(3x+ 2y) = 12f(x+y) +f(x) +f(y)

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for allx, y ∈E1.Combining the last equation with (2.4), we get (2.10) f(x+ 3y) +f(3x+y) = 6f(x+y) + 4f(x) + 4f(y).

Replacingxandy by x+y2 and x−y2 in (2.10), respectively, we have the desired result (1.3).

Now, letf :E1 →E2satisfy the functional equation (1.3). Puttingx= 0 = y in (1.3), we get f(0) = 0. Lettingy = 0 and y = x in (1.3), respectively, we obtain that f(2x) = 4f(x) and f(3x) = 9f(x) for all x ∈ E1. Putting y = 2xin (1.3), we getf(x) =f(−x).Replacingxandybyx+yandx−y, respectively, in (1.3), we have

(2.11) f(3x+y) +f(x+ 3y) = 6f(x+y) + 4f(x) + 4f(y) for allx, y ∈E1.Replacingybyx+yin (1.3), we obtain

(2.12) f(3x+y) +f(x−y) = 6f(x) +f(2x+y) +f(y).

Interchangexwithyin (2.12) to get the relation

(2.13) f(3y+x) +f(x−y) = 6f(y) +f(2y+x) +f(x).

Adding (2.12) to (2.13), we obtain

(2.14) 6f(x+y) + 2f(x−y) =f(2x+y) +f(x+ 2y) + 3f(x) + 3f(y) for allx, y∈E1.Setting−yinstead ofyin (2.14) and using the evenness off, we get the relation

(2.15) 6f(x−y) + 2f(x+y) = f(2x−y) +f(2y−x) + 3f(x) + 3f(y).

Adding (2.14) to (2.15), we obtain the result (1.1).

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3. Stability of (1.3)

From now on, letX be a real vector space and letY be a Banach space unless we give any specific reference. We will investigate the Hyers-Ulam-Rassias sta- bility problem for the functional equation (1.3). Thus we find the condition that there exists a true quadratic function near an approximately quadratic function.

Theorem 3.1. Letφ:X2 →R+be a function such that

(3.1)

X

i=0

φ(2ix,0) 4i

X

i=1

4iφ(x

2i,0), respectively

!

converges and

(3.2) lim

n→∞

φ(2nx,2ny)

4n = 0

n→∞lim 4nφx 2n, y

2n

= 0 for allx, y ∈X.Suppose that a functionf :X →Y satisfies

(3.3) kf(2x+y) +f(2x−y)−f(x+y)−f(x−y)−6f(x)k ≤φ(x, y) for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y which satisfies the equation (1.3) and the inequality

kf(x)−T(x)k ≤ 1 8

X

i=0

φ(2ix,0) 4i (3.4)

kf(x)−T(x)k ≤ 1 8

X

i=1

4iφx 2i,0

!

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for allx∈X. The functionT is given by

(3.5) T(x) = lim

n→∞

f(2nx) 4n

T(x) = lim

n→∞4nf x

2n

for allx∈X.

Proof. Puttingy = 0in (3.3) and dividing by8, we have (3.6)

f(2x)

4 −f(x)

≤ 1

8φ(x,0)

for all x ∈ X.Replacingxby2xin (3.6) and dividing by4and summing the resulting inequality with (3.6), we get

(3.7)

f(22x)

42 −f(x)

≤ 1 8

φ(x,0) + φ(2x,0) 4

for allx∈X.Using the induction on a positive integern, we obtain that

f(2nx)

4n −f(x)

≤ 1 8

n−1

X

i=0

φ(2ix,0) 4i (3.8)

≤ 1 8

X

i=0

φ(2ix,0) 4i

for all x ∈ X. In order to prove convergence of the sequence nf(2nx) 4n

o , we

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divide inequality (3.8) by4mand also replacexby2mxto find that forn, m >0,

f(2n2mx)

4n+m −f(2mx) 4m

= 1 4m

f(2n2mx)

4n −f(2mx) (3.9)

≤ 1 8·4m

n−1

X

i=0

φ(2i2mx,0) 4i

≤ 1 8

X

i=0

φ(2i2mx,0) 4m+i .

Since the right hand side of the inequality tends to 0asm tends to infinity, the sequence

nf(2nx) 4n

o

is a Cauchy sequence. Therefore, we may define T(x) = limn→∞2−2nf(2nx) for all x ∈ X. By lettingn → ∞ in (3.8), we arrive at the formula (3.4). To show thatT satisfies the equation (1.3), replace x, y by 2nx,2ny,respectively, in (3.3) and divide by4n,then it follows that

4−nkf(2n(2x+y)) +f(2n(2x−y))−f(2n(x+y))

− f(2n(x−y))−6f(2nx))k ≤4−nφ(2nx,2ny).

Taking the limit asn → ∞,we find thatT satisfies (1.3) for allx, y ∈X.

To prove the uniqueness of the quadratic functionT subject to (3.4), let us assume that there exists a quadratic functionS : X → Y which satisfies (1.3) and the inequality (3.4). Obviously, we have S(2nx) = 4nS(x)andT(2nx) = 4nT(x)for allx∈X andn∈N.Hence it follows from (3.4) that

kS(x)−T(x)k= 4−nkS(2nx)−T(2nx)k

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≤4−n(kS(2nx)−f(2nx)k+kf(2nx)−T(2nx)k)

≤ 1 4

X

i=0

φ(2i2nx,0) 4n+i

for allx ∈ X.By lettingn → ∞in the preceding inequality, we immediately find the uniqueness ofT.This completes the proof of the theorem.

Throughout this paper, letBbe a unital Banach algebra with norm|·|, and let

BB1 andBB2 be left BanachB-modules with normsk·kandk·k, respectively.

A quadratic mappingQ:BB1BB2is calledB-quadratic if Q(ax) =a2Q(x), ∀a∈B,∀x∈BB1.

Corollary 3.2. Letφ :BB1×BB1→R+be a function satisfying (3.1) and (3.2) for allx, y ∈BB1. Suppose that a mappingf :BB1BB2 satisfies

f(2αx+αy) +f(2αx−αy)−α2f(x+y)−α2f(x−y)−6α2f(x)

≤φ(x, y) for allα ∈ B (|α| = 1) and for allx, y ∈ BB1, andf is measurable or f(tx) is continuous in t ∈ R for each fixed x ∈ BB1. Then there exists a unique B-quadratic mapping T : BB1BB2, defined by (3.5), which satisfies the equation (1.3) and the inequality (3.4) for allx∈BB1.

Proof. By Theorem3.1, it follows from the inequality of the statement forα = 1 that there exists a unique quadratic mappingT : BB1BB2 satisfying the inequality (3.4) for allx ∈ BB1.Under the assumption thatf is measurable or

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f(tx)is continuous int∈ Rfor each fixedx∈ BB1, by the same reasoning as the proof of [5], the quadratic mappingT :BB1BB2satisfies

T(tx) =t2T(x), ∀x∈BB1,∀t∈R.

That is, T is R-quadratic. For each fixedα ∈ B (|α| = 1), replacingf by T and setting y= 0in (1.3), we haveT(αx) = α2T(x)for allx ∈BB1.The last relation is also true for α = 0.For each elementa ∈ B (a 6= 0), a =|a| · |a|a. SinceT isR-quadratic andT(αx) =α2T(x)for each elementα∈B(|α|= 1),

T(ax) = T

|a| · a

|a|x

= |a|2·T a

|a|x

= |a|2· a2

|a|2 ·T(x)

= a2T(x), ∀a∈B(a6= 0), ∀x∈BB1.

So the unique R-quadratic mapping T : BB1BB2 is also B-quadratic, as desired. This completes the proof of the corollary.

SinceCis a Banach algebra, the Banach spacesE1andE2 are considered as Banach modules overC. Thus we have the following corollary.

Corollary 3.3. Let E1 andE2 be Banach spaces over the complex fieldC, and letε≥0be a real number. Suppose that a mappingf :E1 →E2 satisfies

kf(2αx+αy) +f(2αx−αy)−α2f(x+y)−α2f(x−y)−6α2f(x)k ≤ε

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for all α ∈ C(|α| = 1) and for all x, y ∈ E1, and f is measurable or f(tx) is continuous in t ∈ R for each fixed x ∈ E1. Then there exists a unique C- quadratic mapping T : E1 → E2 which satisfies the equation (1.3) and the inequality

kf(x)−T(x)k ≤ ε 6 for allx∈E1.

The S. Czerwik [4] theorem for the functional equation (1.1) states that if a function f : G → Y, whereG is an abelian group and Y a Banach space, satisfies the inequalitykf(x+y)+f(x−y)−2f(x)−2f(y)k ≤ε(kxkp+kykp) for p 6= 2and for allx, y ∈ G, then there exists a unique quadratic functionq such thatkf(x)−q(x)k ≤ |4−2εkxkpp|+kf(0)k3 for allx∈G,and for allx∈G− {0}

andkf(0)k = 0ifp < 0.From the main theorem3.1, we obtain the following corollary concerning the stability of the equation (1.3). We note thatpneed not be equal toq.

Corollary 3.4. Let X and Y be a real normed space and a Banach space, respectively, and let ε, p, qbe real numbers such that ε ≥ 0, q > 0and either p, q <2orp, q >2.Suppose that a functionf :X →Y satisfies

kf(2x+y) +f(2x−y)−f(x+y)−f(x−y)−6f(x)k ≤ε(kxkp+kykq) for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y which satisfies the equation (1.3) and the inequality

kf(x)−T(x)k ≤ ε

2|4−2p|kxkp

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for allx∈X and for allx∈X− {0}ifp < 0.The functionT is given by T(x) = lim

n→∞

f(2nx)

4n ifp, q <2

T(x) = lim

n→∞4nfx 2n

ifp, q >2 for allx∈ X.Further, if for each fixedx ∈X the mappingt 7→f(tx)fromR toY is continuous, thenT(rx) =r2T(x)for allr ∈R.

The proof of the last assertion in the above corollary goes through in the same way as that of [4].

The Skof-Cholewa [3] theorem for the functional equation (1.1) states that if a functionf : G → Y, whereGis an abelian group andY a Banach space, satisfies the inequalitykf(x+y) +f(x−y)−2f(x)−2f(y)k ≤εfor allx, y ∈ G, then there exists a unique quadratic functionqsuch thatkf(x)−q(x)k ≤ ε2 for all x ∈ G. But we have a much better possible upper bound concerning the stability theorem for the functional equation (1.3) as follows. The following corollary is an immediate consequence of Theorem3.1.

Corollary 3.5. Let X and Y be a real normed space and a Banach space, respectively, and letε ≥0be a real number. Suppose that a functionf :X → Y satisfies

(3.10) kf(2x+y) +f(2x−y)−f(x+y)−f(x−y)−6f(x)k ≤ε for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y defined by T(x) = limn→∞ f(2nx)

4n which satisfies the equation (1.3) and the inequality

(3.11) kf(x)−T(x)k ≤ ε

6

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for allx∈ X.Further, if for each fixedx ∈X the mappingt 7→f(tx)fromR toY is continuous, thenT(rx) =r2T(x)for allr ∈R.

Remark 3.1. If we writey=xin the inequality of (3.3), we get (3.12) kf(3x)−5f(x)−f(2x)k ≤φ(x, x) +kf(0)k.

Combining (3.12) with (3.6), we have

(3.13) kf(3x)−9f(x)k ≤φ(x, x) + φ(x,0)

2 +kf(0)k.

We can easily show the following relation by induction onntogether with (3.13)

f(3nx)

9n −f(x)

≤ 1 9

n−1

X

i=0

1 9i

φ(3ix,3ix) + φ(3ix,0)

2 +kf(0)k

for allx∈X.

In Theorem3.1, letφ:X2 →R+be a function such that

X

i=0

φ(3ix,3ix) +φ(3ix,0) 9i

X

i=1

9ih φx

3i, x 3i

+φx 3i,0i

, respectively

!

converges and

n→∞lim

φ(3nx,3ny)

9n = 0

n→∞lim 9nφx 3n, y

3n

= 0

for all x, y ∈ X. Note that in the second case f(0) = 0 since φ(0,0) = 0.

Then, using the last inequality and the same argument of Theorem 3.1, we can

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find the unique quadratic function T defined by T(x) = limn→∞3−2nf(3nx) which satisfies (1.3) and the inequality

kf(x)−T(x)k ≤ 1 9

X

i=0

1 9i

φ(3ix,3ix) + φ(3ix,0) 2

+ kf(0)k (3.14) 8

kf(x)−T(x)k ≤ 1 9

X

i=1

9i

"

φx 3i, x

3i

+ φ 3xi,0 2

#!

for allx∈X. Thus we obtain an alternative result of Theorem3.1. In Theorem 3.1, we have a simpler possible upper bound (3.4) than that of (3.14). The advantage of the inequality (3.4) compared to (3.14) is that the right hand side of (3.4) has no term forkf(0)k.

As a consequence of the above Remark3.1, we have the following corollary.

Because of the restricted condition0< p,we havef(0) = 0.

Corollary 3.6. Let X and Y be a real normed space and a Banach space, respectively, and letε≥0,0< p6= 2be real numbers. Suppose that a function f :X →Y satisfies

kf(2x+y) +f(2x−y)−f(x+y)−f(x−y)−6f(x)k ≤ε(kxkp+kykp) for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y which satisfies the equation (1.3) and the inequality

kf(x)−T(x)k ≤ 5ε

2|9−3p|kxkp

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for allx∈X.The functionT is given by

T(x) = lim

n→∞

f(3nx)

9n if0< p <2

T(x) = lim

n→∞9nf x

3n

ifp >2

for allx∈ X.Further, if for each fixedx ∈X the mappingt 7→f(tx)fromR toY is continuous, thenT(rx) =r2T(x)for allr ∈R.

Remark 3.2. If we puty=x= 0in the inequality of (3.10), we get6kf(0)k ≤ ε.Applying Remark3.1to (3.10), we know that there exists a unique quadratic function T : X → Y defined by T(x) = limn→∞ f(3nx)

9n which satisfies the equation (1.3) and the inequality

kf(x)−T(x)k ≤ 3ε

16+kf(0)k 8 ≤ 5ε

24

for allx∈X.But we have a better possible upper bound (3.11) than that of the last inequality.

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4. Stability of (1.4)

We will investigate the Hyers-Ulam-Rassias stability problem for the functional equation (1.4). Thus we find the condition that there exists a true quadratic function near an approximately quadratic function.

Theorem 4.1. Letφ:X2 →R+be a function such that

X

i=0

1 9i

φ(3ix,3ix)

2 + 2φ(3ix,0) (4.1)

X

i=1

9i 1

2φ x

3i, x 3i

+ 2φ

x 3i,0

, respectively

!

converges and

(4.2) lim

n→∞

φ(3nx,3ny)

9n = 0

n→∞lim 9nφx 3n, y

3n

= 0 for allx, y ∈X.Suppose that a functionf :X →Y satisfies

(4.3) kf(2x+y) +f(x+ 2y)−4f(x+y)−f(x)−f(y)k ≤φ(x, y) for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y which satisfies the equation (1.4) and the inequality

kf(x)−T(x)k ≤ 1 9

X

i=0

1 9i

φ(3ix,3ix)

2 + 2φ(3ix,0)

+kf(0)k (4.4) 4

kf(x)−T(x)k ≤ 1 9

X

i=1

9i 1

2φx 3i, x

3i

+ 2φx 3i,0

,

!

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for allx∈X. The functionT is given by

(4.5) T(x) = lim

n→∞

f(3nx) 9n

T(x) = lim

n→∞9nfx 3n

for allx∈X.

Proof. If we writey=xin the inequality of (4.3), we get (4.6) kf(3x)−2f(2x)−f(x)k ≤ 1

2φ(x, x).

Puttingy= 0in (4.3) and multiplying by2, we have

(4.7) k2f(2x)−8f(x)k ≤2φ(x,0) + 2kf(0)k

for allx∈X.Adding the inequality (4.6) with (4.7) and then dividing by9, we get

(4.8)

f(3x)

9 −f(x)

≤ 1 9

φ(x, x)

2 + 2φ(x,0) + 2kf(0)k

for allx∈X.Using the induction onn, we obtain that

f(3nx)

9n −f(x)

≤ 1 9

n−1

X

i=0

1 9i

φ(3ix,3ix)

2 + 2φ(3ix,0) + 2kf(0)k (4.9)

≤ 1 9

X

i=0

1 9i

φ(3ix,3ix)

2 + 2φ(3ix,0)

+kf(0)k 4

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for allx∈X.

Repeating the similar argument of Theorem 3.1, we obtain the desired re- sult. The proof of assertion indicated by parentheses in the theorem is similarly proved and we omit it. In this case,f(0) = 0sinceφ(0,0) = 0by assumption.

This completes the proof of the theorem.

The proof of the following corollary is similar to that of Corollary3.2.

Corollary 4.2. Letφ :BB1×BB1→R+be a function satisfying (4.1) and (4.2) for allx, y ∈BB1. Suppose that a mappingf :BB1BB2 satisfies

(4.10) kf(2αx+αy) +f(αx+ 2αy)

−4α2f(x+y)−α2f(x)−α2f(y)

≤φ(x, y) for allα ∈ B (|α| = 1) and for allx, y ∈ BB1, andf is measurable or f(tx) is continuous in t ∈ R for each fixed x ∈ BB1. Then there exists a unique B-quadratic mapping T : BB1BB2, defined by (4.5), which satisfies the equation (1.4) and the inequality (4.4) for allx∈BB1.

Corollary 4.3. Let E1 andE2 be Banach spaces over the complex fieldC, and letε≥0be a real number. Suppose that a mappingf :E1 →E2 satisfies

f(2αx+αy) +f(αx+ 2αy)−4α2f(x+y)−α2f(x)−α2f(y) ≤ε for all α ∈ C(|α| = 1) and for all x, y ∈ E1, and f is measurable or f(tx) is continuous in t ∈ R for each fixed x ∈ E1. Then there exists a unique C- quadratic mapping T : E1 → E2 which satisfies the equation (1.3) and the inequality

kf(x)−T(x)k ≤ 5ε 16

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for allx∈E1.

In Theorem 4.1, we obtain the alternative result if the conditions of φ are replaced by the following.

Remark 4.1. Letφ :X2 →R+be a function such that

X

i=0

1

4iφ(2ix,0)

X

i=1

4iφx 2i,0

, respectively

!

converges and

n→∞lim

φ(2nx,2ny) 4n = 0

n→∞lim 4nφ x

2n, y 2n

= 0

for allx, y ∈X.Suppose that a functionf :X →Y satisfies

kf(2x+y) +f(x+ 2y)−4f(x+y)−f(x)−f(y)k ≤φ(x, y) for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y which satisfies the equation (1.4) and the inequality

kf(x)−T(x)k ≤ 1 4

X

i=0

1

4iφ(2ix,0) + kf(0)k (4.11) 3

kf(x)−T(x)k ≤ 1 4

X

i=1

4iφx 2i,0

!

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for allx∈X. The functionT is given by

T(x) = lim

n→∞

f(2nx) 4n

T(x) = lim

n→∞4nfx 2n

for allx∈X.

From Remark4.1, we obtain the following corollary concerning the stability of the equation (1.4). We note thatpneed not be equal toqandkf(0)k = 0if p > 0.

Corollary 4.4. Let X and Y be a real normed space and a Banach space, respectively, and let ε, p, qbe real numbers such that ε ≥ 0, q > 0and either p, q <2orp, q >2.Suppose that a functionf :X →Y satisfies

kf(2x+y) +f(x+ 2y)−4f(x+y)−f(x)−f(y)k ≤ε(kxkp+kykq) for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y which satisfies the equation (1.4) and the inequality

kf(x)−T(x)k ≤ ε

|4−2p|kxkp +kf(0)k 3

for allx∈X and for allx∈X− {0}ifp < 0.The functionT is given by

T(x) = lim

n→∞

f(2nx)

4n ifp, q <2

T(x) = lim

n→∞4nfx 2n

ifp, q >2 for allx∈ X.Further, if for each fixedx ∈X the mappingt 7→f(tx)fromR toY is continuous, thenT(rx) =r2T(x)for allr ∈R.

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As a consequence of the above Theorem4.1, we have the following.

Corollary 4.5. Let X and Y be a real normed space and a Banach space, respectively, and letε≥0,0< p6= 2be real numbers. Suppose that a function f :X →Y satisfies

kf(2x+y) +f(x+ 2y)−4f(x+y)−f(x)−f(y)k ≤ε(kxkp+kykp) for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y which satisfies the equation (1.4) and the inequality

kf(x)−T(x)k ≤ 3ε

|9−3p|kxkp for allx∈X.The functionT is given by

T(x) = lim

n→∞

f(3nx)

9n if0< p <2

T(x) = lim

n→∞9nf x

3n

ifp >2

for allx∈ X.Further, if for each fixedx ∈X the mappingt 7→f(tx)fromR toY is continuous, thenT(rx) =r2T(x)for allr ∈R.

The following corollary is an immediate consequence of Theorem4.1.

Corollary 4.6. Let X and Y be a real normed space and a Banach space, respectively, and letε ≥0be a real number. Suppose that a functionf :X → Y satisfies

(4.12) kf(2x+y) +f(x+ 2y)−4f(x+y)−f(x)−f(y)k ≤ε

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for all x, y ∈ X. Then there exists a unique quadratic function T : X → Y defined by T(x) = limn→∞ f(2nx)

4n which satisfies the equation (1.4) and the inequality

(4.13) kf(x)−T(x)k ≤ 5ε

16

for allx∈ X.Further, if for each fixedx ∈X the mappingt 7→f(tx)fromR toY is continuous, thenT(rx) =r2T(x)for allr ∈R.

Remark 4.2. If we puty=x= 0in the inequality of (4.12), we get4kf(0)k ≤ ε.Applying Remark4.1to (4.12), we know that there exists a unique quadratic function T : X → Y defined by T(x) = limn→∞ f(2nx)

4n which satisfies the equation (1.4) and the inequality

kf(x)−T(x)k ≤ ε

3 +kf(0)k 3 ≤ 5ε

12

for allx∈X.But we have a better possible upper bound (4.13) than that of the last inequality.

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References

[1] J. ACZÉL AND J. DHOMBRES, Functional Equations in Several Vari- ables, Cambridge Univ. Press, 1989.

[2] J. BAKER, The stability of the cosine equation, Proc. Amer. Math. Soc., 80 (1980), 411–416.

[3] P.W. CHOLEWA, Remarks on the stability of functional equations, Ae- quationes Math., 27 (1984), 76–86.

[4] S. CZERWIK, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59–64.

[5] S. CZERWIK, The stability of the quadratic functional equation, In Sta- bility of Mappings of Hyers-Ulam Type (edited by Th. M. Rassias and J.

Tabor), Hadronic Press, Florida, 1994, 81–91.

[6] G.L. FORTI, Hyers-Ulam stability of functional equations in several vari- ables, Aequationes Math., 50 (1995), 143–190.

[7] P. G ˇAVRUTA, A generalization of the Hyers-Ulam-Rassias Stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–

436.

[8] A. GRABIEC, The generalized Hyers-Ulam stability of a class of func- tional equations, Publ. Math. Debrecen, 48 (1996), 217–235.

[9] D.H. HYERS, On the stability of the linear functional equation, Proc. Natl.

Acad. Sci., 27 (1941), 222–224.

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[10] D.H. HYERS, G. ISAC AND Th. M. RASSIAS, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.

[11] D.H. HYERS, G. ISAC ANDTh. M. RASSIAS, On the asymptoticity as- pect of Hyers-Ulam stability of mappings, Proc. Amer. Math. Soc., 126 (1998), 425–430.

[12] D.H. HYERSANDTh. M. RASSIAS, Approximate homomorphisms, Ae- quationes Math., 44 (1992), 125–153.

[13] K.W. JUN ANDY.H. LEE, On the Hyers-Ulam-Rassias stability of a pex- iderized quadratic inequality, Math. Ineq. Appl., 4(1) (2001), 93–118.

[14] S.M. JUNG, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl., 222 (1998), 126–137.

[15] S.M. JUNG, On the Hyers-Ulam-Rassias stability of a quadratic functional equation, J. Math. Anal. Appl., 232 (1999), 384–393.

[16] Pl. KANNAPPAN, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368–372.

[17] Th. M. RASSIAS, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.

[18] Th. M. RASSIAS, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 264–284.

[19] F. SKOF, Proprietà locali e approssimazione di operatori, Rend. Sem. Mat.

Fis. Milano, 53 (1983), 113–129.

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[20] S. M. ULAM, Problems in Modern Mathematics, Chap. VI, Science Ed., Wiley, New York, 1960.

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