Hyers–Ulam stability of a generalized Apollonius type quadratic mapping

J. Math. Anal. Appl. 322 (2006) 371–381 www.elsevier.com/locate/jmaa

Hyers–Ulam stability of a generalized Apollonius type quadratic mapping ✩ Chun-Gil Park a,∗ , Themistocles M. Rassias b a Department of Mathematics, Chungnam National University, Daejeon 305–764, Republic of Korea b Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

Received 29 March 2005 Available online 12 October 2005 Submitted by R.P. Agarwal

Abstract Let X, Y be linear spaces. It is shown that if a mapping Q : X → Y satisfies the following functional equation:  n   n   n   n      zi − xi zi − yi +Q Q i=1

i=1

i=1

i=1

 n   n   n        ( ni=1 xi ) + ( ni=1 yi ) 1 xi − yi zi − = Q + 2Q 2 2 i=1

i=1

(0.1)

i=1

then the mapping Q : X → Y is quadratic. We moreover prove the Hyers–Ulam stability of the functional equation (0.1) in Banach spaces. © 2005 Elsevier Inc. All rights reserved. Keywords: Hyers–Ulam stability; Quadratic mapping of Apollonius type

1. Introduction The stability problem of functional equations originated from a question of S.M. Ulam [21] concerning the stability of group homomorphisms: Let (G1 , ∗) be a group and let (G2 , , d) be ✩

Supported by Korea Research Foundation Grant KRF-2005-070-C00009.

* Corresponding author.

E-mail addresses: [email protected], [email protected] (C.-G. Park), [email protected] (Th.M. Rassias). 0022-247X/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.09.027

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a metric group with the metric d(·,·). Given  > 0, does there exist a δ() > 0 such that if a mapping h : G1 → G2 satisfies the inequality   d h(x ∗ y), h(x)  h(y) < δ for all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with   d h(x), H (x) <  for all x ∈ G1 ? If the answer is affirmative, we would say the equation of homomorphism H (x ∗ y) = H (x)  H (y) stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? D.H. Hyers [9] gave a first affirmative answer to the question of Ulam for Banach spaces. Let X and Y be Banach spaces. Assume that f : X → Y satisfies   f (x + y) − f (x) − f (y)  ε for all x, y ∈ X and some ε  0. Then there exists a unique additive mapping T : X → Y such that   f (x) − T (x)  ε for all x ∈ X. Th.M. Rassias [17] succeeded in extending the result of Hyers by weakening the condition for the Cauchy difference to be unbounded. A number of mathematicians were attracted to this result of Th.M. Rassias and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Th.M. Rassias in his 1978 paper is called the Hyers–Ulam stability. G.L. Forti [5] and P. Gˇavruta [8] have generalized the result of Th.M. Rassias, which permitted the Cauchy difference to become arbitrary unbounded. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem. A large list of references can be found, for example, in the papers [2,6,7,10–19]. Now, a square norm on an inner product space satisfies the important parallelogram equality   x + y2 + x − y2 = 2 x2 + y2 for all vectors x, y. The following functional equation, which was motivated by this equation, Q(x + y) + Q(x − y) = 2Q(x) + 2Q(y),

(1.1)

is called a quadratic functional equation, and every solution of Eq. (1.1) is said to be a quadratic mapping. F. Skof [20] proved the Hyers–Ulam stability of the quadratic functional equation (1.1) for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. In [4], S. Czerwik proved the Hyers–Ulam stability of the quadratic functional equation. C. Borelli and G.L. Forti [3] generalized the stability result as follows: let G be an abelian group, E a Banach space. Assume that a mapping f : G → E satisfies the functional inequality   f (x + y) + f (x − y) − 2f (x) − 2f (y)  ϕ(x, y) for all x, y ∈ G, and ϕ : G × G → [0, ∞) is a function such that

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373

∞   1  i Φ(x, y) := ϕ 2 x, 2i y < ∞ 4i+1 i=0

for all x, y ∈ G. Then there exists a unique quadratic mapping Q : G → E with the properties   f (x) − Q(x)  Φ(x, x) for all x ∈ G. Jun and Lee [14] proved the Hyers–Ulam stability of the Pexiderized quadratic equation f (x + y) + g(x − y) = 2h(x) + 2k(y) for mappings f, g, h and k. In an inner product space, the equality 2   x +y 1  z − x2 + z − y2 = x − y2 + 2 z −  2 2  holds, and is called the Apollonius’ identity. The following functional equation, which was motivated by this equation,

1 x+y Q(z − x) + Q(z − y) = Q(x − y) + 2Q z − , (1.2) 2 2 is quadratic. For this reason, the functional equation (1.2) is called a quadratic functional equation of Apollonius type, and each solution of the functional equation (1.2) is said to be a quadratic mapping of Apollonius type. The quadratic functional equation and several other functional equations are useful to characterize inner product spaces [1]. In this paper, employing the above equality (1.2), we introduce the new functional equation, which is called the generalized Apollonius type quadratic functional equation and whose solution of the functional equation is said to be a generalized Apollonius type quadratic mapping,  n   n   n   n      Q zi − xi zi − yi +Q i=1



1 = Q 2

i=1 n  i=1



xi −



n  i=1

 yi

i=1



+ 2Q

i=1 n  i=1



zi

   ( ni=1 xi ) + ( ni=1 yi ) − . 2

(1.3)

As a special case, if n = 1 in (1.3), then the functional equation (1.3) reduces to

x+y 1 . Q(z − x) + Q(z − y) = Q(x − y) + 2Q z − 2 2 We are going to show that the generalized Apollonius type quadratic functional equation (1.3) is quadratic, and prove the Hyers–Ulam stability of generalized Apollonius type quadratic mappings in Banach spaces. 2. Hyers–Ulam stability of a generalized Apollonius type quadratic mapping Throughout this section, let X be a normed space with norm  ·  and Y a Banach space with norm  · .

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Lemma 2.1. If a mapping f : X → Y satisfies f (0) = 0 and  n   n   n   n      f zi − xi zi − yi +f i=1

1 = f 2



i=1 n 





xi −

i=1

n 



i=1



+ 2f

yi

i=1

i=1 n 



zi

i=1

   ( ni=1 xi ) + ( ni=1 yi ) − 2

(2.1)

for all x1 , . . . , xn ∈ X, then the mapping f : X → Y is quadratic. Proof. Suppose that a mapping f : X → Y satisfies the equality (2.1). Letting z2 = · · · = zn = x2 = · · · = xn = y2 = · · · = yn = 0 and y1 = −x1 , we get 1 f (z1 − x1 ) + f (z1 + x1 ) = f (2x1 ) + 2f (z1 ) 2 for all x1 , z1 ∈ X. Letting z1 = x1 in (2.2), we get

(2.2)

1 f (2x1 ) = f (2x1 ) + 2f (x1 ) 2 for all x1 ∈ X. So f (2x1 ) = 4f (x1 ) for all x1 ∈ X. It follows from (2.2) that f (z1 + x1 ) + f (z1 − x1 ) = 2f (z1 ) + 2f (x1 ) for all x1 , z1 ∈ X. Thus the mapping f : X → Y is quadratic.

2

Given a mapping f : X → Y , we define Df : X 3n → Y by Df (z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )  n   n   n   n      zi − xi zi − yi =f +f i=1

1 − f 2



n  i=1



i=1

xi −



n 

i=1

 yi



+ 2f

i=1

i=1

n  i=1

 zi

   ( ni=1 xi ) − ( ni=1 yi ) − 2

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Theorem 2.2. Let f : X → Y be a mapping satisfying f (0) = 0 for which there exists a function ϕ : X 3n → [0, ∞) such that ϕ(z ˜ 1 , . . . , zn , x1 , . . . , xn , y1 , · · · , yn )

∞  z1 zn x1 xn y1 yn j = 4 ϕ j , . . . , j , j , . . . , j , j , . . . , j < ∞, 2 2 2 2 2 2 j =0   Df (z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )  ϕ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )

(2.3) (2.4)

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375

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Then there exists a unique generalized Apollonius type quadratic mapping Q : X → Y such that    x x f (x) − Q(x)  2ϕ˜ , . . . , , 0, . . . , 0 (2.5)   n n n times 2n times

for all x ∈ X. Proof. Letting z1 = · · · = zn = x1 = · · · = xn = x and y1 = · · · = yn = 0 in (2.4), we get     f (nx) − 1 f (nx) − 2f n x   ϕ(x, . . . , x , 0, . . . , 0)      2 2  2n times

n times

for all x ∈ X. So  

  f (x) − 4f 1 x   2ϕ x , . . . , x , 0, . . . , 0    2  n n 2n times

(2.6)

n times

for all x ∈ X. Hence 



 m−1  l   x x j 4 f x − 4 m f x   , . . . , 2 · 4 ϕ , 0, . . . , 0  j j   2l 2m  2 n 2 n j =l

2n times

(2.7)

n times

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.3) and (2.7) that the sequence {4d f (x/2d )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4d f (x/2d )} converges. So one can define the mapping Q : X → Y by

x Q(x) := lim 4d f d d→∞ 2 for all x ∈ X. By (2.3) and (2.4),   DQ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) 

  z1 zn x1 xn y1 yn   Df = lim 4d  , . . . , , , . . . , , , . . . , d→∞  2d 2d 2d 2d 2d 2d 

z1 zn x1 xn y1 yn  lim 4d ϕ d , . . . , d , d , . . . , d , d , . . . , d = 0 d→∞ 2 2 2 2 2 2 for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. So DQ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = 0. By Lemma 2.1, the mapping Q : X → Y is quadratic. Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5). So there exists a generalized Apollonius type quadratic mapping Q : X → Y satisfying (2.5). Now let Q : X → Y be another generalized Apollonius type quadratic mapping satisfying (2.5). Then we have

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     Q(x) − Q (x) = 4d Q x − Q x   d d 2 2  

 



   x x   + Q x − f x   4d  Q − f    d d d d 2 2 2 2 

x x  2 · 4d ϕ˜ , . . . , d , 0, . . . , 0 , dn 2 2 n    n times 2n times

which tends to zero as d → ∞ for all x ∈ X. So we can conclude that Q(x) = Q (x) for all x ∈ X. This proves the uniqueness of Q. 2 Corollary 2.3. Let p > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and  n  n n      p p p Df (z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )  θ zj  + xj  + yj  j =1

j =1

j =1

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Then there exists a unique generalized Apollonius type quadratic mapping Q : X → Y such that   f (x) − Q(x) 

(2p

2p+2 θ xp − 4)np−1

for all x ∈ X. Proof. Define



ϕ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = θ

n  j =1

and apply Theorem 2.2.

zj  + p

n 

xj  + p

j =1

n 

 yj 

p

,

j =1

2

Theorem 2.4. Let f : X → Y be a mapping satisfying f (0) = 0 for which there exists a function ϕ : X 3n → [0, ∞) satisfying (2.4) such that ϕ(z ˜ 1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) =

∞   1  j ϕ 2 z1 , . . . , 2j zn , 2j x1 , . . . , 2j xn , 2j y1 , . . . , 2j yn < ∞ 4j

(2.8)

j =1

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Then there exists a unique generalized Apollonius type quadratic mapping Q : X → Y such that

  f (x) − Q(x)  2ϕ˜ x , . . . , x , 0, . . . , 0 (2.9)   n n n times 2n times

for all x ∈ X.

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377

Proof. It follows from (2.6) that  

  f (x) − 1 f (2x)  1 ϕ 2x , . . . , 2x , 0, . . . , 0   2   4 n n n times 2n times

for all x ∈ X. Hence  

j m  1  l    2j x 2 2 x  f 2 x − 1 f 2m x   ,..., ϕ , 0, . . . , 0  4l    4m 4j  n n  n times j =l+1

(2.10)

2n times

for all nonnegativeintegers mand l with m > l and all x ∈ X. It follows from (2.8) and (2.10) that the sequence 41d f (2d x) is a Cauchy sequence for all x ∈ X. Since Y is complete, the   sequence 41d f (2d x) converges. So one can define the mapping Q : X → Y by 1  d  f 2 x d→∞ 4d

Q(x) := lim

for all x ∈ X. By (2.8) and (2.4),   DQ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )  1  = lim d Df 2d z1 , . . . , 2d zn , 2d x1 , . . . , 2d xn , 2d y1 , . . . , 2d yn  d→∞ 4  1   lim d ϕ 2d z1 , . . . , 2d zn , 2d x1 , . . . , 2d xn , 2d y1 , . . . , 2d yn = 0 d→∞ 4 for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. So DQ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = 0. By Lemma 2.1, the mapping Q : X → Y is quadratic. Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.9). So there exists a generalized Apollonius type quadratic mapping Q : X → Y satisfying (2.9). The rest of the proof is similar to the proof of Theorem 2.2. 2 Corollary 2.5. Let p < 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and  n  n n      p p p Df (z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )  θ zj  + xj  + yj  j =1

j =1

j =1

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Then there exists a unique generalized Apollonius type quadratic mapping Q : X → Y such that   f (x) − Q(x)  for all x ∈ X.

2p+2 θ xp (4 − 2p )np−1

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Proof. Define



ϕ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = θ

n 

zj  + p

j =1

and apply Theorem 2.4.

n 

xj  + p

j =1

n 

 yj 

p

,

j =1

2

Theorem 2.6. Let f : X → Y be a mapping satisfying f (0) = 0 for which there exists a function ϕ : X 3n → [0, ∞) satisfying (2.4) such that ϕ(z ˜ 1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) ∞   1  j = ϕ 2 z1 , . . . , 2j zn , 2j x1 , . . . , 2j xn , 2j y1 , . . . , 2j yn < ∞ 4j

(2.11)

j =0

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Then there exists a unique generalized Apollonius type quadratic mapping Q : X → Y such that

  f (x) − Q(x)  1 ϕ˜ x , . . . , x , − x , . . . , − x (2.12) 2 n n n  n 2n times

n times

for all x ∈ X. Proof. Letting z1 = · · · = zn = x1 = · · · = xn = x and y1 = · · · = yn = −x in (2.4), we get     f (2nx) − 1 f (2nx) − 2f (nx)  ϕ(x, . . . , x , −x, . . . , −x )       2 2n times

n times

for all x ∈ X. So  

  f (x) − 1 f (2x)  1 ϕ x , . . . , x , − x , . . . , − x   2 4 n n n  n 2n times

(2.13)

n times

for all x ∈ X. Hence   

j 1  l    m−1 2j x 2j x 1 2j x 2 x  f 2 x − 1 f 2m x   , . . . , , . . . , − ϕ , −  4l  4m 2 · 4j n  n n   n

(2.14)

j =l

2n times

n times

for all nonnegativeintegers mand l with m > l and all x ∈ X. It follows from (2.11) and (2.14) that the sequence 41d f (2d x) is a Cauchy sequence for all x ∈ X. Since Y is complete, the   sequence 41d f (2d x) converges. So one can define the mapping Q : X → Y by 1  d  f 2 x d→∞ 4d

Q(x) := lim

for all x ∈ X. By (2.11) and (2.4),

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379

  DQ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = lim

  1 Df 2d z1 , . . . , 2d zn , 2d x1 , . . . , 2d xn , 2d y1 , . . . , 2d yn 

d→∞ 4d

 1  d ϕ 2 z1 , . . . , 2d zn , 2d x1 , . . . , 2d xn , 2d y1 , . . . , 2d yn = 0 d d→∞ 4 for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. So  lim

DQ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = 0. By Lemma 2.1, the mapping Q : X → Y is quadratic. Moreover, letting l = 0 and passing the limit m → ∞ in (2.14), we get (2.12). So there exists a generalized Apollonius type quadratic mapping Q : X → Y satisfying (2.12). The rest of the proof is similar to the proof of Theorem 2.2. 2 Corollary 2.7. Let p < 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and  n  n n      p p p Df (z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )  θ zj  + xj  + yj  j =1

j =1

j =1

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Then there exists a unique generalized Apollonius type quadratic mapping Q : X → Y such that   6θ f (x) − Q(x)  xp (4 − 2p )np−1 for all x ∈ X. Proof. Define



ϕ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = θ

n  j =1

and apply Theorem 2.6.

zj  + p

n  j =1

xj  + p

n 

 yj 

p

,

j =1

2

Theorem 2.8. Let f : X → Y be a mapping satisfying f (0) = 0 for which there exists a function ϕ : X 3n → [0, ∞) satisfying (2.4) such that ϕ(z ˜ 1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )

∞  z1 zn x1 xn y1 yn = 4j ϕ j , . . . , j , j , . . . , j , j , . . . , j < ∞ 2 2 2 2 2 2

(2.15)

j =1

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Then there exists a unique generalized Apollonius type quadratic mapping Q : X → Y such that

  f (x) − Q(x)  1 ϕ˜ x , . . . , x , − x , . . . , − x (2.16) 2 n n n  n 2n times

for all x ∈ X.

n times

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Proof. It follows from (2.13) that  

  f (x) − 4f x   2ϕ x , . . . , x , − x , . . . , − x  2  2n 2n 2n 2n 2n times

n times

for all x ∈ X. Hence 



 m    l x x x 4j x 4 f x − 4 m f x   ϕ j ,..., j , − j ,...,− j  2l 2m  2 2 n 2 n 2 n 2 n j =l+1 2n times

(2.17)

n times

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.15) and (2.17) that the sequence {4d f (x/2d )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4d f (x/2d )} converges. So one can define the mapping Q : X → Y by

x Q(x) := lim 4d f d d→∞ 2 for all x ∈ X. By (2.15) and (2.4),   DQ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) 

  z1 zn x1 xn y1 yn   = lim 4d  Df , . . . , , , . . . , , , . . . , d→∞  2d 2d 2d 2d 2d 2d 

z1 zn x1 xn y1 yn  lim 4d ϕ d , . . . , d , d , . . . , d , d , . . . , d = 0 d→∞ 2 2 2 2 2 2 for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. So DQ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = 0. By Lemma 2.1, the mapping Q : X → Y is quadratic. Moreover, letting l = 0 and passing the limit m → ∞ in (2.17), we get (2.16). So there exists a generalized Apollonius type quadratic mapping Q : X → Y satisfying (2.16). The rest of the proof is the same as in the proof of Theorem 2.2. 2 Corollary 2.9. Let p > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and  n  n n      p p p Df (z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn )  θ zj  + xj  + yj  j =1

j =1

j =1

for all z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ∈ X. Then there exists a unique generalized Apollonius type quadratic mapping Q : X → Y such that   f (x) − Q(x)  for all x ∈ X.

(2p

6θ xp − 4)np−1

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Proof. Define



ϕ(z1 , . . . , zn , x1 , . . . , xn , y1 , . . . , yn ) = θ

n  j =1

and apply Theorem 2.8.

zj  + p

n  j =1

xj  + p

n 

381

 yj 

p

,

j =1

2

References [1] J. Aczél, J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, 1989. [2] J. Bae, K. Jun, Y. Lee, On the Hyers–Ulam–Rassias stability of an n-dimensional Pexiderized quadratic equation, Math. Inequal. Appl. 7 (2004) 63–77. [3] C. Borelli, G.L. Forti, On a general Hyers–Ulam stability result, Internat. J. Math. Math. Sci. 18 (1995) 229–236. [4] S. Czerwik, The stability of the quadratic functional equation, in: Th.M. Rassias, J. Tabor (Eds.), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 81–91. [5] G.L. Forti, An existence and stability theorem for a class of functional equations, Stochastica 4 (1980) 23–30. [6] G.L. Forti, Hyers–Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995) 143– 190. [7] G.L. Forti, Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations, J. Math. Anal. Appl. 295 (2004) 127–133. [8] P. Gˇavruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436. [9] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941) 222–224. [10] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998. [11] D.H. Hyers, G. Isac, Th.M. Rassias, On the asymptoticity aspect of Hyers–Ulam stability of mappings, Proc. Amer. Math. Soc. 126 (1998) 425–430. [12] K.W. Jun, H.M. Kim, On the Hyers–Ulam stability of a generalized quadratic and additive functional equations, Bull. Korean Math. Soc. 42 (2005) 133–148. [13] K.W. Jun, H.M. Kim, Ulam stability problem for generalized A-quadratic mappings, J. Math. Anal. Appl., in press. [14] K.W. Jun, Y.H. Lee, On the Hyers–Ulam–Rassias stability of a Pexiderized quadratic inequality, Math. Inequal. Appl. 4 (2001) 93–118. [15] C. Park, On the stability of the quadratic mapping in Banach modules, J. Math. Anal. Appl. 27 (2002) 135–144. [16] C. Park, On the Hyers–Ulam–Rassias stability of generalized quadratic mappings in Banach modules, J. Math. Anal. Appl. 291 (2004) 214–223. [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] P. Šemrl, On quadratic functionals, Bull. Austral. Math. Soc. 37 (1987) 27–28. [20] F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano 53 (1983) 113–129. [21] S.M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.