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Later, many different quadratic func-tional equations were solved by numerous authors [2–6].. Results The m-variable quadratic functional equation 1.2 induces the quadratic functional equ

Volume 2007, Article ID 24716, 8 pages

doi:10.1155/2007/24716

Research Article

A Multidimensional Functional Equation Having

Quadratic Forms as Solutions

Won-Gil Park and Jae-Hyeong Bae

Received 7 July 2007; Accepted 3 September 2007

Recommended by Vijay Gupta

We obtain the general solution and the stability of them-variable quadratic functional

equation f (x1+y1, ,x m+y m) + f (x1− y1, ,x m − y m)=2f (x1, ,x m) + 2f (y1, ,

y m) The quadratic form f (x1, ,x m)=1≤ i ≤ j ≤ a ij x i x jis a solution of the given func-tional equation

Copyright © 2007 W.-G Park and J.-H Bae This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distribu-tion, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In this paper, letX and Y be real vector spaces A mapping f is called a quadratic form if

there exista ij ∈ R(1≤ i ≤ j ≤ m) such that

fx1, ,x m

1≤ i ≤ j ≤

for allx1, ,x m ∈ X.

For a mapping f : X m → Y, consider the m-variable quadratic functional equation

fx1+y1, ,x m+y m

+fx1− y1, ,x m − y m

=2fx1, ,x m

+ 2fy1, , y m

.

(1.2) WhenX = Y = R, the quadratic form f :Rm →Rgiven by

fx1, ,x m

1≤ i ≤ j ≤

is a solution of (1.2)

For a mappingg : X → Y, consider the quadratic functional equation

g(x + y) + g(x − y) =2g(x) + 2g(y). (1.4)

In 1989, Acz´el [1] proposed the solution of (1.4) Later, many different quadratic func-tional equations were solved by numerous authors [2–6]

In this paper, we investigate the relation between (1.2) and (1.4) And we find out the general solution and the generalized Hyers-Ulam stability of (1.2)

2 Results

The m-variable quadratic functional equation (1.2) induces the quadratic functional equation (1.4) as follows

Theorem 2.1 Let f : X m → Y be a mapping satisfying ( 1.2 ) and let g : X → Y be the mapping given by

for all x ∈ X, then g satisfies ( 1.4 ).

Proof By (1.2) and (2.1),

g(x + y) + g(x − y) = f (x + y, ,x + y) + f (x − y, ,x − y)

=2f (x, ,x) + 2 f (y, , y) =2g(x) + 2g(y) (2.2)

The quadratic functional equation (1.4) induces them-variable quadratic functional

equation (1.2) with an additional condition

Theorem 2.2 Let a ij ∈ R(1≤ i ≤ j ≤ m) and g : X → Y be a mapping satisfying ( 1.4 ) If

f : X m → Y is the mapping given by

fx1, ,x m

:=

m



i =1

a ii gx i

+1 4



1≤ i<j ≤

a ij

gx i+x j

− gx i − x j

(2.3)

for all x1, ,x m ∈ X, then f satisfies ( 1.2 ) Furthermore, ( 2.1 ) holds if



1≤ i ≤ j ≤

Proof By (1.4) and (2.3),

fx1+y1, ,x m+y m

+fx1− y1, ,x m − y m

=

m



i =1

a ii

gx i+y i

+gx i − y i

+1

4



1≤ i<j ≤ a ij

gx i+y i+x j+y j

− gx i+y i − x j − y j

+1

4



1≤ i<j ≤

a ij

gx i − y i+x j − y j

− gx i − y i − x j+y j

=2

m



i =1

a ii

gx i

+gy i

+1

4



1≤ i<j ≤

a ij

gx i+y i+x j+y j

+gx i − y i+x j − y j

−1

4



1≤ i<j ≤

a ij

gx i+y i − x j − y j

+gx i − y i − x j+y j

=2

m



i =1

a ii

gx i

+gy i

+1

2



1≤ i<j ≤

a ij

gx i+x j

+gy i+y j

− gx i − x j

− gy i − y j

=2fx1, ,x m

+ 2fy1, , y m

(2.5)

for allx1, ,x m,y1, , y m ∈ X.

Lettingx = y =0 andy = x in (1.4), respectively,

for allx ∈ X By (2.3) and the above two equalities,

f (x, ,x) =m

i =1a ii g(x) +1

4



1≤ i<j ≤ a ij

g(2x) − g(0)= 

1≤ i ≤ j ≤ a ij g(x) = g(x) (2.7)

Example 2.3 The function g : R→Rgiven byg(x) = x2 satisfies (1.4) ByTheorem 2.2, the quadratic form f :Rm →Rgiven by

fx1, ,x m

1≤ i ≤ j ≤

satisfies (1.2)

Example 2.4 Let g : C→Cbe the function given byg(z) = zz Then, it satisfies the

qua-dratic functional equation (1.4) If f :Cm → Cis the mapping given by (2.3), that is,

fz1, ,z m

=

m



i =1

z i z i

 i



j =1

a ji+1 2

m



j = i+1 a ij

then f satisfies the m-variable quadratic functional equation (1.2)

Example 2.5 Let M2(R) be the real vector space of all 2×2 real matrices andg : M2(R)→R

the determinant function given by

for allA ∈ M2(R) Then, it satisfies (1.4) Using (2.3), f : M2(R)× M2(R)→Ris given

by f (A,B) =(a11+ (1/2)a12) det (A) + (a22+ (1/2)a12) det (B)(a11,a12,a21,a22∈ R) Also,

f satisfies (1.2)

In the following theorem, we find out the general solution of them-variable quadratic

functional equation (1.2)

Theorem 2.6 A mapping f : X m → Y satisfies ( 1.2 ) if and only if there exist symmetric biadditive mappings S1, ,S m:X2→ Y and biadditive mappings M ij:X2→ Y (1 ≤ i < j ≤

m) such that

fx1, ,x m

=

m



i =1

S i

x i,x i

1≤ i<j ≤

M ij

x i,x j

(2.11)

for all x1, ,x m ∈ X.

Proof We first assume that f is a solution of (1.2) Define f1, , f m:X → Y by f1(x) : =

f (x,0, ,0), , f m(x) : = f (0, ,0,x) for all x ∈ X One can easily verify that f1, , f m

are quadratic By [1], there exist symmetric biadditive mappingsS1, ,S m:X2→ Y such

that f1(x) = S1(x,x), , f m(x) = S m(x,x) for all x ∈ X Define M ij:X2→ Y by

M ij(x, y) : = f (0, ,0,x,0, ,0, y,0, ,0) − f (0, ,0,x,0, ,0,0,0, ,0)

for alli, j with 1 ≤ i < j ≤ m and all x, y ∈ X On the right-hand side of (2.12),x and y

are theith and the jth components, respectively Then, M ijare biadditive for alli, j with

1≤ i < j ≤ m Indeed, by (1.2) and (2.12), we obtain

M ij

x1+x2,y

= f0, ,0,x1+x2, 0, ,0, y,0, ,0− f0, ,0,x1+x2, 0, ,0,0,0, ,0

− f (0, ,0,0,0, ,0, y,0, ,0)

= f0, ,0,x1+x2, 0, ,0, y,0, ,0

−1

2



f0, ,0,x1+x2, 0, ,0, y,0, ,0+ f0, ,0,x1+x2, 0, ,0, − y,0, ,0

=1

2



f0, ,0,x1+x2, 0, ,0, y,0, ,0− f0, ,0,x1+x2, 0, ,0, − y,0, ,0

= f (0, ,0,x1+x2, 0, ,0, y,0, ,0)

−1

2



f0, ,0,x1+x2, 0, ,0, y,0, ,0+ f0, ,0,x1+x2, 0, ,0, − y,0, ,0

= f0, ,0,x1+x2, 0, ,0, y,0, ,0− f0, ,0,x1+x2, 0, ,0,0,0, ,0

− f (0, ,0,0,0, ,0, y,0, ,0)

=1

2



2f0, ,0,x1+x2, 0, ,0, y,0, ,0+ 2f (0, ,0,0,0, ,0, y,0, ,0)

−2f0, ,0,x1+x2, 0, ,0,0,0, ,0−2f (0, ,0,0,0, ,0, y,0, ,0)

=1

2



f0, ,0,x1+x2, 0, ,0,2y,0, ,0− f0, ,0,x1+x2, 0, ,0,0,0, ,0

−2f (0, ,0,0,0, ,0, y,0, ,0)

=1

2

f0, ,0,x1+x2, 0, ,0,2y,0, ,0+f0, ,0,x1− x2, 0, ,0,0,0, ,0

−1

2



f0, ,0,x1+x2, 0, ,0,0,0, ,0+f0, ,0,x1− x2, 0, ,0,0,0, ,0

−2f (0, ,0,0,0, ,0, y,0, ,0)

= f0, ,0,x1, 0, ,0, y,0, ,0+f0, ,0,x2, 0, ,0, y,0, ,0

− f0, ,0,x1, 0, ,0,0,0, ,0− f0, ,0,x2, 0, ,0,0,0, ,0

−2f (0, ,0,0,0, ,0, y,0, ,0)

= f0, ,0,x1, 0, ,0, y,0, ,0

−f0, ,0,x1, 0, ,0,0,0, ,0+f (0, ,0,0,0, ,0, y,0, ,0)

+f0, ,0,x2, 0, ,0, y,0, ,0

−f0, ,0,x2, 0, ,0,0,0, ,0+f (0, ,0,0,0, ,0, y,0, ,0

= M ij

0, ,0,x1, 0, ,0, y,0, ,0+M ij

0, ,0,x2, 0, ,0, y,0, ,0

(2.13)

for allx1,x2,y ∈ X Similarly,

M ij

0, ,0,x,0, ,0, y1+y2, 0, ,0

= M ij

0, ,0,x,0, ,0, y1, 0, ,0+M ij

0, ,0,x,0, ,0, y2, 0, ,0 (2.14)

for allx, y1,y2∈ X.

Conversely, we assume that there exist symmetric biadditive mappingsS1, ,S m:X2→ Y

and biadditive mappingsM ij:X2→ Y (1 ≤ i < j ≤ m) such that

fx1, ,x m

=

m



i =1

S i

x i,x i

1≤ i<j ≤

M ij

x i,x j

(2.15)

for allx1, ,x m ∈ X Since M ij(1≤ i < j ≤ m) are biadditive and S1, ,S mare symmetric biadditive,

fx1+y1, ,x m+y m

+ fx1− y1, ,x m − y m

=

m



i =1

S i

x i+y i,x i+y i

1≤ i<j ≤

M ij

x i+y i,x j+y j

+

m



i =1

S i

x i − y i,x i − y i

1≤ i<j ≤

M ij

x i − y i,x j − y j

=

m



i =1



S i

x i,x i

+ 2S i

x i,y i) +S i

y i,y i

1≤ i<j ≤



M ij

x i,x j

+M ij

x i,y j

+M ij

y i,x j

+M ij

y i,y j

+

m



i =1



S i

x i,x i

−2S i

x i,y i

+S i

y i,y i

1≤ i<j ≤



M ij

x i,x j

− M ij

x i,y j

− M ij

y i,x j

+M ij

y i,y j

=2

m



i =1

S i

x i,x i

1≤ i<j ≤

M ij

x i,x j

+ 2

m



i =1

S i

y i,y i

1≤ i<j ≤

M ij

y i,y j

=2fx1, ,x m

+ 2fy1, , y m

(2.16)

LetY be complete and let ϕ : X2m →[0,∞) be a function satisfying

ϕx1, ,x m,y1, , y m

:=

∞



j =0

1

4j+1 ϕ2j x1, ,2 j x m, 2j y1, ,2 j y m

< ∞ (2.17) for allx1, ,x m,y1, , y m ∈ X.

Theorem 2.7 Let f : X m → Y be a mapping such that

fx1+y1, ,x m+y m

+fx1− y1, ,x m − y m

−2fx1, ,x m

−2fy1, , y m ϕx1, ,x m,y1, , y m (2.18)

for all x1, ,x m,y1, , y m ∈ X Then, there exists a unique m-variable quadratic mapping

F : X m → Y such that

fx1, ,x m

− Fx1, ,x m ϕx1, ,x m,x1, ,x m

(2.19)

for all x1, ,x m ∈ X The mapping F is given by

Fx1, ,x m

:=lim

j →∞

1

4j f2j x1, ,2 j x m

(2.20)

for all x1, ,x m ∈ X.

Proof Letting y1= x1, , y m = x min (2.18), we have

fx1, ,x m

−1

4

f (0, ,0) + f2x1, ,2x m 1

4ϕx1, ,x m,x1, ,x m

(2.21) for allx1, ,x m ∈ X Thus, we obtain

1

4j f2j x1, ,2 j x m

− 1

4j+1



f (0, ,0) + f2j+1 x1, ,2 j+1 x m

≤ 1

4j+1 ϕ2j x1, ,2 j x m, 2j x1, ,2 j x m (2.22)

for allx1, ,x m ∈ X and all j For given integers l, n (0 ≤ l < n), we get

1

4l f2l x1, ,2 l x m

− 1

4n



f (0, ,0) + f2n x1, ,2 n x m

n −1



j = l

1

4j+1 ϕ2j x1, ,2 j x m, 2j x1, ,2 j x m (2.23)

for all x1, ,x m ∈ X By (2.23), the sequence{(1/4 j)f (2 j x1, ,2 j x m)}is a Cauchy se-quence for allx1, ,x m ∈ X Since Y is complete, the sequence {(1/4 j)f (2 j x1, ,2 j x m)}

converges for allx1, ,x m ∈ X Define F : X m → Y by

Fx1, ,x m

:=lim

j →∞

1

4j f2j x1, ,2 j x m

(2.24)

for allx1, ,x m ∈ X By (2.18), we have

1

4j f2j

x1+y1



, ,2 j

x m+y m

+ 1

4j f2j

x1− y1



, ,2 j

x m − y m

− 2

4j f2j x1, ,2 j x m

− 2

4j f2j y1, ,2 j y m 1

4j ϕ2j x1, ,2 j x m, 2j y1, ,2 j y m

(2.25) for allx1, ,x m,y1, , y m ∈ X and all j Letting j →∞and using (2.17), we see thatF

sat-isfies (1.2) Settingl =0 and takingn →∞in (2.23), one can obtain the inequality (2.19)

IfG : X m → Y is another m-variable quadratic mapping satisfying (2.19), we obtain

Fx1, ,x m

− Gx1, ,x m

= 1

4n F2n x1, ,2 n x m

− G2n x1, ,2 n x m

≤ 1

4n F2n x1, ,2 n x m

− f2n x1, ,2 n x m

+ 1

4n f2n x1, ,2 n x m

− G2n x1, ,2 n x m

≤ 2

4n ϕ2n x1, ,2 n x m, 2n x1, ,2 n x m

−→0 asn −→ ∞

(2.26)

for allx1, ,x m ∈ X Hence, the mapping F is the unique m-variable quadratic mapping,

References

[1] J Acz´el and J Dhombres, Functional Equations in Several Variables, vol 31 of Encyclopedia of

Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.

[2] J.-H Bae and K.-W Jun, “On the generalized Hyers-Ulam-Rassias stability of ann-dimensional

quadratic functional equation,” Journal of Mathematical Analysis and Applications, vol 258,

no 1, pp 183–193, 2001.

[3] J.-H Bae and W.-G Park, “On the generalized Hyers-Ulam-Rassias stability in Banach modules over aC ∗ -algebra,” Journal of Mathematical Analysis and Applications, vol 294, no 1, pp 196–

205, 2004.

[4] J.-H Bae and W.-G Park, “On stability of a functional equation withn-variables,” Nonlinear Analysis: Theory, Methods & Applications, vol 64, no 4, pp 856–868, 2006.

[5] S.-M Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic

property,” Journal of Mathematical Analysis and Applications, vol 222, no 1, pp 126–137, 1998 [6] W.-G Park and J.-H Bae, “On a bi-quadratic functional equation and its stability,” Nonlinear

Analysis: Theory, Methods & Applications, vol 62, no 4, pp 643–654, 2005.

Won-Gil Park: National Institute for Mathematical Sciences, 385-16 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea

Email address:wgpark@nims.re.kr

Jae-Hyeong Bae: Department of Applied Mathematics, Kyung Hee University, Yongin 449-701, South Korea

Email address:jhbae@khu.ac.kr

quadratic functional equation, ” Journal of Mathematical Analysis and Applications,...