Volume 2007, Article ID 24716,8pages 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,...,xm+ym) + f(x1−y1,...,xm−ym)=2f(x1,...,xm) + 2f(y1,...,
ym).The quadratic form f(x1,...,xm)=1≤i≤j≤maijxixjis 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, letXandYbe real vector spaces. A mapping f is called aquadratic formif there existaij∈R(1≤i≤j≤m) such that
fx1,...,xm=
1≤i≤j≤m
aijxixj (1.1)
for allx1,...,xm∈X.
For a mapping f :Xm→Y, consider them-variable quadratic functional equation
fx1+y1,...,xm+ym+fx1−y1,...,xm−ym=2fx1,...,xm+ 2fy1,...,ym. (1.2)
WhenX=Y=R, the quadratic form f :Rm→Rgiven by
fx1,...,xm=
1≤i≤j≤m
aijxixj (1.3)
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.
Theorem2.1. Let f :Xm→Ybe a mapping satisfying (1.2) and letg:X→Ybe the mapping given by
g(x) :=f(x,...,x) (2.1)
for allx∈X, thengsatisfies (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) + 2f(y,...,y)=2g(x) + 2g(y) (2.2)
for allx,y∈X.
The quadratic functional equation (1.4) induces them-variable quadratic functional equation (1.2) with an additional condition.
Theorem2.2. Letaij∈R(1≤i≤j≤m)andg:X→Y be a mapping satisfying (1.4). If
f :Xm→Yis the mapping given by
fx1,...,xm:=
m
i=1
aiigxi+1 4
1≤i<j≤m
aijgxi+xj−gxi−xj (2.3)
for allx1,...,xm∈X, thenf satisfies (1.2). Furthermore, (2.1) holds if
1≤i≤j≤m
Proof. By (1.4) and (2.3),
fx1+y1,...,xm+ym+fx1−y1,...,xm−ym
=
m
i=1
aiigxi+yi+gxi−yi
+1 4
1≤i<j≤maij
gxi+yi+xj+yj−gxi+yi−xj−yj
+1 4
1≤i<j≤m
aijgxi−yi+xj−yj−gxi−yi−xj+yj
=2 m
i=1
aiigxi+gyi
+1 4
1≤i<j≤m
aijgxi+yi+xj+yj+gxi−yi+xj−yj
−1 4
1≤i<j≤m
aijgxi+yi−xj−yj+gxi−yi−xj+yj
=2 m
i=1
aiigxi+gyi
+1 2
1≤i<j≤m
aijgxi+xj+gyi+yj−gxi−xj−gyi−yj
=2fx1,...,xm+ 2fy1,...,ym
(2.5)
for allx1,...,xm,y1,...,ym∈X.
Lettingx=y=0 andy=xin (1.4), respectively,
g(0)=0, g(2x)=4g(x) (2.6)
for allx∈X. By (2.3) and the above two equalities,
f(x,...,x)=m
i=1aiig(x) + 1 4
1≤i<j≤maij
g(2x)−g(0)=
1≤i≤j≤maijg(x)=g(x) (2.7)
for allx∈X.
Example 2.3. The functiong:R→Rgiven byg(x)=x2 satisfies (1.4). ByTheorem 2.2, the quadratic form f :Rm→Rgiven by
fx1,...,xm= 1≤i≤j≤m
aijxiyj (2.8)
Example 2.4. Letg: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,...,zm= m
i=1
zizi
i
j=1
aji+1 2
m
j=i+1
aij , (2.9)
then f satisfies them-variable quadratic functional equation (1.2).
Example 2.5. LetM2(R) be the real vector space of all 2×2 real matrices andg:M2(R)→R
the determinant function given by
g(A)=det (A) (2.10)
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 :Xm→Y satisfies (1.2) if and only if there exist symmetric biadditive mappingsS1,...,Sm:X2→Y and biadditive mappingsMij:X2→Y(1≤i < j≤
m)such that
fx1,...,xm= m
i=1
Sixi,xi+ 1≤i<j≤m
Mijxi,xj (2.11)
for allx1,...,xm∈X.
Proof. We first assume that f is a solution of (1.2). Define f1,...,fm:X→Y by f1(x) :=
f(x, 0,..., 0),...,fm(x) := f(0,..., 0,x) for allx∈X. One can easily verify that f1,...,fm are quadratic. By [1], there exist symmetric biadditive mappingsS1,...,Sm:X2→Y such that f1(x)=S1(x,x),...,fm(x)=Sm(x,x) for allx∈X. DefineMij:X2→Yby
Mij(x,y) :=f(0,..., 0,x, 0,..., 0,y, 0,..., 0)−f(0,..., 0,x, 0,..., 0, 0, 0,..., 0)
−f(0,..., 0, 0, 0,..., 0,y, 0,..., 0) (2.12)
1≤i < j≤m. Indeed, by (1.2) and (2.12), we obtain
Mijx1+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
f
0,..., 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 =Mij0,..., 0,x1, 0,..., 0,y, 0,..., 0+Mij0,..., 0,x2, 0,..., 0,y, 0,..., 0
for allx1,x2,y∈X. Similarly,
Mij0,..., 0,x, 0,..., 0,y1+y2, 0,..., 0
=Mij0,..., 0,x, 0,..., 0,y1, 0,..., 0+Mij0,..., 0,x, 0,..., 0,y2, 0,..., 0 (2.14)
for allx,y1,y2∈X.
Conversely, we assume that there exist symmetric biadditive mappingsS1,...,Sm:X2→Y and biadditive mappingsMij:X2→Y(1≤i < j≤m) such that
fx1,...,xm= m
i=1
Sixi,xi+ 1≤i<j≤m
Mijxi,xj (2.15)
for allx1,...,xm∈X. SinceMij(1≤i < j≤m) are biadditive andS1,...,Smare symmetric biadditive,
fx1+y1,...,xm+ym+ fx1−y1,...,xm−ym
=
m
i=1
Sixi+yi,xi+yi+ 1≤i<j≤m
Mijxi+yi,xj+yj
+ m
i=1
Sixi−yi,xi−yi+ 1≤i<j≤m
Mijxi−yi,xj−yj
=
m
i=1
Sixi,xi+ 2Sixi,yi) +Siyi,yi
+
1≤i<j≤m
Mijxi,xj+Mijxi,yj+Mijyi,xj+Mijyi,yj
+ m
i=1
Sixi,xi−2Sixi,yi+Siyi,yi
+
1≤i<j≤m
Mijxi,xj−Mijxi,yj−Mijyi,xj+Mijyi,yj
=2
m
i=1
Sixi,xi+ 1≤i<j≤m
Mijxi,xj
+ 2
m
i=1
Siyi,yi+
1≤i<j≤m
Mijyi,yj
=2fx1,...,xm+ 2fy1,...,ym
(2.16)
for allx1,...,xm,y1,...,ym∈X.
LetYbe complete and letϕ:X2m→[0,∞) be a function satisfying
ϕx1,...,xm,y1,...,ym:=
∞
j=0 1 4j+1ϕ
2jx1,..., 2jxm, 2jy1,..., 2jym<∞ (2.17)
Theorem2.7. Let f :Xm→Ybe a mapping such that
f
x1+y1,...,xm+ym
+fx1−y1,...,xm−ym
−2fx1,...,xm−2fy1,...,ym≤ϕx1,...,xm,y1,...,ym (2.18)
for allx1,...,xm,y1,...,ym∈X. Then, there exists a uniquem-variable quadratic mapping
F:Xm→Y such that
f
x1,...,xm−Fx1,...,xm≤ϕx1,...,xm,x1,...,xm (2.19)
for allx1,...,xm∈X. The mappingFis given by
Fx1,...,xm:=limj
→∞
1 4jf
2jx1,..., 2jxm (2.20)
for allx1,...,xm∈X.
Proof. Lettingy1=x1,...,ym=xmin (2.18), we have
fx1,...,xm−1 4
f
(0,..., 0) +f2x1,..., 2xm≤1 4ϕ
x
1,...,xm,x1,...,xm (2.21)
for allx1,...,xm∈X. Thus, we obtain
41jf
2jx1,..., 2jxm− 1 4j+1
f(0,..., 0) +f2j+1x1,..., 2j+1xm
≤ 1
4j+1ϕ
2jx1,..., 2jxm, 2jx1,..., 2jxm
(2.22)
for allx1,...,xm∈Xand all j. For given integersl,n(0≤l < n), we get
41lf
2lx1,..., 2lxm− 1 4n
f(0,..., 0) +f2nx1,..., 2nxm n−1
j=l 1 4j+1ϕ
2jx1,..., 2jxm, 2jx1,..., 2jxm
(2.23)
for all x1,...,xm∈X. By (2.23), the sequence{(1/4j)f(2jx1,..., 2jxm)}is a Cauchy se-quence for allx1,...,xm∈X. SinceYis complete, the sequence{(1/4j)f(2jx1,..., 2jxm)} converges for allx1,...,xm∈X. DefineF:Xm→Yby
Fx1,...,xm:=limj
→∞
1 4jf
for allx1,...,xm∈X. By (2.18), we have
41jf
2jx1+y1
,..., 2jxm+ym+ 1 4jf
2jx1−y1
,..., 2jxm−ym
− 2 4jf
2jx1,..., 2jxm− 2 4jf
2jy1,..., 2jym≤ 1 4jϕ
2jx1,..., 2jxm, 2jy1,..., 2jym (2.25)
for allx1,...,xm,y1,...,ym∈Xand allj. 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:Xm→Y is anotherm-variable quadratic mapping satisfying (2.19), we obtain
F
x1,...,xm−Gx1,...,xm
= 1
4nF
2nx1,..., 2nxm−G2nx1,..., 2nxm
≤ 1
4nF
2nx1,..., 2nxm−f2nx1,..., 2nxm
+ 1 4nf
2nx1,..., 2nxm−G2nx1,..., 2nxm
≤ 2
4nϕ
2nx1,..., 2nxm, 2nx1,..., 2nxm−→0 asn−→ ∞
(2.26)
for allx1,...,xm∈X. Hence, the mappingFis the uniquem-variable quadratic mapping,
as desired.
References
[1] J. Acz´el and J. Dhombres,Functional Equations in Several Variables, vol. 31 ofEncyclopedia 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:[email protected]
Jae-Hyeong Bae: Department of Applied Mathematics, Kyung Hee University, Yongin 449-701, South Korea
