Volume 2007, Article ID 10725,15pages doi:10.1155/2007/10725
Research Article
Stability Problem of Ulam for Euler-Lagrange
Quadratic Mappings
Hark-Mahn Kim, John Michael Rassias, and Young-Sun Cho
Received 26 May 2007; Revised 9 August 2007; Accepted 9 November 2007
Recommended by Ondrej Dosly
We solve the generalized Hyers-Ulam stability problem for multidimensional Euler-Lagrange quadratic mappings which extend the original Euler-Euler-Lagrange quadratic map-pings.
Copyright © 2007 Hark-Mahn Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In 1940, Ulam [1] proposed, at the University of Wisconsin, the following problem: “give conditions in order for a linear mapping near an approximately linear mapping to exist.” In 1968, Ulam proposed the general Ulam stability problem: “when is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” 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 “how do the solutions of the inequality differ from those of the given functional equation?” If the answer is affirmative, we would say that the equation is stable. In 1978, Gruber [2] remarked that Ulam problem is of particular interest in probability theory and in the case of functional equations of different types. We wish to note that stability properties of different functional equations can have applications to unrelated fields. For instance, Zhou [3] used a stability property of the functional equationf(x−y) +f(x+y)=2f(x) to prove a conjecture ofZ. Ditzian about the relationship between the smoothness of a mapping and the degree of its approximation by the associated Bernstein polynomials.
to become 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 [7–9] and references therein.
We note that J. M. Rassias introduced theEuler-Lagrange quadratic mappings, moti-vated from the following pertinent algebraic equation
a1x1+a2x22
+a2x1−a1x2 2
=a2
1+a22x1 2
+x2 2
. (1.1)
Thus the second author of this paper introduced and investigated the stability problem of Ulam for the relativeEuler-Lagrange functional equation
fa1x1+a2x2
+fa2x1−a1x2
=a2 1+a22
fx1
+fx2
(1.2)
in the publications [10–12]. Analogous quadratic mappings were introduced and inves-tigated through J. M. Rassias publications [13–15]. Therefore, these mappings could be namedEuler-Lagrange mappingsand the corresponding Euler-Lagrange equations might be calledEuler-Lagrange equations. Before 1992, these mappings and equations were not known at all in functional equations and inequalities. However, a completely different kind of Euler-Lagrange partial differential equations is known in calculus of variations. Already, some mathematicians have employed these Euler-Lagrange mappings [16–22].
In addition, J. M. Rassias [23] generalized the above functional equation (1.2) as fol-lows. LetXandYbe real linear spaces. Then a mappingQ:X→Yis calledquadratic with respect toaif the functional equation
Q
n
i=1 aixi
+
1≤i< j≤n
Qajxi−aixj= n
i=1 a2
i
n
i=1
Qxi (1.3)
holds for all vector (x1,...,xn)∈Xn, wherea:=(a1,...,an)∈Rnof nonzero reals, andn≥
2 is arbitrary, but fixed, such that 0< m:=ni=1a2i =[1 + (n2)]/n. In this case, a mapping
Qa:Xn→Y defined by
Qax1,...,xn
:=
n
i=1Q
aixi
n
i=1a2i
(1.4)
is called thesquare of the quadratic weighted mean ofQwith respect toa.
For everyx∈R, setQ(x)=x2. Then the mappingQa:Rn→Ris quadratic such that
Qa(x,...,x)=x2. Denoting byx2
wthe quadratic weighted mean, we note that the
above-mentioned mapping Qa is an analogous case to the square of the quadratic weighted mean employed in mathematical statistics:x2
w=
n
i=1wix2i/
n
i=1wiwith weightswi=a2i,
dataxi, andQ(aixi)=(aixi)2fori=1,...,n, wheren≥2 is arbitrary, but fixed.
2. Stability of (1.3) in quasi-Banach spaces
We will investigate under what conditions it is then possible to find a true quadratic mapping of Euler-Lagrange near an approximate Euler-Lagrange quadratic mapping with small error.
We recall some basic facts concerning quasi-Banach spaces and some preliminary re-sults.
Definition 2.1(see [24,25]). LetX be a linear space. Aquasinorm·is a real-valued function onXsatisfying the following:
(1)x ≥0 for allx∈Xandx =0 if and only ifx=0; (2)λx = |λ|·xfor allλ∈Rand allx∈X;
(3) there is a constantKsuch thatx+y ≤K(x+y) for allx,y∈X.
The smallest possibleK is called themodulus of concavityof·. The pair (X,·) is called aquasinormed spaceif·is a quasinorm onX. Aquasi-Banach spaceis a complete quasinormed space.
A quasinorm·is called ap-norm(0< p≤1) if
x+yp≤ xp+yp (2.1)
for allx,y∈X. In this case, a quasi-Banach space is called ap-Banach space.
Clearly, p-norms are continuous, and in fact, if·is ap-norm onX, then the for-mulad(x,y) := x−yp defines a translation invariant metric forX, and·p is a p
-homogeneousF-norm. The Aoki-Rolewicz theorem [24,25] guarantees that each quasi-norm is equivalent to some p-norm for some 0< p≤1. In this section, we are going to prove the generalized Ulam stability of mappings satisfying approximately (1.3) in quasi-Banach spaces, and inp-Banach spaces, respectively. LetXbe a quasinormed space and Y a quasi-Banach space with the modulus of concavityK≥1 of·.
Given a mappingf :X→Y, we define a difference operatorDaf :Xn→Yfor notational
convenience as
Daf
x1,...,xn
:= f
n
i=1 aixi
+
1≤i< j≤n
fajxi−aixj
− n
i=1 a2
i
n
i=1 fxi
, (2.2)
which is called the approximate remainder of the functional equation (1.3) and acts as a perturbation of the equation, wherea:=(a1,...,an)∈Rnof nonzero reals, andn≥2 is
arbitrary, but fixed, such that 0< m:=n
i=1a2i ∈ {[1 + (n2)]/n, √
K}.
Lemma 2.2 [23]. LetQ:X→Y be a Euler-Lagrange quadratic mapping satisfying (1.3). ThenQsatisfies the equation
Qmpx=m2pQ(x) (2.3)
for allx∈Xandp∈N,where0< m:=n
Theorem 2.3. Assume that there exists a mapping ϕ:Xn→[0,∞)for which a mapping
f :X→Y satisfies
Daf
x1,...,xn≤ϕx1,...,xn (2.4)
and the series
Φx1,...,xn
:=
∞
i=0
Kiϕmix1,...,mix n
m2i <∞ (2.5)
for all x1,...,xn∈X. Then there exists a unique Euler-Lagrange quadratic mappingQ:
X→Ysuch thatQsatisfies (1.3), that is,
DaQx1,...,xn=0 (2.6)
for allx1,...,xn∈X, and the inequality
f(x)−Q(x)≤K
mΦ(x, 0,..., 0) + K m2Φ
a1x,...,anx
+K(n−1)(m+ 1)|n−2m|f(0) 2m2−K
(2.7)
holds for allx∈X,where
f(0)=0, ifm <√K, f(0)≤ ϕ(0,..., 0)
mn−[1 + (n2)]
, ifm >√K. (2.8)
The mappingQis given by
Q(x)=lim
k→∞
fmkx
m2k (2.9)
for allx∈X.
Proof. Substitution ofxi=0 (i=1,...,n) in the functional inequality (2.4) yields that
mn−
1 + n
2
f(0)≤ϕ(0,..., 0). (2.10)
Thus we note that ifm <√K, thenϕ(0,..., 0)=0 by the convergence ofΦ(0,..., 0), and so f(0)=0. Substitutingx1=x,xj=0 (j=2,...,n) in the functional inequality (2.4), we
obtain
n
i=1 faix
−m f(x) + n−1 2
−m(n−1)
f(0)≤ϕ(x, 0,..., 0) (2.11)
or
fa(x,...,x)−f(x) +
1 m
n−1 2
−(n−1)
f(0)≤ϕ(x, 0,..., 0)
for allx∈X.In addition, replacingxibyaixin (2.4), one gets the inequality
f(mx) + n2
f(0)−m
n
i=1 faix
≤ϕa1x,...,anx
(2.13)
or
f(mmx2 )+ 1 m2
n 2
f(0)−fa(x,...,x)≤ϕ
a1x,...,anx
m2 (2.14)
for allx∈X.From this inequality and (2.12) as well as the triangle inequality, we get the basic inequality
f(mmx2 )−f(x) +
n(n−1) 2m2 +
(n−1)(n−2)
2m −(n−1)
f(0)
≤ϕ(x, 0,..., 0)
m +
ϕa1x,...,anx
m2
(2.15)
or
f(mmx2 )−f(x)
≤ε(x) :=mϕ(x, 0,..., 0) +ϕ
a1x,...,anx
m2 +
(n−1)(m+ 1)|n−2m|f(0) 2m2
(2.16)
for allx∈X.
By induction onl∈N, we prove the general functional inequality
fmm2llx−f(x)
≤Kl−1ε
ml−1x m2(l−1) +K
l−2
i=0
Kiεmix
m2i (2.17)
for allx∈Xand all nonnegative integerl.In fact, we calculate the inequality
fmm2(ll+1+1)x−f(x) ≤Kf
ml+1x m2(l+1) −
f(mx) m2
+Kf(mx) m2 −f(x)
≤ K
m2
Kl−1εmlx
m2(l−1) +K
l−2
i=0
Kiεmi+1x m2i
+Kε(x)
=Klε
mlx
m2l +K l−1
i=0
Kiεmix
m2i
(2.18)
for allx∈X.
It follows from (2.5) and (2.17) that a sequence{fl(x)}of mappings fl(x) :=f(mlx)/
m2lis Cauchy in the quasi-Banach spaceY, and it thus converges. Therefore, we see that
a mappingQ:X→Y defined by
Q(x) :=lim
l→∞ fmlx
exists for allx∈X.Taking the limitl→∞in (2.17), we find that
f(x)−Q(x)≤K∞
i=0
Kiεmix
m2i
=K
mΦ(x, 0,..., 0) + K m2Φ
a1x,...,anx
+K(n−1)(m+ 1)|n−2m|f(0) 2m2−K
(2.20)
for allx∈X. Therefore, the mappingQnear the approximate mappingf :X→Y of (1.3) satisfies the inequality (2.7). In addition, it is clear from (2.4) that the following inequality
1 m2lDaf
mlx1,...,mlx
n≤m12lϕ
mlx1,...,mlx n
(2.21)
holds for allx1,...,xn∈Xand alll∈N.Taking the limitl→∞, we see that the mappingQ
satisfies the equationDaQ(x1,...,xn)=0, and soQis Euler-Lagrange quadratic mapping.
Let ˇQ:X→Ybe another Euler-Lagrange quadratic mapping satisfying the equation
DaQˇ
x1,...,xn
=0 (2.22)
and the inequality (2.7). To prove the before-mentioned uniqueness, we employ (2.4) so that
Q(x)=m−2lQmlx, Qˇ(x)=m−2lQˇmlx (2.23)
hold for allx∈Xandl∈N.Thus from the last equality and inequality (2.7), one proves that
Q(x)−Qˇ(x)= 1 m2lQ
mlx−Qˇmlx
≤ K
m2lQ
mlx−fmlx+fmlx−Qˇmlx
≤ 2K2
m2Kl
∞
i=0
mKi+lϕmi+lx, 0,..., 0+Ki+lϕa1mi+lx,...,a nmi+lx
m2(i+l)
+K
2(n−1)(m+ 1)|n−2m|f(0)
m2−Km2l
(2.24)
for allx∈Xand alll∈N.Therefore, froml→∞, one establishes
Q(x)−Qˇ(x)=0 (2.25)
for allx∈X, completing the proof of uniqueness.
Theorem 2.4. Assume that there exists a mapping ϕ:Xn→[0,∞)for which a mapping
f :X→Y satisfies
Daf
x1,...,xn≤ψ
x1,...,xn
and the series
Ψx1,...,xn
:=
∞
i=1 Kim2iψ
x1 mi,...,
xn
mi
<∞ (2.27)
for all x1,...,xn∈X. Then there exists a unique Euler-Lagrange quadratic mappingQ:
X→Ysuch thatQsatisfies (1.3), that is,
DaQ
x1,...,xn
=0 (2.28)
for allx1,...,xn∈X, and the inequality
f(x)−Q(x)≤ 1
mΨ(x, 0,..., 0) + 1 m2Ψ
a1x,...,anx
+K(n−1)(m+ 1)|n−2m|f(0) 21−Km2
(2.29)
holds for allx∈X, where
f(0)=0, ifm >√1
K, f(0)≤
ϕ(0,..., 0) mn−[1 + (n
2)]
, ifm <√1
K. (2.30)
The mappingQis given by
Q(x)=lim
k→∞m 2kf
x mk
(2.31)
for allx∈X.
Proof. We note that ifm >1/√K, thenψ(0,..., 0)=0 by the convergence ofΨ(0,..., 0), and sof(0)=0. Using the same arguments as those of (2.12)–(2.17), we prove the general functional inequality
f(x)−m2lf
x ml
≤l
−1
i=1 Kim2iε
x mi
+Kl−1m2lε
x ml
(2.32)
for allx∈Xand all nonnegative integerl >1, where
ε(x) :=mψ(x, 0,..., 0) +ψ
a1x,...,anx
m2 +
(n−1)(m+ 1)|n−2m|f(0)
2m2 . (2.33)
The rest of the proof goes through by the same way as that ofTheorem 2.3.
Corollary2.5. LetᏭbe a normed space andᏮa Banach space, and letθ,pbe positive real numbers withp=2. Assume that a mappingf :Ꮽ→Ꮾsatisfies
D
af
x1,...,xn≤θx1p+···+xnp
for allx1,...,xn∈Ꮽ. Then there exists a unique Euler-Lagrange quadratic mappingQ:
Ꮽ→Ꮾsuch that
DaQ
x1,...,xn
=0 (2.35)
for allx1,...,xn∈Ꮽ, and
f(x)−Q(x)≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
θxpm+n i=1aip
m2−mp ifm >1, 0< p <2,
orm <1,p >2, θxpm+n
i=1aip
mp−m2 ifm <1, 0< p <2, orm >1,p >2
(2.36)
for allx∈Ꮽ.
Remark 2.6. We remark that inCorollary 2.5the casep=2 is not discussed. The Euler-Lagrange type quadratic functional equation (1.3) is not stable as we will see in the follow-ing example withn=2. This counterexample is a modification of the example contained in [26,27].
Let us define a mapping f :R→Rby
f(x)=∞
n=0 ϕ2nx
4n , (2.37)
where the mappingϕ:R→Ris given by
ϕ(x)= ⎧ ⎨ ⎩
1 if|x| ≥1;
x2 if|x|<1. (2.38)
Then the mapping f satisfies the inequality f
a1x1+a2x2
+ fa2x1−a1x2
−a2 1+a22
fx1
+fx2
≤32 3
1 +a2
1+a22 2
x2+y2 (2.39)
for allx,y∈R, but there exist no Euler-Lagrange quadratic mappingQ:R→R, and a constantb >0 such that
f(x)−Q(x)≤bx2 (2.40)
for allx∈R.
In fact, forx=y=0 or forx,y∈Rsuch thatx2+y2≥1/4(1 +a2
1+a22), it is clear that f
a1x1+a2x2
+ fa2x1−a1x2
−a21+a22
fx1
+fx2
≤8 3
1 +a21+a22
≤32 3
1 +a21+a22 2
because|f(x)| ≤4/3 for allx∈R.Now, we consider the case 0< x2+y2<1/4(1 +a2 1+ a22).Choose a positive integerk∈Nsuch that
1 4k+11 +a2
1+a22
≤x2+y2< 1 4k1 +a2
1+a22
. (2.42)
Then one has 4k−1x2<1/4|a
i|2, 4k−1y2<1/4|ai|2, and so
2k−1x, 2k−1y, 2k−1a 1x+a2y
, 2k−1a2x−a1y
∈(−1, 1). (2.43)
Therefore, we have
2nx, 2ny, 2na1x+a2y, 2na2x−a1y∈(−1, 1), (2.44)
and hence
ϕa1x1+a2x2
+ϕa2x1−a1x2
−a2 1+a22
ϕx1
+ϕx2
=0 (2.45)
for eachn=0, 1,...,k−1.Thus we obtain, using (2.42) and (2.45), f
a1x1+a2x2
+fa2x1−a1x2
−a21+a22
fx1
+fx2
≤∞
n=0 1 4nϕ
2na1x1+a2x2
+ϕ2na2x1−a1x2
−a2 1+a22
ϕ2nx1
+ϕ2nx2
≤∞
n=k
1 4nϕ
2na1x1+a2x2
+ϕ2na2x1−a1x2
−a2 1+a22
ϕ2nx1
+ϕ2nx2
≤ ∞
n=k
21 +a21+a22
4n =
321 +a21+a22
3·4k+1 ≤
321 +a21+a22 2
3
x2+y2,
(2.46)
which yields the inequality (2.39).
Now, assume that there exist an Euler-Lagrange quadratic mappingQ:R→Rand a constantb >0 such that
f(x)−Q(x)≤bx2 (2.47)
for allx∈R.Since|Q(x)| ≤ |f(x)|+bx2≤4/3 +bx2is locally bounded, the mappingQ is of the formQ(x)=cx2,x∈Rfor some constantc[28]. Hence one obtains
f(x)≤
b+|c|x2 (2.48)
for allx∈R.On the other hand, form∈Nwithm > b+|c|andx∈(0, 1/2m−1), we have 2nx∈(0, 1) for alln≤m−1, and so
f(x)= ∞
n=0 ϕ2nx
4n ≥ m−1
n=0
2nx2 4n =mx
2>b+|c|x2, (2.49)
Corollary2.7. LetᏭbe a normed space,Ꮾa Banach space, andθ,pipositive real
num-bers such thatp:=n
i=1pi=2.Assume that a mapping f :Ꮽ→Ꮾsatisfies
Daf
x1,...,xn≤θ n
i=1 xipi
(2.50)
for allx1,...,xn∈Ꮽ. Then there exists a unique Euler-Lagrange quadratic mappingQ:
Ꮽ→Ꮾsuch that
DaQx1,...,xn=0 (2.51)
for allx1,...,xn∈Ꮽand
f(x)−Q(x)≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
θxpn i=1aipi
m2−mp ifm >1, 0< p <2,
orm <1, p >2, θxpn
i=1aipi
mp−m2 ifm <1, 0< p <2, orm >1, p >2
(2.52)
for allx∈Ꮽ.
In casen=2, we have the Hyers-Ulam stability result as a special case of Theorems2.3
and2.4for the Euler-Lagrange type quadratic functional equation (1.2).
Corollary 2.8. Let Ꮽ be a linear space,Ꮾa Banach space, and0≤θ a real number. Assume that a mappingf :Ꮽ→Ꮾsatisfies
D
af
x1,...,xn≤θ (2.53)
for allx1,...,xn∈Ꮽ. Then there exists a unique Euler-Lagrange quadratic mappingQ:
Ꮽ→Ꮾsuch that
DaQ
x1,...,xn
=0 (2.54)
for allx1,...,xn∈Ꮽ, and the inequality
f(x)−Q(x)≤ θ
|m−1|+
θ(n−1)|n−2m|f(0)
2|m−1| (2.55)
for allx∈Ꮽ.
3. Stability of (1.3) in Banach modules
In the last part of this paper, letBbe a unital Banach algebra with norm|·|, and letBM1 andBM2be left BanachB-modules with norms·and·, respectively.
Theorem3.1. Assume that there exists a mappingϕ:BM21→[0,∞)for which a mapping f : BM1→BM2satisfies
Da,uf
x1,...,xn
:=f
n
i=1 aiuxi
+
1≤i< j≤n
fajuxi−aiuxj
−
n
i=1 a2i
u2
n
i=1 fxi
≤ϕ
x1,...,xn
(3.1)
for allx1,...,xn∈BM1and allu∈B(1) := {u∈B| |u| =1}, and the series (2.5) withK=1
converges for allx1,...,xn∈BM1. If f is measurable, or for each fixedx∈BM1, the
map-pingRt→f(tx)∈BM2is continuous, then there exists a unique Euler-Lagrange quadratic
mappingQ:BM1→BM2such that the following equations
DaQ
x1,...,xn
=0, Q(bx)=b2Q(x) (3.2)
and the inequality (2.7) withK=1hold for allx,x1,...,xn∈BM1and allb∈B.
Proof. From (3.1) withu=1, it follows byTheorem 2.3that there exists a unique Euler-Lagrange quadratic mappingQ:BM1→BM2such that
DaQ
x1,...,xn
=0 (3.3)
for allx1,...,xn∈BM1and the inequality (2.7) withK=1 for allx∈BM1.
The mappingQis given by
Q(x)=lim
k→∞
fmkx
m2k (3.4)
for allx∈BM1.
Furthermore, suppose that f is measurable, or for each fixedx∈BM1, the mapping
f(tx) is continuous with respect tot∈R. Then for any continuous linear functionalL defined onBM2, letΦ:R→Rbe given by
Φ(t) :=LQ(tx) (3.5)
fort∈R, wherexis fixed. ThenΦis a quadratic mapping and, moreover, is also measur-able since it is the pointwise limit of the sequence
Φk(t) :=m−2kL
fmktx. (3.6)
Hence it has the formΦ(t)=t2Φ(1) for allt∈R[7]. Therefore, one obtains that for each fixedx∈BM1and allt∈R,
LQ(tx)=Φ(t)=t2Φ(1)=t2LQ(x)=Lt2Q(x), (3.7)
which implies the condition
Q(tx)=t2Q(x), ∀x∈
That is,QisR-quadratic. Replacingx1,...,xnbymkx1,...,mkxnin (3.1), respectively, and
dividing it bym2k, we figure out
Da,uf
mkx1,...,mkx n
m2k ≤
ϕmkx1,...,mkx n
m2k (3.9)
for allu∈B(1) and for allx1,...,xn∈BM1.Taking the limitk→∞, one obtains by condi-tion (2.5) that
Da,uQ
x1,...,xn
=0 (3.10)
for allx1,...,xn∈BM1 and allu∈B(1).Substitutingx1=x,xj=0 (j=2,...,n) in the
functional inequality (3.3), we obtain
n
i=1 Qaix
−mQ(x)=0, (3.11)
for allx∈BM1.In addition, replacingxibyaixin (3.10), one gets the inequality
Q(mux)−mu2
n
i=1 Qaix
=0 (3.12)
or
mQ(ux)−u2n
i=1 Qaix
=0 (3.13)
for allx∈BM1and allu∈B(1).Associating the last two equations, we have
Q(ux)−u2Q(x)=0 (3.14)
for allx∈BM1and allu∈B(1).The last equality is also true foru=0 vacuously. Now, for each elementb∈B(b=0), we figure out
Q(bx)=Q
|b|· b
|b|x
= |b|2· Q
b
|b|x
= |b|2· b2
|b|2·Q(x)=b 2Q(x)
(3.15)
for allb∈B(b=0) and allx∈BM1.Thus the mappingQsatisfies
Q(bx)=b2Q(x) (3.16)
for allb∈Band for allx∈BM1, as desired. This completes the proof of the theorem.
Theorem3.2. Assume that a mapping f :BM1→BM2satisfies D
a,uf
x1,...,xn≤ϕ
x1,...,xn
(3.17)
for allx1,...,xn∈BM1and allu∈B(1), and the series (2.27) withK=1converges for all x1,...,xn∈BM1. Iff is measurable, or for each fixedx∈BM1, the mappingRt→f(tx)∈
BM2is continuous, then there exists a unique Euler-Lagrange quadratic mappingQ:BM1→
BM2such that the following equations
DaQ
x1,...,xn
=0, Q(bx)=b2Q(x) (3.18)
and the inequality (2.29) withK=1hold for allx,x1,...,xn∈BM1and allb∈B.
SinceCis a Banach algebra, the Banach spacesM1andM2are considered as Banach modules overC. Thus we have the following corollary.
Corollary3.3. Letϕbe a mapping defined as inTheorem 3.1. LetM1andM2be Banach
spaces over the complex fieldC.Suppose that a mapping f :M1→M2satisfies Da,uf
x1,...,xn≤ϕ
x1,...,xn
(3.19)
for allu∈C(1)and allx1,...,xn∈M1. If f is measurable or the mappingRt→f(tx)∈ M2is continuous for each fixedx∈M1, then there exists a unique Euler-Lagrange quadratic
mappingQ:M1→M2such that the following equations
DaQx1,...,xn=0, Q(cx)=c2Q(x) (3.20)
hold for allx1,...,xn∈M1and allc∈C,and the inequality (2.7) withK=1holds for all x∈M1.
Theorem3.4. Assume that there exists a mappingϕ:BM21→[0,∞)for which a mapping f :BM1→BM2satisfies
u2f
n
i=1 aixi
+
1≤i< j≤n
u2fa
jxi−aixj− n
i=1 a2i
n
i=1 fuxi
≤ϕx1,...,xn (3.21)
for all x1,...,xn∈BM1 and allu∈B(1), and the series (2.5) with K=1 converges for
allx1,...,xn∈BM1. If f is measurable or the mapping Rt→f(tx)∈BM2 is
continu-ous for each fixedx∈BM1,then there exists a unique Euler-Lagrange quadratic mapping
Q:BM1→BM2such that the following equations
DaQx1,...,xn=0, Q(bx)=b2Q(x) (3.22)
and the inequality (2.7) withK=1hold for allx,x1,...,xn∈BM1and allb∈B.
Proof. The proof of this theorem is similar to that ofTheorem 3.1.
Acknowledgment
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Hark-Mahn Kim: Department of Mathematics, College of Natural Sciences, Chungnam National University, 220 Yuseong-Gu, Daejeon 305-764, South Korea
Email address:[email protected]
John Michael Rassias: Pedagogical Department E. E., National and Capodistrian University of Athens, Section of Mathematics and Informatics, 4 Agamemnonos St., Aghia Paraskevi, Athens 15342, Greece
Email address:[email protected]
Young-Sun Cho: Department of Mathematics, College of Natural Sciences, Chungnam National University, 220 Yuseong-Gu, Daejeon 305-764, South Korea
