Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 957541, 9pages
doi:10.1155/2011/957541
Research Article
On the Stability of Quadratic Double Centralizers
and Quadratic Multipliers: A Fixed Point Approach
Abasalt Bodaghi,1Idham Arif Alias,2and Madjid Eshaghi Gordji3
1Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran
2Laboratory of Theoretical Studies, Institute for Mathematical Research, University Putra Malaysia UPM,
43400 Serdang, Selangor Darul Ehsan, Malaysia
3Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
Correspondence should be addressed to Abasalt Bodaghi, abasalt.bodaghi@gmail.com
Received 3 December 2010; Revised 11 January 2011; Accepted 18 January 2011
Academic Editor: Michel Chipot
Copyright q2011 Abasalt Bodaghi 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.
We prove the superstability of quadratic double centralizers and of quadratic multipliers on
Banach algebras by fixed point methods. These results show that we can remove the conditions
of being weakly commutative and weakly without order which are used in the work of M. E.
Gordji et al. 2011for Banach algebras.
1. Introduction
In 1940, Ulam 1raised the following question concerning stability of group homomor-
phisms: under what condition does there exist an additive mapping near an approximately additive
mapping? Hyers 2answered the problem of Ulam for Banach spaces. He showed that for
two Banach spaces Xand Y,if>0andf:X→Ysuch that
fxy−fx−fy
≤, 1.1
for all x, y ∈X, then there exist a unique additive mapping T:X→Ysuch that
fx−Tx
≤, x∈X
.1.2
2 Journal of Inequalities and Applications
The work has been extended to quadratic functional equations. Consider f:X→Yto be
a mapping such that ftxis continuous in t∈R, for all x∈X. Assume that there exist
constants ≥0andp∈0,1such that
fxy−fx−fy
≤xp
y
p,x∈X
.1.3
Th. M. Rassias in 3showed with the above conditions for f, there exists a unique R-linear
mapping T:X→Ysuch that
fx−Tx
≤2
2−2pxp,x∈X
.1.4
G˘
avrut¸a then generalized the Rassias’s result in 4.
A square norm on an inner product space satisfies the important parallelogram
equality
xy
2
x−y
22x2
y
2.1.5
Recall that the functional equation
fxyfx−y2fx2fy1.6
is called quadratic functional equation. In addition, every solution of functional eqaution
1.6is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic
functional equation was proved by Skof 5for mappings f:X→Y, where Xis a normed
space and Yis a Banach space. Cholewa 6noticed that the theorem of Skof is still true
if the relevant domain Xis replaced by an abelian group. Indeed, Czerwik in 7proved
the Cauchy-Rassias stability of the quadratic functional equation. Since then, the stability
problems of various functional equation have been extensively investigated by a number of
authors e.g, 8–13.
One should remember that the functional equation is called stable if any approximately
solution to the functional equation is near to a true solution of that functional equation, and is
super superstable if every approximately solution is an exact solution of it see 14. Recently,
the first and third authors in 15investigated the stability of quadratic double centralizer:
the maps which are quadratic and double centralizer. Later, Eshaghi Gordji et al. introduced
a new concept of the quadratic double centralizer and the quadratic multipliers in 16,and
established the stability of quadratic double centralizer and quadratic multipliers on Banach
algebras. They also established the superstability for those which are weakly commutative
and weakly without order. In this paper, we show that the hypothesis on Banach algebras
being weakly commutative and weakly without order in 16can be eliminated, and prove
the superstability of quadratic double centralizers and quadratic multipliers on a Banach
algebra by a method of fixed point.
Journal of Inequalities and Applications 3
2. Stability of Quadratic Double Centralizers
A linear mapping L:A→Ais said to be left centralizer on Aif LabLab, for all a, b ∈A.
Similarly, a linear mapping R:A→Asatisfying RabaRb, for all a, b ∈Ais called
right centralizer on A.Adouble centralizer on Ais a pair L, R, where Lis a left centralizer, R
is a right centralizer and aLbRab, for all a, b ∈A. An operator T:A→Ais said to be
amultiplier if aTbTab, for all a, b ∈A.
Throughout this paper, let Abe a complex Banach algebra. Recall that a mapping
L:A→Ais a quadratic left centralizer if Lis a quadratic homogeneous mapping, that is L
is quadratic and Lλaλ2La, for all a∈Aand λ∈C,andLabLab2, for all a, b ∈A.
A mapping R:A→Ais a quadratic right centralizer if Ris a quadratic homogeneous
mapping and Raba2Rb, for all a, b ∈A. Also, a quadratic double centralizer of
an algebra Ais a pair L, Rwhere Lis a quadratic left centralizer, Ris a quadratic right
centralizer and a2LbRab2, for all a, b ∈Asee 16for details.
It is proven in 8; that for the vector spaces Xand Yand the fixed positive integer k,
the map f:X→Yis quadratic if and only if the following equality holds:
2fkx ky
22fkx −ky
2k2fxk2fy.2.1
We thus can show that fis quadratic if and only if for a fixed positive integer k, the following
equality holds:
fkx kyfkx −ky2k2fx2k2fy.2.2
Before proceeding to the main results, we will state the following theorem which is useful to
our purpose.
Theorem 2.1 The alternative of fixed point 17.Suppose that we are given a complete
generalized metric space X, dand a strictly contractive mapping T:X→Xwith Lipschitz
constant L. Then for each given x∈X,eitherdTnx, Tn1x∞, for all n≥0, or else exists a
natural number n0such that
1dTnx, Tn1x<∞, for all n≥n0,
2the sequence {Tnx}is convergent to a fixed point y∗of T,
3y∗is the unique fixed point of Tin the set Λ{y∈X:dTn0x, y<∞},
4dy, y∗≤1/1−Ldy, Ty, for all y∈Λ.
Theorem 2.2. Let fj:A→Abe continuous mappings with fj00(j0,1), and let φ:A6→
0,∞be continuous in the first and second variables such that
fjλa λb cdfjλa −λb cd−2λ2fjafjb
−21−jfjcd21−jjc2fjdju2f0v−f1uv2
≤a, b, c, d, u, v,
2.3
4 Journal of Inequalities and Applications
for all λ∈T{λ∈C:|λ|1}and, for all a, b, c, d, u, v ∈A,j 0,1. If there exists a constant m,
0<m<1such that
φa, b, c, d, u, v≤4mMinφa
2,b
2,c
2,d,u
2,v
2,φ
a
2,b
2,c,d
2,u
2,v
2,2.4
for all a, b, c, d, u, v ∈A, then there exists a unique double quadratic centralizer L, Ron A
satisfying
f0a−La
≤1
41−mφa, a, 0,0,0,0,2.5
f1a−Ra
≤1
41−mφa, a, 0,0,0,0,2.6
for all a∈A.
Proof. From 2.4, it follows that
lim
i4−iφ2ia, 2ib, 2ic, d, 2iu, 2iv0,2.7
for all a, b, c, d, u, v ∈A. Putting j0,λ1,ab, c duv0 and replacing aby 2a
in 2.3,weget
f02a−4f0a
≤φa, a, 0,0,0,0,2.8
for all a∈A. By the above inequality, we have
1
4f02a−f0a
≤1
4φa, a, 0,0,0,0,2.9
for all a∈A. Consider the set X:{g:A→A|g00}and introduce the generalized
metric on X:
dh, g:infC∈R:
ga−ha
≤Cφa, a, 0,0,0,0,∀a∈A
.2.10
It is easy to show that X, dis complete. Now, we define the linear mapping Q:X→Xby
Qha1
4h2a,2.11
for all a∈A.Giveng,h ∈X,letC∈Rbe an arbitrary constant with dg, h≤C,thatis
ga−ha
≤Cφa, a, 0,0,0,0,2.12
Journal of Inequalities and Applications 5
for all a∈A. Substituting aby 2ain the inequality 2.12and using 2.4and 2.11, we have
Qga−Qha
1
4
g2a−h2a
≤1
4Cφ2a, 2a, 0,0,0,0
≤Cmφa, a, 0,0,0,0,
2.13
for all a∈A. Hence, dQg, Qh≤Cm. Therefore, we conclude that dQg, Qh≤mdg,h,
for all g,h ∈X. It follows from 2.9that
dQf0,f
0≤1
4.2.14
By Theorem 2.1,Qhas a unique fixed point L:A→Ain the set X1{h∈
X, df0,h<∞}. On the other hand,
lim
n→∞
f02na
4nLa,2.15
for all a∈A.ByTheorem 2.1 and 2.14,weobtain
df0,L
≤1
1−mdQf0,L
≤1
41−m,2.16
that is, the inequality 2.5is true, for all a∈A. Now, substitute 2naand 2nbby aand b
respectively, put cduv0andj0in2.15. Dividing both sides of the resulting
inequality by 2n, and letting ngoes to infinity, it follows from 2.7and 2.3that
Lλa λbLλa −λb2λ2La2λ2Lb,2.17
for all a, b ∈Aand λ∈T. Putting λ1in2.17we have
LabLa−b2La2Lb,2.18
for all a, b ∈A. Hence Lis a quadratic mapping.
Letting b0in2.17,wegetLλaλ2La, for all a, b ∈Aand λ∈T. We can show
from 2.18that Lrar2Lafor any rational number r. It follows from the continuity of f0
and φthat for each λ∈R,Lλaλ2La.So,
LλaLλ
|λ||λ|aλ2
|λ|2L|λ|aλ2
|λ|2|λ|2Laλ2La,2.19