We say a functional equation (xi) is stable if any function g satisfying the equation (xi) approximately is near to true solution of (xi), moreover, a functional equation (xi) is superstable if any function g satisfying the equation (xi) approximately is a true solution of (xi). In the present paper, we investigate the stability and the superstability of double centralizers and of multipliers on Banach algebras by using the fixed point methods

���������

HANYANG Repository

U.P.B. Sci. Bull., Series A, Vol. 73, Iss. 2, 2011 ISSN 1223-7027
A FIXED POINT APPROACH TO THE STABILITY OF
DOUBLE JORDAN CENTRALIZERS AND JORDAN
MULTIPLIERS ON BANACH ALGEBRAS
M. Eshaghi Gordji1, A. Bodaghi2, C. Park3
We say a functional equation () is stable if any function g satisfying
the equation () approximately is near to true solution of (); moreover, a
functional equation () is superstable if any function g satisfying the equa-
tion () approximately is a true solution of (): In the present paper, we
investigate the stability and the superstability of double centralizers and of
multipliers on Banach algebras by using the xed point methods.
Keywords: Double centralizer; Multiplier; Hyers-Ulam stability.
MSC2000: 39B82, 39B52.
1. Introduction
In this paper, we assume that A is a complex Banach algebra. A linear
mapping L : A! A is said to be left Jordan centralizer on A if L(a2) = L(a)a
for all a 2 A. Similarly, a linear mapping R : A ! A satisfying that R(a2) =
aR(a) for all a 2 A is called right Jordan centralizer on A. A double Jordan
centralizer on A is a pair (L;R), where L is a left Jordan centralizer, R is
a right Jordan centralizer and aL(b) = R(a)b for all a; b 2 A. For example,
(Lc; Rc) is a double Jordan centralizer, where Lc(a) := ca and Rc(a) := ac (see
[9] and [11]).
A mapping T : A ! A is said to be a Jordan multiplier (see [12]) if
aT (a) = T (a)a for all a 2 A. Clearly, if Al(A) = f0g (Ar(A) = f0g, respec-
tively) then T is a left (right) Jordan centralizer (see [14]).
In 1940, Ulam [26] proposed the following question concerning stability of
group homomorphisms: under what condition does there is an additive mapping
near an approximately additive mapping? Hyers [10] answered the problem of
Ulam for the case where G1 and G2 are Banach spaces. A generalized version
of the theorem of Hyers for approximately linear mapping was given by Th.
1Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran;,
Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University,
Semnan, Iran,, e-mail:madjid.eshaghi@gmail.com
2Department of Mathematics, Islamic Azad University, Garmsar Branch, Garmsar, Iran,
e-mail: abasalt.bodaghi@gmail.com
3Department of Mathematics, Research Institute for Natural Sciences, Hanyang Univer-
sity, Seoul 133-791, South Korea, e-mail:baak@hanyang.ac.kr
65
66 M. Eshaghi Gordji, A. Bodaghi, C. Park
M. Rassias [25]. After that, several functional equations have been extensively
investigated by a number of authors (for instances, [4]{[7], [13] and [15]{[24]).
In 2003, Cadariu and Radu applied the xed point method to the investi-
gation of the Jensen functional equation [1] (see also [2, 3, 8]). They presented
a short and a simple proof ( no \direct method ", initiated by Hyers in 1941) for
the Hyers-Ulam stability of the Jensen functional equation [1], for the Cauchy
functional equation [3] and for some other functional equations [1, 2, 3].
We need the following known xed point theorem which is useful for our
goals.
Theorem 1.1. (The alternative of xed point [5]) Suppose (
; d) be a com-
plete generalized metric space and let J : 
 ! 
 be a strictly contractive
mapping with Lipschitz constant L < 1. Then for each element x 2 
, either
d(Jnx; Jn+1x) = 1 for all n  0; or there exists a natural number n0 such
that:
() d(Jnx; Jn+1x) <1 for all n  n0;
() the sequence fJnxg is convergent to a xed point y of J;
(  ) y is the unique xed point of J in the set
 = fy 2 
 : d(Jn0x; y) <1g;
(  ) d(y; y)  1
1 Ld(y; Jy) for all y 2 .
Moreover, we will use the following known lemma in the proof of main
results of our paper. We do not remove the proof of lemma. We suppose that
T1 := fz 2 C : jzj = 1g.
Lemma 1.1. Let n0 2 N be a positive integer number and let X; Y be linear
vector spaces on C. Suppose that f : X ! Y is an additive mapping. Then
f is C linear if and only if f(x) = f(x) for all x in X and  in T11
no
:=
fei ; 0    2
no
g:
Proof. Suppose that f is additive and f(x) = f(x) for all x in X and all 
in T 11
no
. Now, let  2 T 1. Then we have  = ei such that 0    2. We set
1 = e
i
no ;
thus 1 belongs to T
1
1
no
and
f(x) = f(no1 x) = 
no
1 f(x) = f(x)
for all x in X. If  belongs to nT 1 = fnz ; z 2 T 1g then by additivity of f,
f(x) = f(x) for all x in X and  in nT 1. If t 2 (0;1) then by archimedean
property there exists a natural number n such that the point (t; 0) lies in the
interior of circle with center at origin and radius n in C. Put
t1 := t+
p
n2   t2 i 2 nT 1
A xed point approach to the stability of ... 67
and
t2 := t 
p
n2   t2 i 2 nT 1:
We have t = t1+t2
2
and
f(tx) = f(
t1 + t2
2
x) =
t1 + t2
2
f(x) = tf(x)
for all x in X.
If  2 C, then
 = jjei1 ;
therefore
f(x) = f(jjei1x) = jjei1f(x) = f(x)
for all x in X. In the other words f is C linear. The converse is clear. 
From now on, we suppose that n0 2 N is a positive integer number, and
that
T11
no
:= fei ; 0    2
no
g:
2. Stability of double centralizers
For every x; y 2 A; we put x0   y0 = 0; x0y = y and also denote
n timesz }| {
A A ::: A by An. We establish the Hyers-Ulam stability of double Jor-
dan centralizers as follows.
Theorem 2.1. Let fj : A! A be mappings with fj(0) = 0 (j = 0; 1), and let
' : A4 ! [0;1) be a function such that
kfj(x+ y + z2)  fj(x)  fj(y)  [j(zfj(z))j + (1  j)(fj(z)z)1 j]
+f1 j(a)a  afj(a)k  '(x; y; z; a) (1)
for all  2 T11
no
: If
lim
n!1
'(2nx; 2ny; 2nz; 2na)
2n
= 0; (2)
and there exists a constant K in which 0 < K < 1 such that
 (2x)  2K (x) (3)
for all x 2 A, then there exists a unique double Jordan centralizer (L;R) on A
satisfying
kf0(x)  L(x)k   (x)
2(1 K) ; (4)
kf1(x) R(x)k   (x)
2(1 K) (5)
68 M. Eshaghi Gordji, A. Bodaghi, C. Park
for all x 2 A, where  (x) = '(x; x; 0; 0).
Proof. We consider the set 
 as follows:

 = fh : A  ! Ajh(0) = 0g:
We also dene the generalized metric on 
:
d(g; h) := inffC 2 [0;1] : kg(x)  h(x)k  C (x) for all x 2 Ag:
One can show that (
; d) is a complete metric space. Now, we dene a mapping
J : 
  ! 
 by
Jh(x) =
1
2
h(2x) (6)
for all x 2 A. Given g; h 2 
, let C 2 [0;1] be an arbitrary constant with
d(g; h)  C, i.e.,
kg(x)  h(x)k  C (x) (7)
for all x 2 A. Substituting x by 2x in the inequality (7) and using from (3)
and (6), we have
kJg(x)  Jh(x)k  1
2
kg(2x)  h(2x)k  1
2
C (2x)  CK (x)
for all x 2 A, and so d(Jg; Jh)  CK. This shows that J is strictly contractive
on 
. Hence we can conclude that
d(Jg; Jh)  Kd(g; h)
for all g; h 2 
. Now, we prove that for all h 2 
, d(Jh; h) < 1. Letting
j = 0;  = 1; x = y; z = a = 0 in (1), we obtain kf0(2x)   2f0(x)k   (x) for
all x 2 A. Thus
k1
2
f0(2x)  f0(x)k  1
2
 (x) (8)
for all x 2 A.
It follows from (8) that d(Jf; f)  1
2
. By Theorem 1.1, the sequence
fJnf0g converges to a unique xed point L : A ! A in the set 
1 = fh 2

; d(f; h) <1g, i.e.,
lim
n!1
f0(2
nx)
2n
= L(x) (9)
for all x 2 A, and so
d(f0; L)  1
1 Kd(Jf0; L) 
1
2(1 K) :
Thus the inequality (4) holds for all x 2 A. Now, replace 2nx and 2ny by x
and y, respectively, and put z = a = 0 in (1). If we divide both sides of the
A xed point approach to the stability of ... 69
resulting inequality by 2n, and letting n tend to innity, then it follows from
(1), (2) and (9) that
L(x) = L(x);
for all x 2 A and all  2 T11
no
. It follows from Lemma 1.1 that L is C linear.
It is routine to show that L(a2) = L(a)2 from (1), and so it is a left Jordan
centralizer on A.
According the above argument, one can show that there exists a unique
mapping R : A! A which is a point of T such that
lim
n!1
f1(2
nx)
2n
= R(x) (10)
for all x 2 A: Indeed, R belongs to the set 
1. If we put i = 0; x = y = z = 0
and substitute a by 2na in (1) and we divide the both sides of the obtained
inequality by 2n, then we get
kaf0(2
na)
2n
  f1(2
na)
2n
ak  '(0; 0; 0; 2
na)
2n
:
Passing to the limit as n!1, and from (2) we conclude that aL(a) = R(a)a,
for all a 2 A. 
Corollary 2.1. Let r and  be nonnegative real numbers such that r < 1, and
let fj : A! A be mappings with fj(0) = 0 (j = 0; 1) such that
kfj(x+ y + z2)  fj(x)  fj(y)  [j(zfj(z))j + (1  j)(fj(z)z)1 j]
 afj(a) + f1 j(a)ak  (kxkr + kykr + kzkr + kakr) (11)
for all  2 T11
no
and all x; y; z; a 2 A: Then there exists a unique double
centralizer (L;R) on A satisfying
kf0(x)  L(x)k  
2  2r ;
kf1(x) R(x)k  
2  2r
for all x; y 2 A.
Proof. By putting K = 2r 1 and
'(x; y; z; w; a) = (kxkr + kykr + kzkr + kakr)
for all x; y; z; a 2 A in Theorem 2.1, we obtain the desired result. 
In the following corollary, we prove the superstability of double central-
izers under some conditions.
70 M. Eshaghi Gordji, A. Bodaghi, C. Park
Corollary 2.2. Let r and  be nonnegative real numbers such that r < 1
6
, and
let fj : A! A be mappings with fj(0) = 0 (j = 0; 1) such that
kfj(x+ y + z2)  fj(x)  fj(y)  [j(zfj(z))j + (1  j)(fj(z)z)1 j]
 afj(a) + f1 j(a)ak  (kxkrkykrkzkrkakr) (12)
for all  2 T11
no
and all x; y; z; a 2 A: Then (f0; f1) is a double centralizer.
Proof. It follows from Theorem 2.1 by taking
'(x; y; z; a) = (kxkrkykrkzkrkakr)
for all x; y; z; a 2 A. 
3. Stability Of multipliers
In this section, we investigate the Hyers-Ulam stability and the super-
stability of multipliers on Banach algebras. First we need the next theorem
which is the main key to investigation of the stability and superstability.
Theorem 3.1. let f : A ! A be a mapping with f(0) = 0 and let  : A3 !
[0;1) be a function such that
kf(x+ y)  f(x)  f(y)  f(z)z + zf(z)k  (x; y; z) (13)
for all  2 T11
no
and all x; y; z 2 A: If
lim
n!1
'(2nx; 2ny; 2nz)
2n
= 0; (14)
and there exists a constant K, 0 < K < 1, such that
 (2x)  2K (x) (15)
for all x 2 A, then there exists a unique multiplier T on A satisfying
kf(x)  L(x)k  1
2(1 K) (x) (16)
for all x 2 A, where  (x) = '(x; x; 0).
Proof. First, similar to the proof of Theorem 2.1, we Consider the set 
 :=
fh : A! A j h(0) = 0g and introduce the generalized metric d on 
 as follows:
d(g; h) := inffC 2 R+ : kg(x)  h(x)k  C (x) for all x 2 Ag:
Again, similar to the proof of Theorem 2.1, the space 
 equipped to the metric
d is complete. Now we dene a mapping J : 
! 
 by
J(h)(x) =
1
2
h(2x)
A xed point approach to the stability of ... 71
for all x 2 A. By the same reasoning as in the proof of Theorem 2.1, J is
strictly contractive on 
. Putting  = 1; x = y and z = 0 in (13), we obtain
kf(2x)  2f(x)k   (x)
for all x 2 A: So
k1
2
f(2x)  f(x)k  1
2
 (x) (17)
for all x 2 A: The inequality (17) shows that
d(Jf; f)  1
2
:
By Theorem 1.1, J has a unique xed point in the set 
1 := fh 2 
 : d(f; h) <
1g. Let T be the xed point of J. Then T is the unique mapping with
T (2x) = 2T (x)
for all x 2 A such that there exists C 2 (0;1) such that
kT (x)  f(x)k  K (x)
for all x 2 A. On the other hand, we have
limn!1d(Jn(f); h) = 0:
Thus
limn!1
1
2n
f(2nx) = h(x) (18)
for all x 2 A. Hence
d(f; T )  1
1 Kd(T; J(f)) 
1
2(1  L) : (19)
This implies the inequality (16). From (13), (14) and (18) we obtain
kT (x+ y)  T (x)  T (y)k = limn!1 1
2n
kf(2n(x+ y)) + f(2n(x))  f(2ny)k
 limn!1 1
2n
(2nx; 2ny; 0) = 0
for all x; y 2 A: So T (x+ y) = T (x) + T (y) for all x; y 2 A: Thus T is Cauchy
additive. If we put y = x; z = 0 in (13), we can conclude that
k2f(x)  f(2x)k   (x)
for all x 2 A. It follows that
kT (2x)  2T (x)k = limn!1 1
2n
kf(22nx)  2f(2nx)k
 limn!1 1
2n
(2nx; 2nx; 0) = 0
for all  2 T and all x 2 A: So T (x) = T (x) for all  2 T11
no
and all x 2 A.
Linearity of T follows from the proof of Theorem 2.1. If we substitute z by
72 M. Eshaghi Gordji, A. Bodaghi, C. Park
2nz in (13), and put x = y = 0 and we divide the both sides of the obtained
inequality by 2n, we get
kz f(2
nz)
2n
  f(2
nz)
2n
zk  (0; 0; 2
nz)
2n
:
Passing to the limit as n!1, and from (16) we conclude that zT (z) = T (z)z
for all z 2 A. 
Corollary 3.1. Let r and  be nonnegative real numbers such that r < 1, and
let f : A! A be a mapping with f(0) = 0 such that
kf(x+ y)  f(x)  f(y)  f(z)z + zf(z)k  (kxkr + kykr + kzkr)
for all  2 T11
no
and all x; y; z 2 A: Then there exists a unique multiplier T on
A satisfying
kf(x)  T (x)k  
2  2r
for all x 2 A.
Proof. The proof follows from Theorem 3.1 by taking
(x; y; z) = (kxkr + kykr + kzkr)
for all x; y; z 2 A and by putting K = 2r 1. 
Now, we have the following result for the superstability of multipliers.
Corollary 3.2. Let r and  be nonnegative real numbers such that r < 1
4
, and
let f : A! A be a mapping with f(0) = 0 such that
kf(x+ y)  f(x)  f(y)  f(z)z + zf(z)k  (kxkrkykrkzkr)
for all  2 T11
no
and all x; y; z 2 A: Then f is a multiplier on A.
Proof. It follows from Theorem 3 by taking
(x; y; z) = (kxkrkykrkzkr)
for all x; y; z 2 A. 
Acknowledgement
The third author was supported by Basic Science Research Program
through the National Research Foundation of Korea funded by the Ministry
of Education, Science and Technology (NRF-2009-0070788).
A xed point approach to the stability of ... 73
REFERENCES
[1] L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation,
J. Inequal. Pure Appl. Math. 4, Art. 4 (2003).
[2] L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equa-
tions, An. Univ. Timisoara, Ser. Mat. Inform. 41 (2003), 25{48.
[3] L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a xed
point approach, Grazer Math. Ber. 346 (2004), 43{52.
[4] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math.
27 (1984), 76{86.
[5] J. B. Diaz and B. Margolis, A xed point theorem of the alternative for contractions
on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305{309.
[6] M. Eshaghi Gordji, M. Ramezani, A. Ebadian and C. Park, Quadratic double cen-
tralizers and quadratic multipliers, Ann. Univ. Ferrara Sez. VII Sci. Mat., (2011), Doi:
10.1007/s11565-011-0115-7..
[7] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers
on Banach algebras, J. Comput. Anal. Appl. Vol.13, NO.4, (2011) 724{729.
[8] M. Eshaghi Gordji and A. Najati, Approximately J-homomorphisms: A xed point
approach, Journal of Geometry and Physics 60 (2010), 809{814.
[9] G. Hochschild, Cohomology and representations of associative algebras, Duke Math.
J. 14 (1947), 921{948.
[10] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci.
U.S.A. 27 (1941), 222{224.
[11] B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. Soc.
14 (1964), 299{320.
[12] R. Larsen, Introduction to the Theory of Multipliers, Springer-Verlag, New York, 1971.
[13] T. Miura, G. Hirasawa and S. E. Takahashi Stability of multipliers on Banach algebras,
Internat. J. Math. Math. Sci. 45 (2004), 2377{2381.
[14] T. W. Parlem, Banach Algebras and General Theory of -Algebras, Vol. I, Algebras
and Banach Algebras, Encyclopedia of Mathematics and its Applications 49, Cambridge
University Press, Cambridge, 1994.
[15] C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal.
Appl. 275 (2002), 711{720.
[16] C. Park, On an approximate automorphism on a C-algebra, Proc. Amer. Math. Soc.
132 (2004), 1739{1745.
[17] C. Park, Homomorphisms between Poisson JC-algebras, Bull. Braz. Math. Soc. 36
(2005), 79{97.
[18] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional
equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID
50175 (2007).
74 M. Eshaghi Gordji, A. Bodaghi, C. Park
[19] C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C-
algebras: A xed point approach, Abstract and Applied Analysis 2009, Art. ID 360432
(2009).
[20] J. M. Rassias, On approximation of approximately linear mappings by linear mappings,
J. Funct. Anal. 46 (1982), 126{130.
[21] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268{273.
[22] J. M. Rassias and M. J. Rassias, Rened Ulam stability for Euler-Lagrange type
mappings in Hilbert spaces, Internat. J. Appl. Math. Stat. 7 (2007), 126{132.
[23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer.
Math. Soc. 72 (1978), 297{300.
[24] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math.
Anal. Appl. 251 (2000), 264{284.
[25] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer.
Math. Soc. 72 (1978), 297{300.
[26] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New
York, 1940.


A FIXED POINT APPROACH TO THE STABILITY OF DOUBLE JORDAN CENTRALIZERS AND JORDAN MULTIPLIERS ON BANACH ALGEBRAS

http://repository.hanyang.ac.kr/bitstream/20.500.11754/70437/1/61.pdf

A FIXED POINT APPROACH TO THE STABILITY OF DOUBLE JORDAN CENTRALIZERS AND JORDAN MULTIPLIERS ON BANACH ALGEBRAS

Abstract

Similar works

Full text

Available Versions

HANYANG Repository