Volume 2008, Article ID 872190,11pages doi:10.1155/2008/872190
Research Article
Stability of the Cauchy-Jensen Functional Equation
in
C
∗
-Algebras: A Fixed Point Approach
Choonkil Park1and Jong Su An2
1Department of Mathematics, Hanyang University, Seoul 133-791, South Korea
2Department of Mathematics Education, Pusan National University, Pusan 609-735, South Korea
Correspondence should be addressed to Jong Su An,[email protected]
Received 3 April 2008; Accepted 14 May 2008
Recommended by Andrzej Szulkin
we prove the Hyers-Ulam-Rassias stability of C∗-algebra homomorphisms and of generalized derivations onC∗-algebras for the following Cauchy-Jensen functional equation 2fxy/2z fx fy 2fz, which was introduced and investigated by Baak2006. The concept of Hyers-Ulam-Rassias stability originated from the stability theorem of Th. M. Rassias that appeared in 1978.
Copyrightq2008 C. Park and J. S. An. 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 and preliminaries
The stability problem of functional equations originated from a question of Ulam 1
concerning the stability of group homomorphisms. Hyers 2gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki3
for additive mappings and by Rassias4for linear mappings by considering an unbounded Cauchy difference.
Theorem 1.1see4. Letf :E→Ebe a mapping from a normed vector spaceEinto a Banach space Esubject to the inequality
fxy−fx−fy≤
xpyp 1.1
for allx, y ∈E, whereandpare constants with >0andp <1. Then, the limit
Lx lim
n→∞
f2nx
exists for allx∈EandL:E→Eis the unique additive mapping which satisfies
fx−Lx≤ 2
2−2px
p 1.3
for allx∈E. Also, if for eachx∈Ethe mappingftxis continuous int∈R, thenLisR-linear.
The above inequality1.1has provided a lot of influence in the development of what is now known as aHyers-Ulam-Rassias stabilityof functional equations. A generalization of Th. M. Rassias’ theorem was obtained by G˘avrut¸a5by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The result of G˘avrut¸a5is a special case of a more general theorem, which was obtained by Forti6. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problemsee7–18.
J. M. Rassias19following the spirit of the innovative approach of Th. M. Rassias4for the unbounded Cauchy difference proved a similar stability theorem in which he replaced the factorxpyp byxp·yqforp, q ∈Rwithpq /1see also20for a number of other
new results.
Theorem 1.2see19–21. LetXbe a real normed linear space andY a real complete normed linear space. Assume that f : X→Y is an approximately additive mapping for which there exist constants θ≥0andp∈R− {1}such thatfsatisfies inequality
fxy−fx−fy≤θ·xp/2·
yp/2 1.4
for allx, y ∈X. Then, there exists a unique additive mappingL:X→Y satisfying
fx−Lx≤ θ
2p−2x
p
1.5
for all x ∈ X. If, in addition, f : X→Y is a mapping such that the transformation t→ftx is continuous int∈Rfor each fixedx∈X, thenLis anR-linear mapping.
We recall two fundamental results in fixed point theory.
Theorem 1.3see22. LetX, dbe a complete metric space and letJ:X→Xbe strictly contractive, that is,
dJx, Jy≤Lfx, y, ∀x, y ∈X 1.6
for some Lipschitz constantL <1. Then, the following conditions hold.
1The mappingJhas a unique fixed pointx∗Jx∗. 2The fixed pointx∗is globally attractive, that is,
lim
n→∞J
nxx∗ 1.7
3One has the following estimation inequalities:
dJnx, x∗≤Lndx, x∗,
dJnx, x∗≤ 1 1−Ld
Jnx, Jn1x,
dx, x∗≤ 1
1−Ldx, Jx
1.8
for all nonnegative integersnand allx∈X.
LetXbe a set. A functiond:X×X→0,∞is called ageneralized metriconXifdsatisfies the following conditions:
1dx, y 0 if and only ifxy; 2dx, y dy, x,for allx, y∈X;
3dx, z≤dx, y dy, z,for allx, y, z∈X.
Theorem 1.4 see23. Let X, dbe a complete generalized metric space and let J : X→X be a strictly contractive mapping with Lipschitz constantL <1. Then for each given elementx∈X, either
dJnx, Jn1x∞ 1.9
for all nonnegative integersnor there exists a positive integern0such that
1dJnx, Jn1x<∞, for all n≥n 0;
2the sequence{Jnx}converges to a fixed pointy∗ofJ;
3y∗is the unique fixed point ofJin the setY {y∈X|dJn0x, y<∞}; 4dy, y∗≤1/1−Ldy, Jy, for ally∈Y.
This paper is organized as follows. In Section 2, using the fixed point method, we prove the Hyers-Ulam-Rassias stability ofC∗-algebra homomorphisms for the Cauchy-Jensen functional equation.
InSection 3, using the fixed point method, we prove the Hyers-Ulam-Rassias stability of generalized derivations onC∗-algebras for the Cauchy-Jensen functional equation.
Throughout this paper, assume that A is aC∗-algebra with norm·A and that B is a
C∗-algebra with norm·B.
2. Stability ofC∗-algebra homomorphisms
For a given mappingf :A→B, we define
Cμfx, y, z:2μf
xy
2 z
−fμx−fμy−2fμz, 2.1
for allμ∈T1:{ν∈C:|ν|1}and allx, y, z∈A.
Theorem 2.1. Letf :A→Bbe a mapping for which there exists a functionϕ:A3→0,∞such that
lim
j→∞
1 2jϕ
2jx,2jy,2jz0, 2.2
Cμfx, y, z
B ≤ϕx, y, z, 2.3 fxy−fxfy
B ≤ϕx, y,0, 2.4 fx∗−fx∗
B ≤ϕx, x, x 2.5
for allμ∈T1and allx, y, z∈A. If there exists anL <1such thatϕx, x, x≤2Lϕx/2, x/2, x/2
for allx∈A, then there exists a uniqueC∗-algebra homomorphismH :A→Bsuch that
fx−Hx B ≤
1
4−4Lϕx, x, x 2.6
for allx∈A.
Proof. Consider the set
X:{g:A→B} 2.7
and introduce thegeneralized metriconXas follows:
dg, h infC∈R :gx−hxB ≤Cϕx, x, x, ∀x∈A . 2.8
It is easy to show thatX, dis complete.
Now, we consider the linear mappingJ:X→Xsuch that
Jgx: 1
2g2x 2.9
for allx∈A.
By22, Theorem 3.1,
dJg, Jh≤Ldg, h 2.10
for allg, h∈X.
Lettingμ1 andyzxin2.3, we get
2f2x−4fxB ≤ϕx, x, x 2.11
for allx∈A. So
fx−1
2f2xB ≤
1
4ϕx, x, x 2.12
for allx∈A. Hence,df, Jf≤1/4.
1H is a fixed point ofJ, that is,
H2x 2Hx 2.13
for allx∈A. The mappingHis a unique fixed point ofJ in the set
Y g ∈X:df, g<∞ . 2.14
This implies thatH is a unique mapping satisfying2.13such that there existsC ∈ 0,∞satisfying
Hx−fx
B ≤Cϕx, x, x 2.15
for allx∈A.
2dJnf, H→0 asn→ ∞. This implies the equality
lim
n→∞
f2nx
2n Hx 2.16
for allx∈A.
3df, H≤1/1−Ldf, Jf, which implies the inequality
df, H≤ 1
4−4L. 2.17
This implies that inequality2.6holds.
It follows from2.2,2.3, and2.16that
2H
xy
2 z
−Hx−Hy−2Hz
B
lim
n→∞ 1 2n2f
2n−1xy 2nz−f2nx−f2ny−2f2nzB
≤ lim
n→∞ 1 2nϕ
2nx,2ny,2nz0
2.18
for allx, y, z∈A. So
2H
xy
2 z
Hx Hy 2Hz 2.19
for all x, y, z ∈ A. By24, Lemma 2.1, the mapping H : A→Bis Cauchy additive, that is, Hxy Hx Hy,for allx, y∈A.
By a similar method to the proof of11, one can show that the mappingH : A→Bis
C-linear.
It follows from2.4that
Hxy−HxHy
B nlim→∞
1 4nf
4nxy−f2nxf2nyB
≤ lim
n→∞
1 4nϕ
2nx,2ny,0≤ lim
n→∞
1 2nϕ
2nx,2ny,00
for allx, y ∈A. So
Hxy HxHy 2.21
for allx, y ∈A.
It follows from2.5that
Hx∗−Hx∗B lim
n→∞ 1 2nf
2nx∗−f2nx∗B ≤ lim
n→∞ 1 2nϕ
2nx,2nx,2nx0 2.22
for allx∈A. So
Hx∗Hx∗ 2.23
for allx∈A.
Thus,H:A→Bis aC∗-algebra homomorphism satisfying2.6, as desired.
Corollary 2.2. Letr <1andθbe nonnegative real numbers, and letf :A→Bbe a mapping such that
Cμfx, y, z B ≤θ
xr
AyrAzrA
,
fxy−fxfy B ≤θ
xr
AyrA
,
f
x∗−fx∗B ≤3θxrA
2.24
for allμ∈ T1and allx, y, z ∈A. Then, there exists a uniqueC∗-algebra homomorphismH : A→B
such that
fx−Hx B ≤
3θ 4−2r1x
r
A 2.25
for allx∈A.
Proof. The proof follows fromTheorem 2.1by taking
ϕx, y, z:θxrAyrAzrA 2.26
for allx, y, z∈A. Then,L2r−1and we get the desired result.
Theorem 2.3. Letf:A→Bbe a mapping for which there exists a functionϕ:A3→0,∞satisfying 2.3,2.4, and2.5such that
lim
j→∞4 jϕ
x 2j,
y 2j,
z 2j
0 2.27
for allx, y, z ∈A. If there exists anL < 1such thatϕx, x, x≤ 1/2Lϕ2x,2x,2xfor allx ∈A, then there exists a uniqueC∗-algebra homomorphismH :A→Bsuch that
fx−Hx B ≤
L
4−4Lϕx, x, x 2.28
Proof. We consider the linear mappingJ :X→Xsuch that
Jgx:2g
x 2
2.29
for allx∈A.
It follows from2.11that
fx−2f
x 2
B
≤ 1 2ϕ
x 2,
x 2,
x 2
≤ L
4ϕx, x, x 2.30
for allx∈A. Hencedf, Jf≤L/4.
By Theorem 1.4, there exists a mapping H : A→B such that the following conditions hold.
1H is a fixed point ofJ, that is,
H2x 2Hx 2.31
for allx∈A. The mappingHis a unique fixed point ofJ in the set
Y g ∈X:df, g<∞ . 2.32
This implies thatH is a unique mapping satisfying2.31such that there existsC ∈ 0,∞satisfying
Hx−fx
B ≤Cϕx, x, x 2.33
for allx∈A.
2dJnf, H→0 asn→ ∞. This implies the equality
lim
n→∞2 nf
x 2n
Hx 2.34
for allx∈A.
3df, H≤1/1−Ldf, Jf, which implies the inequality
df, H≤ L
4−4L, 2.35
which implies that inequality2.28holds.
The rest of the proof is similar to the proof ofTheorem 2.1.
Corollary 2.4. Letr >2, letθbe nonnegative real numbers, and letf :A→Bbe a mapping satisfying 2.24. Then, there exists a uniqueC∗-algebra homomorphismH :A→Bsuch that
fx−Hx B ≤
3θ 2r1−4x
r
A 2.36
for allx∈A.
Proof. The proof follows fromTheorem 2.3by taking
ϕx, y, z:θxrAyrAzrA 2.37
3. Stability of generalized derivations onC∗-algebras
For a given mappingf :A→A, we define
Cμfx, y, z:2μf
xy
2 z
−fμx−fμy−2fμz 3.1
for allμ∈T1and allx, y, z∈A.
Definition 3.1see25. A generalized derivationδ:A→Ais involutiveC-linear and fulfills
δxyz δxyz−xδyzxδyz 3.2
for allx, y, z∈A.
We prove the Hyers-Ulam-Rassias stability of derivations on C∗-algebras for the functional equationCμfx, y, z 0.
Theorem 3.2. Letf :A→Abe a mapping for which there exists a functionϕ:A3→0,∞satisfying
2.2such that
Cμfx, y, z
A ≤ϕx, y, z, 3.3 fxyz−fxyzxfyz−xfyz
A ≤ϕx, y, z, 3.4
f
x∗−fx∗A ≤ϕx, x, x 3.5
for allμ∈T1and allx, y, z∈A. If there exists anL <1such thatϕx, x, x≤2Lϕx/2, x/2, x/2 for allx∈A, then there exists a unique generalized derivationδ:A→Asuch that
fx−δx A≤
1
4−4Lϕx, x, x 3.6 for allx∈A.
Proof. By the same reasoning as the proof ofTheorem 2.1, there exists a unique involutiveC -linear mappingδ:A→Asatisfying3.6. The mappingδ :A→Ais given by
δx lim
n→∞
f2nx
2n 3.7
for allx∈A.
It follows from3.4that
δxyz−δxyzxδyz−xδyz A
lim
n→∞
1 8nf
8nxyz−f4nxy·2nz2nxf2ny·2nz−2nxf4nyzA
≤ lim
n→∞ 1 8nϕ
2nx,2ny,2nz≤ lim
n→∞ 1 2nϕ
2nx,2ny,2nz0
3.8
for allx, y, z∈A. So
δxyz δxyz−xδyzxδyz 3.9
Corollary 3.3. Let r<1, Letθbe nonnegative real numbers, and letf :A→Abe a mapping such that
Cμfx, y, zA≤θ·xAr/3·yr/A3·zr/A3, fxyz−fxyzxfyz−xfyz
A≤θ·x r/3
A ·y
r/3
A ·z r/3
A , fx∗−fx∗A≤θ·xr
A
3.10
for allμ∈ T1and allx, y, z ∈A. Then, there exists a unique generalized derivationδ : A→Asuch
that
fx−δx A≤
θ 4−2r1x
r
A 3.11
for allx∈A.
Proof. The proof follows fromTheorem 3.2by taking
ϕx, y, z:θ·xr/A3·yr/A3·zr/A3 3.12
for allx, y, z∈A. Then,L 2r−1and we get the desired result.
Theorem 3.4. Letf :A→Abe a mapping for which there exists a functionϕ:A3→0,∞satisfying
3.3,3.4, and3.5such that
lim
j→∞8 jϕ
x 2j,
y 2j,
z 2j
0 3.13
for allx, y, z ∈A. If there exists anL < 1such thatϕx, x, x≤ 1/2Lϕ2x,2x,2xfor allx ∈A, then there exists a unique generalized derivationδ:A→Asuch that
fx−δx A≤
L
4−4Lϕx, x, x 3.14
for allx∈A.
Proof. The proof is similar to the proofs of Theorems2.3and3.2.
Corollary 3.5. Letr >3, letθbe nonnegative real numbers, and letf:A→Abe a mapping satisfying 3.10. Then, there exists a unique generalized derivationδ:A→Asuch that
fx−δx A≤
θ 2r1−4x
r
A 3.15
for allx∈A.
Proof. The proof follows fromTheorem 3.4by taking
ϕx, y, z:θ·xr/A3·yr/A3·zr/A3 3.16
Acknowledgments
The first author was supported by Korea Research Foundation Grant KRF-2007-313-C00033. The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.
References
1S. M. Ulam,A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, NY, USA, 1960.
2D. H. Hyers, “On the stability of the linear functional equation,”Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941.
3T. Aoki, “On the stability of the linear transformation in Banach spaces,”Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.
4Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,”Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
5P. G˘avrut¸a, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,”Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994. 6G. L. Forti, “An existence and stability theorem for a class of functional equations,”Stochastica, vol. 4,
no. 1, pp. 23–30, 1980.
7L. C˘adariu and V. Radu, “Fixed point methods for the generalized stability of functional equations in a single variable,”Fixed Point Theory and Applications, vol. 2008, Article ID 749392, 15 pages, 2008. 8C. Park, “On the stability of the linear mapping in Banach modules,”Journal of Mathematical Analysis
and Applications, vol. 275, no. 2, pp. 711–720, 2002.
9C. Park, “Lie∗-homomorphisms between LieC∗-algebras and Lie∗-derivations on LieC∗-algebras,”
Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 419–434, 2004.
10C. Park, “Homomorphisms between LieJC∗-algebras and Cauchy-Rassias stability of LieJC∗-algebra derivations,”Journal of Lie Theory, vol. 15, no. 2, pp. 393–414, 2005.
11C. Park, “Homomorphisms between Poisson JC∗-algebras,” Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79–97, 2005.
12C. Park, “Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras,”Fixed Point Theory and Applications, vol. 2007, Article ID 50175, 15 pages, 2007. 13C. Park, “Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and
isomorphisms betweenC∗-algebras,”Mathematische Nachrichten, vol. 281, no. 3, pp. 402–411, 2008. 14C. Park, “Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point
approach,”Fixed Point Theory and Applications, vol. 2008, Article ID 493751, 9 pages, 2008.
15C. Park and J. Hou, “Homomorphisms between C∗-algebras associated with the Trif functional equation and linear derivations on C∗-algebras,”Journal of the Korean Mathematical Society, vol. 41, no. 3, pp. 461–477, 2004.
16Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,”
Aequationes Mathematicae, vol. 39, pp. 292–293, 309, 1990.
17Th. M. Rassias, “The problem of S.M. Ulam for approximately multiplicative mappings,”Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 352–378, 2000.
18Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264–284, 2000.
19J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Bulletin des Sciences Math´ematiques. 2e S´erie, vol. 108, no. 4, pp. 445–446, 1984.
20J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268–273, 1989.
21J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.
22L. C˘adariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,”Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 7 pages, 2003.
24C. Baak, “Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces,” Acta Mathematica Sinica, vol. 22, no. 6, pp. 1789–1796, 2006.
