Stability of a Generalized Euler Lagrange Type Additive Mapping and Homomorphisms in Algebras

Advances in Difference Equations Volume 2009, Article ID 273165,22pages doi:10.1155/2009/273165

Research Article

Stability of a Generalized Euler-Lagrange

Type Additive Mapping and Homomorphisms in

C

∗

_-Algebras

Abbas Najati

_{and Choonkil Park}

1_{Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,}

Ardabil, 56199-11367, Iran

2_{Department of Mathematics, Research Institute for Natural Sciences, Hanyang University,}

Seoul, 133-791, South Korea

Correspondence should be addressed to Choonkil Park,[email protected]

Received 17 June 2009; Revised 30 July 2009; Accepted 4 August 2009

Recommended by Patricia J. Y. Wong

LetX, Y be Banach modules over a C∗-algebra and let r1, . . . , rn ∈ R be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unitalC∗-algebra:n_j1f−rjxj

1≤i≤n,i /jrixi 2ni1rifxi nf

_n

i1rixi. We show that

ifn_i1ri/0,ri, rj/0 for some 1 ≤ i < j ≤nand a mappingf :X → Y satisfies the functional equation mentioned above then the mappingf :X → Yis Cauchy additive. As an application, we investigate homomorphisms in unitalC∗-algebras.

Copyrightq2009 A. Najati and C. Park. 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. Hyers2gave a first affirmative partial

answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki

3for additive mappings and by Th. M. Rassias4for linear mappings by considering an

unbounded Cauchy difference.

Theorem 1.1Th. M. Rassias4. Letf : E → Ebe a mapping from a normed vector spaceE into a Banach spaceEsubject to the inequality

_f

for allx, y∈E, whereandpare constants with >0andp <1. Then the limit

Lx lim

n→ ∞

f2nx

2n 1.2

exists for allx∈EandL:E → Eis the unique additive mapping which satisfies

_fx−Lx≤ 2 2−2px

p _1.3

for allx ∈ E. Ifp < 0, then1.1holds forx, y /0and1.3forx /0. Also, if for eachx ∈ Ethe mappingt →ftxis continuous int∈R, thenLisR-linear.

Theorem 1.2J. M. Rassias5–7. LetXbe a real normed linear space andY a real Banach space. Assume thatf :X → Y is a mapping for which there exist constantsθ≥0andp, q∈Rsuch that rpq /1andfsatisfies the functional inequality

_f

xy−fx−fy≤θxpyq 1.4

for allx, y∈X. Then there exists a unique additive mappingL:X → Y satisfying

_fx−Lx≤ θ

|2r−2|x

r _1.5

for allx ∈X. If, in addition,f :X → Y is a mapping such that the transformationt → ftxis continuous int∈Rfor each fixedx∈X,thenLis linear.

The paper of Th. M. Rassias4has provided a lot of influence in the development

of what we call the generalized Hyers-Ulam stability of functional equations. In 1994, a

generalization of Theorems 1.1 and 1.2 was obtained by G˘avrut¸a 8, who replaced the

boundsεxpypandθxpyqby a general control functionϕx, y. The functional equation

fxyfx−y2fx 2fy 1.6

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The generalized Hyers-Ulam stability problem for

the quadratic functional equation was proved by Skof9for mappingsf:X → Y, whereX

is a normed space andY is a Banach space. Cholewa10noticed that the theorem of Skof is

still true if the relevant domainXis replaced by an Abelian group. Czerwik11proved the

generalized Hyers-Ulam stability of the quadratic functional equation. J. M. Rassias12,13

introduced and investigated the stability problem of Ulam for the Euler-Lagrange quadratic

mappings1.6and

fa1x1a2x2 fa2x1−a1x2

a2 1a22

_fx

Grabiec 14 has generalized these results mentioned above. In addition, J. M. Rassias

15generalized the Euler-Lagrange quadratic mapping 1.7 and investigated its stability

problem. Thus these Euler-Lagrange type equations mappings are called as

Euler-Lagrange-Rassias functional equationsmappings.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problemsee4–8,12,13,15–55.

Recently, C. Park and J. Park45introduced and investigated the following additive

functional equation of Euler-Lagrange type:

n

whose solution is said to be a generalized additive mapping of Euler-Lagrange type.

In this paper, we introduce the following additive functional equation of

Euler-Lagrange type which is somewhat different from1.8:

n

Euler-Lagrange type additive mapping.

We investigate the generalized Hyers-Ulam stability of the functional equation1.9

in Banach modules over a C∗-algebra. These results are applied to investigate C∗-algebra

homomorphisms in unitalC∗-algebras.

Throughout this paper, assume thatAis a unitalC∗-algebra with norm · Aand unit

2. Generalized Hyers-Ulam Stability of the Functional Equation

1.9

in Banach Modules Over a

C

-Algebra

Lemma 2.1. LetXandYbe linear spaces and letr1, . . . , rn be real numbers withnk1rk/0and

ri, rj/0for some1≤i < j ≤n.Assume that a mappingL:X → Ysatisfies the functional equation

1.9for allx1, . . . , xn ∈ X.Then the mappingLis Cauchy additive. Moreover,Lrkx rkLxfor allx∈ Xand all1≤k≤n.

Proof. Sincen_k1r_k/0,puttingx1 · · · xn 0 in1.9, we getL0 0.Without loss of

generality, we may assume thatr1, r2/0.Lettingx3· · ·xn0 in1.9, we get

L−r1x1r2x2 Lr1x1−r2x2 2r1Lx1 2r2Lx2 2Lr1x1r2x2 2.1

for allx1, x2∈ X.Lettingx20 in2.1, we get

2r1Lx1 Lr1x1−L−r1x1 2.2

for allx1 ∈ X.Similarly, by puttingx10 in2.1, we get

2r2Lx2 Lr2x2−L−r2x2 2.3

for allx1 ∈ X.It follows from2.1,2.2and2.3that

L−r1x1r2x2 Lr1x1−r2x2 Lr1x1 Lr2x2−L−r1x1−L−r2x2 2Lr1x1r2x2

2.4

for allx1, x2∈ X.Replacingx1andx2byx/r1andy/r2in2.4, we get

L−xyLx−yLx Ly−L−x−L−y2Lxy 2.5

for allx, y∈ X.Lettingy −xin2.5, we get thatL−2x L2x 0 for allx∈ X.So the

mappingLis odd. Therefore, it follows from2.5that the mappingLis additive. Moreover,

letx∈ Xand 1≤k≤n.Settingxkxandxl0 for all 1≤l≤n, l /k,in1.9and using the

oddness ofL,we get thatLrkx rkLx.

Using the same method as in the proof ofLemma 2.1, we have an alternative result of

Lemma 2.1whenn_k1r_k0.

Lemma 2.2. LetXandYbe linear spaces and letr1, . . . , rn be real numbers withri, rj/0for some

1 ≤i < j ≤ n.Assume that a mappingL :X → YwithL0 0satisfies the functional equation

1.9for allx1, . . . , xn ∈ X.Then the mappingLis Cauchy additive. Moreover,Lrkx rkLxfor allx∈ Xand all1≤k≤n.

We investigate the generalized Hyers-Ulam stability of a generalized Euler-Lagrange type additive mapping in Banach spaces.

for allx∈X.Replacingxby 2kxin2.18and dividing both sides of2.18by 2k1,we get desired inequality2.9.

It follows from2.7and2.8that

To prove the uniqueness, letT :X → Y be another generalized Euler-Lagrange type

additive mapping withT0 0 satisfying2.9. ByLemma 2.2, the mappingT is additive.

Theorem 2.4. Let f : X → Y be a mapping satisfying f0 0 for which there is a function ϕ:Xn → 0,∞satisfying2.6,2.7and

_Du,r

1,...,rnfx1, . . . , xn≤ϕx1, . . . , xn 2.26

for allx1, . . . , xn ∈ X and allu ∈ UA.Then there exists a uniqueA-linear generalized

Euler-Lagrange type additive mappingL : X → Y satisfying2.9 for allx ∈ X.Moreover, Lrkx

rkLxfor allx∈Xand all1≤k≤n.

Proof. By Theorem 2.3, there exists a unique generalized Euler-Lagrange type additive

mappingL : X → Y satisfying2.9 and moreoverLrkx rkLxfor allx ∈ X and all

1≤k≤n.

By the assumption, for eachu∈UA, we get

Du,r1,...,rnL0, . . . ,0, x ith

,0· · · ,0

Y

lim

k→ ∞ 1 2k

Du,r1,...,rnf0, . . . ,0,2kx ith

,0· · ·,0

Y

≤ lim

k→ ∞ 1 2kϕ

⎛

⎝_{0, . . . ,0, 2}k_x

ith

,0· · ·,0 ⎞ ⎠₀

2.27

for allx∈X. So

riuLx Lriux 2.28

for allu∈UAand allx∈X.SinceLrix riLxfor allx∈Xandri/0,

Lux uLx 2.29

for allu∈UAand allx∈X.

By the same reasoning as in the proofs of41,43,

LaxbyLax LbyaLx bLy 2.30

for alla, b ∈ Aa, b /0and allx, y ∈X.SinceL0x 0 0Lxfor allx∈X,the unique

generalized Euler-Lagrange type additive mappingL:X → Yis anA-linear mapping.

Corollary 2.5. Letδ ≥0,{k}k∈Jand{pk}k∈J be real numbers such thatk ≥0and0< pk <1for

allk ∈J,whereJ ⊆ {1,2, . . . , n}.Assume that a mappingf :X → Y withf0 0satisfies the inequality

_Du,r

1,...,rnfx1, . . . , xn_Y ≤δ

k∈J

kxkpk

for allx1, . . . , xn ∈ X and allu ∈ UA. Then there exists a uniqueA-linear generalized

Euler-Lagrange type additive mappingL:X → Y such that

_fx−Lx

Y ≤

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Mijx, i, j∈J;

Mix, i∈J, j /∈J;

Mjx, j∈J, i /∈J;

M, i, j /∈J.

2.32

for allx∈X,where

Mijx 9₂δ

k∈{i,j}

121−pk k

2−2pkrpk k

xpk X,

Mix 9₂δ

121−pi_i

2−2pirpi i

xpi X,

Mjx 9₂δ

121−pj j

2−2pjrpj j

xpj

X, M

9 2δ.

2.33

Moreover,Lr_kx r_kLxfor allx∈Xand all1≤k≤n.

Proof. Defineϕx1, . . . , xn:δk∈JkxkpXk,and applyTheorem 2.4.

Corollary 2.6. Letδ, ≥0, p, q >0withλ pq <1.Assume that a mappingf :X → Y with f0 0satisfies the inequality

_{Du,r1,...,rnfx1, . . . , xnY ≤δxi}p

XxjqX 2.34

for allx1, . . . , xn ∈ X and allu ∈ UA.Then there exists a uniqueA-linear generalized Euler-Lagrange type additive mappingL:X → Y such that

_fx−Lx

Y ≤

9

2δ

121−λ

22−2λr_ipr_jqx

λ

X 2.35

for allx∈X.Moreover,Lr_kx r_kLxfor allx∈Xand all1≤k≤n.

Proof. Defineϕx1, . . . , xn:δxip_Xxjq_X.ApplyingTheorem 2.4, we obtain the desired

Theorem 2.7. Let f : X → Y be a mapping satisfying f0 0 for which there is a function

Proof. By a similar method to the proof ofTheorem 2.3, we have the following inequality

for allx∈X.Replacingxbyx/2k1in2.40and multiplying both sides of2.40by 2k,we get

2k1f

_x

2k1

−2kf _x

2k

Y ≤2 k_Ψ x

2k1

2.43

for allx∈Xand allk∈Z.Therefore, we have

2k1f

_x

2k1

−2mf x 2m

Y

≤k lm

2l1f

_x

2l1

−2lf _x

2l

Y ≤ k

lm

2lΨ

_x

2l1

2.44

for all x ∈ X and all integersk ≥ m. It follows from2.42and 2.44 that the sequence

{2kfx/2k}is Cauchy inY for allx∈X,and thus converges by the completeness ofY.Thus

we can define a mappingL:X → Yby

Lx lim

k→ ∞2

k_fx

2k

2.45

for allx∈X.Lettingm0 in2.44and taking the limit ask → ∞in2.44, we obtain the desired inequality2.39.

The rest of the proof is similar to the proof ofTheorem 2.3.

Theorem 2.8. Letf :X → Y be a mapping withf0 0for which there is a functionφ :Xn →

0,∞satisfying2.36,2.37and

_{Du,r1,...,rnfx1, . . . , xn≤φx1, . . . , xn} 2.46

for allx1, . . . , xn ∈ X and allu ∈ UA.Then there exists a uniqueA-linear generalized Euler-Lagrange type additive mappingL : X → Y satisfying2.39for allx ∈ X.Moreover,Lrkx

rkLxfor allx∈Xand all1≤k≤n.

Proof. The proof is similar to the proof ofTheorem 2.4.

Corollary 2.9. Let{k}_k∈J and{pk}_k∈J be real numbers such thatk ≥0andpk >1for allk ∈J,

whereJ⊆ {1,2, . . . , n}.Assume that a mappingf:X → Y withf0 0satisfies the inequality

_Du,r

1,...,rnfx1, . . . , xn_Y ≤

k∈J

kxkpk

for allx1, . . . , xn ∈ X and allu ∈ UA.Then there exists a uniqueA-linear generalized

Euler-Lagrange type additive mappingL:X → Y such that

_fx−Lx

Y ≤

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Nijx, i, j∈J;

Nix, i∈J, j /∈J;

Njx, j ∈J, i /∈J;

N, i, j /∈J.

2.48

for allx∈X,where

Nijx

k∈{i,j}

121−pk_k

2pk−₂rpk k

xpk X,

Nix

121−pi_i

2pi−₂rpi i

xpi X,

Njx

121−pj_j

2pj−₂rpj j

xpj X.

2.49

Moreover,Lrkx rkLxfor allx∈Xand all1≤k≤n.

Proof. Defineφx1, . . . , xn : k∈JkxkpXk.ApplyingTheorem 2.8, we obtain the desired

result.

Corollary 2.10. Let≥0, p, q >0withλ pq >1.Assume that a mappingf :X → Y with f0 0satisfies the inequality

_Du,r

1,...,rnfx1, . . . , xn_Y ≤xip_Xxjq_X 2.50

for allx1, . . . , xn ∈ X and allu ∈ UA.Then there exists a uniqueA-linear generalized

Euler-Lagrange type additive mappingL:X → Y such that

_fx−Lx

Y ≤

121−λ

22λ−2r_ipr_jqx

λ

X 2.51

for allx∈X.Moreover,Lr_kx r_kLxfor allx∈Xand all1≤k≤n.

Proof. Define φx1, . . . , xn : xipXxjqX. Applying Theorem 2.8, we obtain the desired

result.

Remark 2.11. In Theorems 2.7 and 2.8 and Corollaries 2.9 and 2.10 one can assume that _n

k1rk/0 instead off0 0.

For the casep1· · ·pn1 in Corollaries2.5and2.9, using an idea from the example

Example 2.12. Letφ:C → Cbe defined by

Consider the functionf:C → Cby the formula

fx: ∞

n0

2−nφ2nx. 2.53

It is clear thatfis continuous and bounded by 2 onC. We prove that

for allm0,1, . . . , k.From the definition offand2.56, we have

$$_{Dμ,r1,...,rnfx1, . . . , xn}$$_≤4

nn

i1

|ri|

_∞

mk1 1 2m

8

nn

i1

|ri|

1 2k1

≤8

nn

i1

|ri|

_n

i1

|ri|1|xi|.

2.59

Thereforefsatisfies2.54. LetL:C → Cbe an additive mapping such that

$$_fx−Lx$$_{≤β|x| 2.60}

for allx∈C.Then there exists a constantc ∈Csuch thatLx cxfor all rational numbers

x.So we have

$$_fx$$_≤

β|c||x| 2.61

for all rational numbersx.Letm∈Nwithm > β|c|.Ifxis a rational number in0,21−m_,

then 2nx∈0,1for alln0,1, . . . , m−1.So

fx≥m−1 n0

2−nφ2nx mx >β|c||x| 2.62

which contradicts with2.61.

3. Homomorphisms in Unital

C

-Algebras

In this section, we investigateC∗-algebra homomorphisms in unitalC∗-algebras. We will use the following lemma in the proof of the next theorem.

Theorem 3.2. Let ≥ 0and{pk}_k∈J be real numbers such thatpk > 0for all k ∈ J,whereJ ⊆ {1,2, . . . , n}and|J| ≥ 3.Letf : A → Bbe a mapping withf0 0for which there is a function ϕ:An → 0,∞satisfying2.7and

_Dμ,r

1,...,rnfx1, . . . , xn_B≤

%

k∈J

xkp_Ak, _3.1

f2ku∗−f2ku∗

B≤ϕ

⎛

⎝₂k_{u, . . . ,2}k_u

ntimes ⎞

⎠_{, 3.2}

f2kux−f2kufx

B≤ϕ

⎛

⎝₂k_{ux, . . . ,2}k_ux

ntimes ⎞

⎠ _3.3

for allx, x1, . . . , xn∈A,for allu∈UA,allk∈Nand allμ∈S1.Then the mappingf :A → Bis

aC∗-algebra homomorphism.

Proof. Since|J| ≥3,lettingμ1 andxk0 for all 1≤k≤n, k /i, j,in3.1, we get

f−rixirjxjfrixi−rjxj2rifxi 2rjfxj2frixirjxj 3.4

for all x_i, x_j ∈ A.By the same reasoning as in the proof of Lemma 2.1, the mappingf is

additive andfrkx rkfxfor allx∈Aandki, j.So by lettingxi xandxk0 for all

1≤k≤n, k /i,in3.1, we get thatfμx μfxfor allx∈Aand allμ∈S1_{. Therefore, by} Lemma 3.1, the mappingfisC-linear. Hence it follows from2.7,3.2and3.3that

_fu∗_−fu∗

B_klim_{→ ∞}

1 2k

f2ku∗−f2ku∗

B

≤ lim

k→ ∞ 1 2kϕ

⎛

⎝₂k_{u, . . . ,2}k_u

ntimes ⎞ ⎠_0,

_fux−fufx

B_klim_{→ ∞}

1 2kf

2kux−f2kufxB

≤ lim

k→ ∞ 1 2kϕ

⎛

⎝₂k_{ux, . . . ,2}k_ux

ntimes ⎞ ⎠₀

3.5

for allx∈Aand allu∈UA.Sofu∗ fu∗andfux fufxfor allx∈Aand all

see57, that is,xm_k1λkuk,whereλk∈Canduk∈UAfor all 1≤k≤n,we have

The following theorem is an alternative result ofTheorem 3.2.

Theorem 3.3. Let ≥ 0and {pk}_k∈J be real numbers such thatpk > 0for allk ∈ J,where J ⊆ aC∗-algebra homomorphism.

Proof. Consider theC∗-algebrasAandBas left Banach modules over the unitalC∗-algebra C.ByTheorem 2.4, there exists a uniqueC-linear generalized Euler-Lagrange type additive

mappingH:A → Bdefined by

and allx∈A.Therefore, by the additivity ofHwe have

Hux lim follows from3.11that

for ally ∈A, Hy fyfor ally ∈A,therefore, the mappingf :A → Bis aC∗-algebra homomorphism.

The following theorem is an alternative result ofTheorem 3.5.

Theorem 3.6. Letf :A → Bbe a mapping withf0 0for which there is a functionφ :An →

0,∞satisfying2.36,2.37,3.7and

_{Dμ,r1,...,rnfx1, . . . , xn}

B≤φx1, . . . , xn, 3.14

for allx1, . . . , xn∈Aand allμ∈S1. Assume thatlimk→ ∞2kfe/2kis invertible. Then the mapping

f:A → Bis aC∗-algebra homomorphism.

Corollary 3.7. Let{k}_k∈J and{pk}_k∈Jbe real numbers such thatk ≥0andpk >1 0< pk <1

for allk∈J,whereJ ⊆ {1,2, . . . , n}.Assume that a mappingf:A → Bwithf0 0satisfies the inequalities

_{Dμ,r1,...,rnfx1, . . . , xn}

B≤

k∈Jkxk pk A,

f

_u∗

2m

−f u

2m ∗

B≤

k∈J

k

2mpk

resp.,f2mu∗−f2mu∗_B≤

k∈J

k2mpk

,

fux

2m

−f u

2m

fx

B≤

k∈J

k

2mpkx pk A

resp.,f2mux−f2mufx_B≤

k∈J

k2mpkxp_Ak

,

3.15

for all x1, . . . , xn ∈ A, all u ∈ UA, all m ∈ N and all μ ∈ S1. Assume that

lim_{k→ ∞}2kfe/2k resp.,lim_{k→ ∞}1/2kf2keis invertible. Then the mappingf : A → B is aC∗-algebra homomorphism.

Proof. The result follows fromTheorem 3.6resp.,Theorem 3.5.

Remark 3.8. In Theorem 3.6 and Corollary 3.7, one can assume that n_k1r_k/0 instead of

f0 0.

Theorem 3.9. Letf :A → Bbe a mapping withf0 0for which there is a functionϕ :An →

0,∞satisfying2.6,2.7,3.2,3.3and

_Dμ,r

forμi,1and allx1, . . . , xn∈A. Assume thatlimk→ ∞1/2kf2keis invertible and for each fixed

x∈Athe mappingt →ftxis continuous int∈R. Then the mappingf:A → Bis aC∗-algebra homomorphism.

Proof. Putμ1 in3.16. By the same reasoning as in the proof ofTheorem 2.3, there exists a

unique generalized Euler-Lagrange type additive mappingH:A → Bdefined by

Hx lim

k→ ∞

f2kx

2k 3.17

for allx ∈ A.By the same reasoning as in the proof of4, the generalized Euler-Lagrange

type additive mappingH:A → BisR-linear.

By the same method as in the proof ofTheorem 2.4, we have

Dμ,r1,...,rnH0, . . . ,0, x_jth

,0, . . . ,0

Y

lim

k→ ∞ 1 2k

Dμ,r1,...,rnf0, . . . ,0, 2

k_x

jth

,0, . . . ,0

Y

≤ lim

k→ ∞ 1 2kϕ

⎛ ⎜

⎝0, . . . ,0,2kx

jth

,0, . . . ,0 ⎞ ⎟ ⎠0

3.18

for allx∈A.So

rjμHx Hrjμx 3.19

for allx∈A.SinceHr_jx r_jHxfor allx∈Xandr_j/0,

HμxμHx 3.20

forμi,1 and for allx∈A.

For each elementλ∈Cwe haveλsit,wheres, t∈R. Thus

Hλx Hsxitx sHx tHix

sHx itHx sitHx λHx 3.21

for allλ∈Cand allx∈A.So

for allζ, η∈Cand allx, y∈A.Hence the generalized Euler-Lagrange type additive mapping

H:A → BisC-linear. The rest of the proof is the same as in the proof ofTheorem 3.5.

The following theorem is an alternative result ofTheorem 3.9.

Theorem 3.10. Letf :A → Bbe a mapping withf0 0for which there is a functionφ:An →

0,∞satisfying2.36,2.37,3.7and

_Dμ,r

1,...,rnfx1, . . . , xn_B≤φx1, . . . , xn, 3.23

forμi,1and allx, x1, . . . , xn∈A. Assume thatlimk→ ∞2kfe/2kis invertible and for each fixed

x∈Athe mappingt →ftxis continuous int∈R. Then the mappingf:A → Bis aC∗-algebra homomorphism.

Remark 3.11. InTheorem 3.10, one can assume thatn_k1r_k/0 instead off0 0.

Acknowledgments

The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper. C. Park author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of

Education, Science and TechnologyNRF-2009-0070788.

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14 A. Grabiec, “The generalized Hyers-Ulam stability of a class of functional equations,”Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 217–235, 1996.

15 J. M. Rassias, “On the stability of the general Euler-Lagrange functional equation,”Demonstratio Mathematica, vol. 29, no. 4, pp. 755–766, 1996.

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25 D. H. Hyers, G. Isac, and Th. M. Rassias, “On the asymptoticity aspect of Hyers-Ulam stability of mappings,”Proceedings of the American Mathematical Society, vol. 126, no. 2, pp. 425–430, 1998.

26 K.-W. Jun and H.-M. Kim, “On the Hyers-Ulam stability of a difference equation,” Journal of Computational Analysis and Applications, vol. 7, no. 4, pp. 397–407, 2005.

27 K.-W. Jun and H.-M. Kim, “Stability problem of Ulam for generalized forms of Cauchy functional equation,”Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 535–547, 2005.

28 K.-W. Jun and H.-M. Kim, “Stability problem for Jensen-type functional equations of cubic mappings,”Acta Mathematica Sinica, vol. 22, no. 6, pp. 1781–1788, 2006.

29 K.-W. Jun and H.-M. Kim, “Ulam stability problem for a mixed type of cubic and additive functional equation,”Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 13, no. 2, pp. 271–285, 2006.

30 K.-W. Jun, H.-M. Kim, and J. M. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,”Journal of Difference Equations and Applications, vol. 13, no. 12, pp. 1139–1153, 2007.

31 A. Najati, “Hyers-Ulam stability of ann-Apollonius type quadratic mapping,”Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 14, no. 4, pp. 755–774, 2007.

32 A. Najati, “On the stability of a quartic functional equation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 569–574, 2008.

33 A. Najati and M. B. Moghimi, “Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 399–415, 2008.

34 A. Najati and C. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 763–778, 2007.

35 A. Najati and C. Park, “On the stability of ann-dimensional functional equation originating from quadratic forms,”Taiwanese Journal of Mathematics, vol. 12, no. 7, pp. 1609–1624, 2008.

36 A. Najati and C. Park, “The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms betweenC∗-algebras,”Journal of Difference Equations and Applications, vol. 14, no. 5, pp. 459–479, 2008.

37 C.-G. 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.

38 C.-G. Park, “Linear functional equations in Banach modules over aC∗-algebra,”Acta Applicandae Mathematicae, vol. 77, no. 2, pp. 125–161, 2003.

40 C.-G. Park, “Lie ∗-homomorphisms between Lie C∗-algebras and Lie ∗-derivations on Lie C∗ -algebras,”Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 419–434, 2004.

41 C.-G. 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.

42 C.-G. Park, “Generalized Hyers-Ulam-Rassias stability of n-sesquilinear-quadratic mappings on Banach modules overC∗-algebras,”Journal of Computational and Applied Mathematics, vol. 180, no. 2, pp. 279–291, 2005.

43 C.-G. Park, “Homomorphisms between PoissonJC∗-algebras,”Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79–97, 2005.

44 C.-G. Park and J. Hou, “Homomorphisms betweenC∗-algebras associated with the Trif functional equation and linear derivations onC∗-algebras,”Journal of the Korean Mathematical Society, vol. 41, no. 3, pp. 461–477, 2004.

45 C. Park and J. M. Park, “Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping,”Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1277–1288, 2006.

46 J. M. Rassias and M. J. Rassias, “Asymptotic behavior of alternative Jensen and Jensen type functional equations,”Bulletin des Sciences Math´ematiques, vol. 129, no. 7, pp. 545–558, 2005.

47 M. J. Rassias and J. M. Rassias, “Refined Hyers-Ulam superstability of approximately additive mappings,”Journal of Nonlinear Functional Analysis and Differential Equations, vol. 1, no. 2, pp. 175– 182, 2007.

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49 Th. M. Rassias, “On the stability of the quadratic functional equation and its applications,”

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50 Th. 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.

51 Th. 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.

52 Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,”Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000.

53 Th. M. Rassias and P. ˇSemrl, “On the Hyers-Ulam stability of linear mappings,”Journal of Mathematical Analysis and Applications, vol. 173, no. 2, pp. 325–338, 1993.

54 Th. M. Rassias and K. Shibata, “Variational problem of some quadratic functionals in complex analysis,”Journal of Mathematical Analysis and Applications, vol. 228, no. 1, pp. 234–253, 1998.

55 D. Zhang and H.-X. Cao, “Stability of group and ring homomorphisms,”Mathematical Inequalities & Applications, vol. 9, no. 3, pp. 521–528, 2006.

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