Almost partial generalized Jordan derivations: a fixed point approach

(1)

R E S E A R C H

Open Access

Almost partial generalized Jordan derivations: a

fixed point approach

Madjid Eshaghi Gordji

1

, Choonkil Park

2

and Jung Rye Lee

3*

* Correspondence: [email protected]. kr

3_{Department of Mathematics,} Daejin University, Kyeonggi 487-711, Korea

Full list of author information is available at the end of the article

Abstract

Using fixed point method, we investigate the Hyers-Ulam stability and the

superstability of partial generalized Jordan derivations on Banach modules related to Jensen type functional equations.

Mathematics Subject Classification 2010:Primary, 39B52; 47H10; 47B47; 13N15; 39B72; 17C50; 39B82; 17C65.

Keywords:Hyers-Ulam stability, superstability, partial Jordan derivation, partial gener-alized Jordan derivation, Jensen type functional equation

1. Introduction and preliminaries

The following question posed by Ulam [1] in 1940: “When is it true that a mapping which approximately satisfies a functional equation _E must be somehow close to an exact solution of _E?”. Hyers [2] proved the problem for the Cauchy functional equa-tion. In 1978, Rassias [3] proved the following theorem.

Theorem 1.1. Let f: E ® E’ be a mapping from a normed vector space E into a

Banach space E’subject to the inequality

f(x+y)−f(x)−f(y) ≤ε(xp+yp) (1:1)

for all x, yÎE, whereεand p are constants withε> 0and p< 1.Then there exists a

unique additive mapping T: E®E’such that

f(x)−T(x) ≤ 2ε 2−2px

p ₍₁_:₂₎

for all xÎE. If p < 0then inequality(1.1)holds for all x, y≠0,and(1.2)for x≠0.

Also, if the function taf(tx)from ℝinto E’is continuous in real t for each xÎE, then T isℝ-linear.

In 1991, Gajda [4] answered the question for the case p> 1, which was raised by Rassias. In 1994, a generalization of the Rassias’theorem was obtained by Găvruta as follows [5].

Stability of the Jensen functional equation, 2fx+₂y=f(x) +f(y), where fis a

map-ping between linear spaces, has been investigated by several mathematicians (see [6,7]). During the last decades several stability problems of functional equations have been investigated by a number of mathematicians. See [8-17] and references therein for more detailed information.

(2)

LetA, B be two Banach algebras. Aℂ-linear mappingd: A®Bis called a general-ized Jordan derivation if there exists a Jordan derivation (in the usual sense) δ:A®X

such thatd(a2) =ad(a) +δ(a)afor allaÎA.

Generalized derivations and generalized Jordan derivations first appeared in the con-text of operator algebras [18]. Later, these were introduced in the framework of pure algebra [19,20].

Recently, Badora [21] proved the stability of ring derivations (see also [22,23]). More recently, Eshaghi Gordji and Ghobadipour [24] investigated the stability of generalized Jordan derivations on Banach algebras.

Let A1,. . .,An be normed algebras over the complex fieldℂand let B be a Banach

algebra over ℂ. A mapping dk:A1×A2× · · · ×An→B is called ak-th partial

deri-vation if

dk(x1, . . ., γak+μbk, . . ., xn) =γdk(x1, . . ., ak, . . ., xn)+μdk(x1, . . ., bk, . . ., xn) and there exists a mapping fk:Ak→B such that

dk(x1, . . .,akbk, . . ., xn) =fk(ak)dk(x1, . . ., bk, . . ., xn)+dk(x1, . . ., ak, . . ., xn)fk(bk)

for all ak,bk∈Ak and xi∈Ai(i=k) and allg,μÎℂ.

Chu et al. [25] established the Hyers-Ulam stability of partial derivations.

Definition 1.2. Let A1,. . .,An be normed algebras over the complex fieldℂand let

X be a Banach module over A1,. . .,An−1and An. Then

(i) A mapping dk:A1×A2× · · · ×An→X is called ak-th partial Jordan

deriva-tion of Jensen type if

2dk

x1,. . .,γ ak+γbk

2 ,. . .,xn

=γdk(x1, . . ., ak, . . ., xn)+γdk(x1, . . ., bk, . . ., xn)

and

dk(x1,. . ., a2k,. . ., xn) =akdk(x1, . . ., ak, . . ., xn) +dk(x1, . . ., ak,. . ., xn)ak for all ak,bk∈Ak and xi∈Ai(i=k) and allgÎℂ.

(ii) A mapping _δk:A1×A2× · · · ×An→X is called ak-th partial generalized

Jor-dan derivation of Jensen type if

2δk

x1,. . .,γ ak+γbk

2 ,. . .,xn

=γ δk(x1, . . ., ak, . . ., xn)+γ δk(x1, . . ., bk, . . ., xn)

and there exists a k-th partial Jordan derivation dk:A1×A2× · · · ×An→X such

that

δk(x1, . . .,a2k,. . ., xn) =δk(x1, . . ., ak, . . ., xn)ak+akdk(x1, . . ., ak,. . ., xn)

for all ak,bk∈Ak and xi∈Ai(i=k) and allgÎℂ.

We now introduce one of fundamental results of fixed point theory. For the proof, refer to [26,27]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [28].

(3)

(GM1)d(x, y) = 0 if and only ifx=y;

(GM2)d(x, y) =d(y, x) for allx, yÎ X;

(GM3)d(x, z)≤d(x, y) +d(y, z) for allx, y, zÎX.

Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity.

Let (X, d) be a generalized metric space. An operatorT:X ®X satisfies a Lipschitz condition with Lipschitz constant Lif there exists a constantL≥0 such that

d(Tx, Ty)≤Ld(x, y)

for all x, y Î X. If the Lipschitz constantL is less than 1, then the operator T is called a strictly contractive operator.

We recall the following theorem by Diaz and Margolis [26].

Theorem 1.3.Suppose that we are given a complete generalized metric space(Ω, d)

and a strictly contractive function T: Ω® Ωwith Lipschitz constant L. Then for each

given xÎΩ,either

dTmx,Tm+1x=∞ for all m≥0,

or other exists a natural number m0such that

⋆d(Tmx, Tm+1x) <∞for all m≥m0;

⋆the sequence{Tmx}is convergent to a fixed point y*of T; ⋆y* is the unique fixed point of T in

={y∈:d(Tm0_x_, _y₎<∞}_;

⋆d(y, y∗)≤ ₁₋1_Ld(y, Ty) for all y∈.

The equation (ξ) is called superstable if every approximate solution of (ξ) is an exact solution.

We use the fixed point method to investigate the Hyers-Ulam stability and the superstability of partial generalized Jordan derivations of Jensen type.

2. Main results Forn0Î N, we define

T1 1 nO

:=

eiθ 0≤θ ≤ 2π

no

and we denote T11 1 by T

1_{. Also, we suppose that} A

1,. . .,An are normed algebras

over the complex field ℂand _X is a Banach module over A1,. . .,An−1 and An. We

denote that 0k, 0_X are zero elements of Ak,X, respectively.

Theorem 2.1. Let Tk,Fk:A1× · · · ×An→Xbe mappings with

2Sk

x1,. . .,λ

ak+λbk

2 ,. . .,xn

−λSk(x1,. . .,ak,. . .,xn)−λSk(x1,. . .,bk,. . .,xn)

(4)

there exist functions k:Ak→[0, ∞), ϕk:A2_k →[0, ∞)satisfying

2Sk

x1,. . .,λ

ak+λbk

2 ,. . .,xn

≤ϕk(ak,bk), (2:1)

max{Fk(x1, . . ., ak2,. . .,xn)−akFk(x1,. . .,ak, . . ., xn)−Fk(x1,. . .,ak,. . .,xn)ak,

Tk(x1,. . .,ak2, . . ., xn)−Tk(x1, . . ., ak,. . .,xn)ak−akFk(x1,. . .,ak,. . .,xn)}

≤k(ak)

(2:2)

for Sk Î {Fk, Tk} and for all λ∈T

1 1

n0and all ak,bk∈Ak,

xi∈Ai(i=k). If there

exists a constant0 <L< 1such thatk(ak, bk)≤2Lk(2-1ak, 2-1bk),Ψk(ak)≤2LΨk(2-1ak)

for all ak,bk∈Ak, then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable dk:A1×A2× · · · ×An→Xand a unique partial gener-alized Jordan derivation of Jensen type with respect to k-th variable (related to dk)

Dk:A1×A2× · · · ×An→Xsuch that

max{Fk(x1,x2,. . .,xn)−dk(x1,x2, . . ., xn),Tk(x1,x2,. . .,xn)−Dk(x1,x2, . . ., xn)}

≤ L

1−Lϕk(xk, 0)

for all xi∈Ai(i= 1, 2, . . ., n). Proof. It follows from (2.1) that

2Sk

x1,. . .,λ

ak+λbk 2 ,. . .,xn

−λSk(x1,. . .,ak,. . .,xn)−λSk(x1,. . ., bk,. . .,xn)

≤ϕk(ak,bk),

(2:3)

for Sk Î{Fk, Tk} and for all λ∈T

1 1 n0

:={λ∈C:|λ| = 1} _{and all} _ak_,_bk_∈_A_k_,

xi∈Ai(i=k).

In the inequality (2.3), put Sk =Fk, bk = 0,l = 1 and replace ak with 2xk. Then we

obtain

Fk(x1, . . .,xk,. . ., xn)−2−1Fk(x1, . . ., 2xk, . . ., xn) ≤2−1ϕk(2xk, 0)≤Lϕ(xk, 0) (2:4)

for all xi∈Ai(i= 1, 2, . . ., n). Put :={Gk|Gk:A1×A2× · · · ×An→X} and

defined:Ω×Ω®[0,∞] by

d(Hk, Gk) : = inf{α∈R+;Gk(x1, . . ., xk, . . .,xn)−Hk(x1,. . ., xk, . . ., xn)

≤αϕk(xk, 0)∀xi∈Ai(i= 1, 2, . . .,n)}.

It is easy to show that (Ω, d) is a complete generalized metric space. We define the mapping J:Ω®Ωby

J(Hk)(x1, . . ., xk, . . ., xn) = 2−1Hk(x1, . . ., 2xk, . . ., xn)

for all xi∈Ai(i= 1, 2, . . ., n). Let Gk, HkÎ Ωand let aÎ (0,∞) be arbitrary with

d(Gk, Hk)≤a. From the definition of d, we have

(5)

for all xi∈Ai(i= 1, 2, . . ., n). Hence we have

(JGk)(x1, . . ., xk, . . ., xn)−(JHk)(x1, . . ., xk, . . ., xn) = 2−1Gk(x1, . . ., 2xk, . . ., xn)−Hk(x1, . . ., 2xk, . . ., xn) ≤2−1αϕk(2xk, 0)≤αLϕk(xk, 0)

for all xi∈Ai(i= 1, 2, . . ., n). So

d(J(Gk), J(Hk))≤Ld(Gk, Hk)

for all Gk, HkÎ Ω. It follows from (2.4) that

d(Fk, J(Fk))≤L.

By Theorem 1.3,J has a unique fixed point in the setΩ1:= { HkÎΩ;d(Fk, Hk) <∞}.

Let dkbe the fixed point ofJ. dkis the unique mapping which satisfies

dk(x1, . . ., 2xk, . . ., xn) = 2dk(x1, . . ., xk, . . ., xn)

for all xi∈Ai(i= 1, 2, . . ., n), and there existsaÎ(0, ∞) such that

dk(x1, . . ., xk, . . ., xn)−Fk(x1, . . ., xk, . . ., xn) ≤αϕk(xk, 0)

for all xi∈Ai(i= 1, 2, . . ., n).

On the other hand, we have limm®∞d(Jm(Fk),dk) = 0. It follows that

lim m→∞2

−m_Fk₍_x

1, . . ., 2mxk, . . ., xn) =dk(x1, . . ., xk, . . ., xn)

for all xi∈Ai(i= 1, 2, . . ., n). It follows from that d(Fk, dk)≤ ₁₋1_Ld(Fk, J(Fk)) that

d(Fk, dk)≤ L 1−L.

This means that

Fk(x1, x2, . . ., xn)−dk(x1, x2, . . ., xn)≤ L

1−Lϕk(xk, 0)

for all xi∈Ai(i= 1, 2, . . ., n). By the inequality k(ak, bk)≤ 2Lk(2-1ak, 2-1bk), we

conclude that

lim m→∞2

−m_ϕ₍₂m_ak_{, 2}m_bk_{) = 0}

for all ak,bk∈Ak. In the inequality (2.3), replacingak, bkby 2mak, 2mbk, respectively,

we obtain that

2−m2Fk

x1,. . .,λ

2m_ak₊_λ₂m_bk 2 ,. . .,xn

−Fk(x1, . . ., 2mak, . . ., xn)

−Fk(x1, . . ., 2mbk, . . ., xn)≤2−mϕk(2mak, 2mbk).

Passing the limit m® ∞, we obtain

2dk

x1,. . .,λ

ak+λbk 2 ,. . .,xn

(6)

for all ak,bk∈Ak and all λ∈T 1

1

n0. Now, we show that dk isℂ-linear with respect

to k-th variable. First suppose thatlbelongs to T1. Thenl=eiθfor some 0 ≤θ≤2π.

We set _λ

1=e iθ

no. Then l1belongs to T1

1 no

and

2dk

x1,. . .,λ1

ak+λ1bk 2 ,. . .,xn

=λ1dk(x1, . . .,ak, . . ., xn)+λ1dk(x1, . . ., bk, . . ., xn)

for all ak,bk∈Ak. It is easy to show thatdkis additive with respect tok-th variable.

Moreover, ifl belongs tonT1= {nz zÎ T1} then by additivity ofdkonk-th variable,

we have

dk(x1, . . ., λak, . . ., xn) =λdk(x1, . . ., ak, . . ., xn)

for all ak∈Ak. IftÎ(0,∞), then by Archimedean property ofℂ, there exists annÎ N such that the point (t, 0) lies in the interior of circle with center at origin and radius

n. Let t1=t+ √

n2₋_t2_i_∈_nT1 _and _t

2=t− √

n2₋_t2_i_∈_nT1_{. We have} _t₌ t1+t2

2 . Then

dk(x1, . . ., tak, . . ., xn) =dk

x1,. . ., t1+t2

2 ak,. . .,xn

=t1+t2

2 dk(x1, . . ., ak, . . ., xn)

for all ak∈Ak. LetlÎℂ. Then λ= |λ|eiλ1 and so

dk(x1, . . ., λak, . . ., xn) = |λ|eiλ1dk(x1, . . ., ak, . . ., xn) =λdk(x1, . . ., ak, . . ., xn) for all ak∈Ak. It follows thatdkisℂ-linear with respect tok-th variable.

By the same reasoning as above, we can show that the limit

Dk(x1, . . ., xk, . . ., xn) := lim m→∞2

−m_Tk₍_x

1, . . ., 2mxk, . . ., xn)

exists for all xi∈Ai(i= 1, 2, . . ., n) and that Dkis ℂ-linear with respect to k-th

variable.

By te inequalityΨk(ak)≤2LΨk(2-1ak), we conclude that

lim m→∞2

−m_k₍₂m_ak_{) = 0}

for all ak∈Ak.

Now, by (2.2), we have

Fk(x1, . . ., a2k, . . ., xn)−akFk(x1, . . ., ak, . . ., xn)−Fk(x1, . . ., ak, . . ., xn)ak

≤k(ak)

for all ak∈Ak, xi∈Ai(i=k). Replacing ak by 2mak in the above inequality, we

obtain that

Fk(x1, . . ., 22ma2k, . . ., xn)−2makFK(x1, . . ., 2mak, . . ., xn) −Fk(x1, . . ., 2mak, . . ., xn)2mak ≤k(2mak).

Then we have

(7)

for all ak∈Ak. Passingm® ∞, we obtain

dk(x1,. . ., a2k,. . ., xn) =akdk(x1, . . ., ak, . . ., xn) +dk(x1, . . ., ak,. . ., xn)ak for all ak∈Ak and all xi∈Ai(i=k). This shows thatdk is a partial Jordan

deriva-tion. We have to show that Dkis a partial generalized Jordan derivation related todk.

By (2.2), we have

Tk(x1, . . ., a2k, . . ., xn)−akTk(x1, . . ., ak, . . ., xn)−Fk(x1, . . ., ak, . . ., xn)ak

≤k(ak)

for all ak∈Ak, xi∈Ai(i=k). Replacingakby 2makin the last inequality, we get

Tk(x1, . . ., 22ma2k, . . ., xn)−2makTK(x1, . . ., 2mak, . . ., xn) −Fk(x1, . . ., 2mak, . . ., xn)2mak ≤k(2mak)

for all ak∈Ak, xi∈Ai(i=k). Then we have

2−2mTk(x1, . . ., 22ma2k, . . ., xn)−2−makTK(x1, . . ., 2mak, . . ., xn) −Fk(x1, . . ., 2mak, . . ., xn)2−mak≤2−2mk(2mak)

for all ak∈Ak, xi∈Ai(i=k). Passingm®∞, we obtain that

Dk(x1, . . ., a2k,. . ., xn) =akDk(x1, . . ., ak, . . ., xn) +dk(x1, . . ., ak, . . ., xn)ak for all ak∈Ak and all xi∈Ai(i=k). HenceDkis a partial generalized Jordan

deri-vation related todk.

Corollary 2.2. Let p Î (0, 1) and θ Î [0, ∞) be real numbers. Let

Tk,Fk:A1× · · · ×An→Xbe mappings such that

Tk(x1, . . ., 0k, . . ., xn) =Fk(x1, . . ., 0k, . . ., xn) = 0Xand that

2Sk

x1,. . .,λak+λbk 2 ,. . .,xn

≤ϕk(ak,bk),

max{Fk(x1, . . ., ak2, . . .,xn)−akFk(x1, . . ., ak,. . ., xn)−Fk(x1, . . ., ak,. . ., xn)ak, _Tk₍_x₁_,_{. . .}_,_a2

k, . . ., xn)−Tk(x1,. . .,ak, . . ., xn)ak−akFk(x1, . . ., ak, . . ., xn)}

≤θ(akp)

for SkÎ {Fk, Tk}and for all λ∈T

1 1 n0

and all ak,bk∈Ak, xi∈Ai(i=k).Then there

exist a unique partial Jordan derivation of Jensen type with respect to k-th variable dk:A1×A2× · · · ×An→Xand a unique partial generalized Jordan derivation of

Jensen type with respect to k-th variable (related to dk)

max Fk(x1,x2,. . .,xn)−dk(x1,x2,. . .,xn),Tk(x1,x2,. . .,xn)−Dk(x1,x2,. . .,xn)

≤ 2p 2−2pθxk

p

for all xi∈Ai(i= 1, 2, . . ., n).

Proof. It follows from Theorem 2.1 by puttingΨk(ak) =θ(||ak||p),k(ak, bk) = θ(|| ak

(8)

Theorem 2.3. Let Tk,Fk:A1× · · · ×An→Xbe mappings with Tk(x1, . . ., 0k, . . ., xn) =Fk(x1, . . ., 0k, . . ., xn) = 0X. Assume that there exist

func-tions k:Ak→[0, ∞), ϕk:A2_k→[0, ∞)satisfying

2Sk

x1,. . .,λak+λbk 2 ,. . .,xn

−λSk(x1, . . ., ak, . . ., xn)−λSk(x1, . . ., bk, . . .,xn) , ≤ϕk(ak,bk),

max{Fk(x1,. . .,ak2,. . .,xn)−akFk(x1, . . ., ak, . . ., xn)−Fk(x1, . . ., ak,. . ., xn)ak, Tk(x1, . . ., a2k, . . ., xn)−Tk(x1, . . ., ak,. . .,xn)ak−akFk(x1,. . .,ak, . . ., xn)}

≤k(ak)

for Sk Î {Fk, Tk} and for all λ∈T

1 1 n0

and all ak,bk∈Ak, xi∈Ai(i=k). If there

exists a constant0 <L< 1such thatk(ak, bk)≤2-1Lk(2ak, 2bk),Ψk(ak)≤2-1LΨk(2ak)

for all ak,bk∈Ak, then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable dk:A1×A2× · · · ×An→Xand a unique partial gener-alized Jordan derivation of Jensen type with respect to k-th variable (related to dk)

max{Fk(x1,x2,. . .,xn)−dk(x1,x2, . . ., xn),Tk(x1,x2,. . .,xn)−Dk(x1,x2, . . ., xn)}

≤ L

2−2Lϕk(2xk, 0)

for all xi∈Ai(i= 1, 2, . . ., n).

Proof. The proof is similar to the proof of Theorem 2.1.□

Corollary 2.4. Let p Î (1, ∞) and θ Î [0, ∞) be real numbers. Let

Tk,Fk:A1× · · · ×An→Xbe mappings such that

2Sk

x1,. . .,λ

ak+λbk 2 ,. . .,xn

−λSk(x1,. . ., ak,. . ., xn)−λSk(x1, . . ., bk,. . ., xn)

≤θ(akp₊_bkp_),

and

2Sk

x1,. . .,λ

ak+λbk 2 ,. . .,xn

−λSk(x1, . . .,ak, . . ., xn)−λSk(x1,. . .,bk, . . .,xn)

≤θ(a_kp₊_b_kp_),

max{Fk(x1, . . ., ak2, . . .,xn)−akFk(x1, . . ., ak,. . ., xn)−Fk(x1, . . ., ak,. . ., xn)ak, Tk(x1, . . ., a2k, . . ., xn)−Tk(x1, . . ., ak,. . .,xn)ak−akFk(x1, . . ., ak,. . ., xn)}

≤θ(akp₎

1 1 n0

and all ak,bk∈Ak, xi∈Ai(i=k).Then there

exist a unique partial Jordan derivation of Jensen type with respect to k-th variable dk:A1×A2× · · · ×An→Xand a unique partial generalized Jordan derivation of

Jensen type with respect to k-th variable (related to dk)

max{Fk(x1,x2,. . .,xn)−dk(x1,x2, . . ., xn),Tk(x1,x2,. . .,xn)−Dk(x1,x2, . . ., xn)}

≤ θ 1−2p−1xk

p

(9)

Proof. It follows from Theorem 2.3 by puttingΨk(ak)=θ(||ak||p),k(ak, bk) =θ(||ak|| p

+ ||bk||p) andL= 21-pfor each ak,bk∈Ak.□

Moreover, we have the following result for the superstability of partial generalized Jordan derivations of Jensen type.

Corollary 2.5. Let p∈(0, 1₂)and θ Î [0, ∞) be real numbers. Let

Tk,Fk:A1× · · · ×An→X be mappings such that

Tk(x1, . . ., 0k, . . ., xn) =Fk(x1, . . ., 0k, . . ., xn) = 0X and

2Sk

x1,. . .,λak +λbk 2 ,. . .,xn

−λSk(x1,. . .,ak, . . ., xn)−λSk(x1, . . ., bk, . . .,xn)

≤θ(akpbkp),

max{Fk(x1, . . ., ak2, . . .,xn)−akFk(x1, . . ., ak,. . ., xn)−Fk(x1, . . ., ak,. . ., xn)ak, Tk(x1, . . ., a2k, . . ., xn)−Tk(x1, . . ., ak,. . .,xn)ak−akFk(x1, . . ., ak,. . ., xn)}

≤θ(akp₎

1 1

n0and all ak,bk∈Ak, xi∈Ai(i=k).Then Fkis a

partial Jordan derivation of Jensen type with respect to k-th variable and Tkis a partial

generalized Jordan derivation of Jensen type with respect to k-th variable (related to Fk).

Proof. It follows from Theorem 2.1 by puttingΨk(ak)=θ(||ak||p),k(ak, bk) =θ(||ak|| p

+ ||bk||p), andL= 22p-1.□

Acknowledgements

This work was supported by the Daejin University Research Grants in 2012.

Author details

1_{Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran}2_{Department of Mathematics,} Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea3Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea

Authors’contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 27 October 2011 Accepted: 29 May 2012 Published: 29 May 2012

References

1. Ulam, SM: A Collection of the Mathematical Problems. Interscience Publ., New York (1960)

2. Hyers, DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA.27, 222_–224 (1941). doi:10.1073/ pnas.27.4.222

3. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc.72, 297_–300 (1978). doi:10.1090/S0002-9939-1978-0507327-1

4. Gajda, Z: On stability of additive mappings. Int J Math Math Sci.14, 431_–434 (1991). doi:10.1155/S016117129100056X 5. Găvruta, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl.

184, 431–436 (1994). doi:10.1006/jmaa.1994.1211

6. Cădariu, L, Radu, V: Fixed points and the stability of Jensen’s functional equation. J Inequal Pure Appl Math4, 7 (2003). no. 1, 7 (Article ID 4)

7. Kominek, Z: On a local stability of the Jensen functional equation. Demonstratio Math.22, 499_–507 (1989) 8. Czerwik, S: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor, FL (2003) 9. Jung, S: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press lnc, Palm Harbor,

FL (2001)

10. Najati, A, Kang, J, Cho, Y: Local stability of the pexiderized Cauchy and Jensen_’s equations in fuzzy spaces. J Inequal Appl2011, 8 (2011). Article No. 78. doi:10.1186/1029-242X-2011-8

(10)

12. Rassias, TM: The problem of S.M. Ulam for approximately multiplicative mappings. J Math Anal Appl.246, 352–378 (2000). doi:10.1006/jmaa.2000.6788

13. Rassias, TM: On the stability of functional equations in Banach spaces. J Math Anal Appl.251, 264–284 (2000). doi:10.1006/jmaa.2000.7046

14. Rassias, TM: On the stability of functional equations and a problem of Ulam. Acta Appl Math.62, 23–130 (2000). doi:10.1023/A:1006499223572

15. Rassias, TM,Šemrl, P: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc Am Math Soc.114, 989_–993 (1992). doi:10.1090/S0002-9939-1992-1059634-1

16. Ulam, TM,Šemrl, P: On the Hyers-Ulam stability of linear mappings. J Math Anal Appl.173, 325–338 (1993). doi:10.1006/ jmaa.1993.1070

17. Rassias, TM, Shibata, K: Variational problem of some quadratic functionals in complex analysis. J Math Anal Appl.228, 234–253 (1998). doi:10.1006/jmaa.1998.6129

18. Mathieu, M: Elementary Operators & Applications. World Scientific, NJ (1992)

19. Feng, W, Zhankui, X: Generalized Jordan derivations on semiprime rings. Demonstratio Math.40, 789–798 (2007) 20. Hvala, B: Generalized derivations. Commun Algebra.26, 1147_–1166 (1998). doi:10.1080/00927879808826190 21. Badora, R: On approximate derivations. Math Inequal Appl.9, 167–173 (2006)

22. Eshaghi Gordji, M, Moslehian, MS: A trick for investigation of approximate derivations. Math Commun.15, 99_–105 (2010)

23. Miura, T, Hirasawa, G, Takahasi, SE: A perturbation of ring derivations on Banach algebras. J Math Anal Appl.319, 522–530 (2006). doi:10.1016/j.jmaa.2005.06.060

24. Eshaghi Gordji, M, Ghobadipour, N: Nearly generalized Jordan derivations. Math Slovaca.61, 55–62 (2011). doi:10.2478/ s12175-010-0059-x

25. Chu, H, Ku, S, Park, J: Partial stabilities and partial derivations ofn-variable functions. Nonlinear Anal TMA.72, 1531–1541 (2010). doi:10.1016/j.na.2009.08.038

26. Diaz, JB, Margolis, B: A fixed point theorem of the alternative for the contractions on generaliuzed complete metric space. Bull Am Math Soc.74, 305_–309 (1968). doi:10.1090/S0002-9904-1968-11933-0

27. Rus, IA: Principles and Applications of Fixed Point Theory. (1979)

28. Hyers, DH, Isac, G, Rassias, TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel. (1998)

doi:10.1186/1029-242X-2012-119

Cite this article as:Gordjiet al.:Almost partial generalized Jordan derivations: a fixed point approach.Journal of Inequalities and Applications20122012:119.

Submit your manuscript to a

journal and benefi t from:

7Convenient online submission

7Rigorous peer review

7Immediate publication on acceptance

7Open access: articles freely available online

7High visibility within the fi eld

7Retaining the copyright to your article