Skip to content

Advertisement

  • Research
  • Open Access

Almost partial generalized Jordan derivations: a fixed point approach

Journal of Inequalities and Applications20122012:119

https://doi.org/10.1186/1029-242X-2012-119

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

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 generalized 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 equation. In 1978, Rassias [3] proved the following theorem.

Theorem 1.1. Let f: EE' 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 ( x p + y p )
(1.1)
for all x, y E, where ε and p are constants with ε > 0 and p < 1. Then there exists a unique additive mapping T: EE' such that
f ( x ) - T ( x ) 2 ε 2 - 2 p x p
(1.2)

for all x E. If p < 0 then inequality (1.1) holds for all x, y ≠ 0, and (1.2) for x ≠ 0. Also, if the function t α f(tx) from into E' is continuous in real t for each x E, then T is -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, 2 f x + y 2 = f ( x ) + f ( y ) , where f is a mapping 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 [817] and references therein for more detailed information.

Let A, B be two Banach algebras. A -linear mapping d: AB is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) δ: AX such that d(a2) = ad(a) + δ(a)a for all a A.

Generalized derivations and generalized Jordan derivations first appeared in the context 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 A 1 , , A n be normed algebras over the complex field and let B be a Banach algebra over . A mapping d k : A 1 × A 2 × × A n B is called a k-th partial derivation if
d k ( x 1 , , γ a k + μ b k , , x n ) = γ d k ( x 1 , , a k , , x n ) + μ d k ( x 1 , , b k , , x n )
and there exists a mapping f k : A k B such that
d k ( x 1 , , a k b k , , x n ) = f k ( a k ) d k ( x 1 , , b k , , x n ) + d k ( x 1 , , a k , , x n ) f k ( b k )

for all a k , b k A k and x i A i ( i k ) and all γ, μ .

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

Definition 1.2. Let A 1 , , A n be normed algebras over the complex field and let X be a Banach module over A 1 , , A n - 1 and A n . Then
  1. (i)
    A mapping d k : A 1 × A 2 × × A n X is called a k-th partial Jordan derivation of Jensen type if
    2 d k x 1 , , γ a k + γ b k 2 , , x n = γ d k ( x 1 , , a k , , x n ) + γ d k ( x 1 , , b k , , x n )
     
and
d k ( x 1 , , a k 2 , , x n ) = a k d k ( x 1 , , a k , , x n ) + d k ( x 1 , , a k , , x n ) a k
for all a k , b k A k and x i A i ( i k ) and all γ .
  1. (ii)
    A mapping δ k : A 1 × A 2 × × A n X is called a k-th partial generalized Jordan derivation of Jensen type if
    2 δ k x 1 , , γ a k + γ b k 2 , , x n = γ δ k ( x 1 , , a k , , x n ) + γ δ k ( x 1 , , b k , , x n )
     
and there exists a k-th partial Jordan derivation d k : A 1 × A 2 × × A n X such that
δ k ( x 1 , , a k 2 , , x n ) = δ k ( x 1 , , a k , , x n ) a k + a k d k ( x 1 , , a k , , x n )

for all a k , b k A k and x i A i ( i k ) and all γ .

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].

Let X be a set. A function d: X × X → [0, ] is called a generalized metric on X if and only if d satisfies:

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

(GM2) d(x, y) = d(y, x) for all x, y X;

(GM3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z X.

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 operator T: XX satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that
d ( T x , T y ) L d ( x , y )

for all x, y X. If the Lipschitz constant L 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 → Ω with Lipschitz constant L. Then for each given x Ω, either
d T m x , T m + 1 x = f o r a l l m 0 ,

or other exists a natural number m 0 such that

d(T m x, Tm+1x) < ∞ for all mm0;

the sequence {T m x} is convergent to a fixed point y* of T;

y* is the unique fixed point of T in
Λ = { y Ω : d ( T m 0 x , y ) < } ;

d ( y , y * ) 1 1 - L d ( y , T y ) f o r a l l y Λ .

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

For n0 , we define
T 1 1 n O : = e i θ 0 θ 2 π n o

and we denote T 1 1 1 by T 1 . Also, we suppose that A 1 , , A n are normed algebras over the complex field and X is a Banach module over A 1 , , A n - 1 and A n . We denote that 0 k , 0 X are zero elements of A k , X , respectively.

Theorem 2.1. Let T k , F k : A 1 × × A n X be mappings with T k ( x 1 , , 0 k , , x n ) = F k ( x 1 , , 0 k , , x n ) = 0 X . Assume that there exist functions Ψ k : A k [ 0 , ) , φ k : A k 2 [ 0 , ) satisfying
2 S k x 1 , , λ a k + λ b k 2 , , x n - λ S k ( x 1 , , a k , , x n ) - λ S k ( x 1 , , b k , , x n ) φ k ( a k , b k ) ,
(2.1)
max { F k ( x 1 , , a k 2 , , x n ) - a k F k ( x 1 , , a k , , x n ) - F k ( x 1 , , a k , , x n ) a k , T k ( x 1 , , a k 2 , , x n ) - T k ( x 1 , , a k , , x n ) a k - a k F k ( x 1 , , a k , , x n ) } Ψ k ( a k )
(2.2)
for S k {F k , T k } and for all λ T 1 1 n 0 and all a k , b k A k , x i A i ( i k ) . If there exists a constant 0 < L < 1 such that φ k (a k , b k ) ≤ 2 k (2-1a k , 2-1b k ), Ψ k (a k ) ≤ 2L Ψ k (2-1a k ) for all a k , b k A k , then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable d k : A 1 × A 2 × × A n X and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d k ) D k : A 1 × A 2 × × A n X such that
max { F k ( x 1 , x 2 , , x n ) - d k ( x 1 , x 2 , , x n ) , T k ( x 1 , x 2 , , x n ) - D k ( x 1 , x 2 , , x n ) } L 1 - L φ k ( x k , 0 )

for all x i A i ( i = 1 , 2 , , n ) .

Proof. It follows from (2.1) that
2 S k x 1 , , λ a k + λ b k 2 , , x n - λ S k ( x 1 , , a k , , x n ) - λ S k ( x 1 , , b k , , x n ) φ k ( a k , b k ) ,
(2.3)

for S k {F k , T k } and for all λ T 1 1 n 0 : = { λ : | λ | = 1 } and all a k , b k A k , x i A i ( i k ) .

In the inequality (2.3), put S k = F k , b k = 0, λ = 1 and replace a k with 2x k . Then we obtain
F k ( x 1 , , x k , , x n ) - 2 - 1 F k ( x 1 , , 2 x k , , x n ) 2 - 1 φ k ( 2 x k , 0 ) L φ ( x k , 0 )
(2.4)
for all x i A i ( i = 1 , 2 , , n ) . Put Ω : = { G k | G k : A 1 × A 2 × × A n X } and define d: Ω × Ω → [0, ] by
d ( H k , G k ) : = inf { α + ; G k ( x 1 , , x k , , x n ) - H k ( x 1 , , x k , , x n ) α φ k ( x k , 0 ) x i A i ( i = 1 , 2 , , n ) } .
It is easy to show that (Ω, d) is a complete generalized metric space. We define the mapping J: Ω → Ω by
J ( H k ) ( x 1 , , x k , , x n ) = 2 - 1 H k ( x 1 , , 2 x k , , x n )
for all x i A i ( i = 1 , 2 , , n ) . Let G k , H k Ω and let α (0, ) be arbitrary with d(G k , H k ) ≤ α. From the definition of d, we have
G k ( x 1 , , x k , , x n ) - H k ( x 1 , , x k , , x n ) α φ k ( x k , 0 )
for all x i A i ( i = 1 , 2 , , n ) . Hence we have
( J G k ) ( x 1 , , x k , , x n ) - ( J H k ) ( x 1 , , x k , , x n ) = 2 - 1 G k ( x 1 , , 2 x k , , x n ) - H k ( x 1 , , 2 x k , , x n ) 2 - 1 α φ k ( 2 x k , 0 ) α L φ k ( x k , 0 )
for all x i A i ( i = 1 , 2 , , n ) . So
d ( J ( G k ) , J ( H k ) ) L d ( G k , H k )
for all G k , H k Ω. It follows from (2.4) that
d ( F k , J ( F k ) ) L .
By Theorem 1.3, J has a unique fixed point in the set Ω1 := { H k Ω; d(F k , H k ) < ∞}. Let d k be the fixed point of J. d k is the unique mapping which satisfies
d k ( x 1 , , 2 x k , , x n ) = 2 d k ( x 1 , , x k , , x n )
for all x i A i ( i = 1 , 2 , , n ) , and there exists α (0, ) such that
d k ( x 1 , , x k , , x n ) - F k ( x 1 , , x k , , x n ) α φ k ( x k , 0 )

for all x i A i ( i = 1 , 2 , , n ) .

On the other hand, we have limmd(J m (F k ), d k ) = 0. It follows that
lim m 2 - m F k ( x 1 , , 2 m x k , , x n ) = d k ( x 1 , , x k , , x n )
for all x i A i ( i = 1 , 2 , , n ) . It follows from that d ( F k , d k ) 1 1 - L d ( F k , J ( F k ) ) that
d ( F k , d k ) L 1 - L .
This means that
F k ( x 1 , x 2 , , x n ) - d k ( x 1 , x 2 , , x n ) L 1 - L φ k ( x k , 0 )
for all x i A i ( i = 1 , 2 , , n ) . By the inequality φ k (a k , b k ) ≤ 2 k (2-1a k , 2-1b k ), we conclude that
lim m 2 - m φ ( 2 m a k , 2 m b k ) = 0
for all a k , b k A k . In the inequality (2.3), replacing a k , b k by 2 m a k , 2 m b k , respectively, we obtain that
2 - m 2 F k x 1 , , λ 2 m a k + λ 2 m b k 2 , , x n - F k ( x 1 , , 2 m a k , , x n ) - F k ( x 1 , , 2 m b k , , x n ) 2 - m φ k ( 2 m a k , 2 m b k ) .
Passing the limit m, we obtain
2 d k x 1 , , λ a k + λ b k 2 , , x n = λ d k ( x 1 , , a k , , x n ) + λ d k ( x 1 , , b k , , x n )
for all a k , b k A k and all λ T 1 1 n 0 . Now, we show that d k is -linear with respect to k-th variable. First suppose that λ belongs to T1. Then λ = e for some 0 ≤ θ ≤ 2π. We set λ 1 = e i θ n o . Then λ1 belongs to T 1 n o 1 and
2 d k x 1 , , λ 1 a k + λ 1 b k 2 , , x n = λ 1 d k ( x 1 , , a k , , x n ) + λ 1 d k ( x 1 , , b k , , x n )
for all a k , b k A k . It is easy to show that d k is additive with respect to k-th variable. Moreover, if λ belongs to nT1 = {nz z T1} then by additivity of d k on k-th variable, we have
d k ( x 1 , , λ a k , , x n ) = λ d k ( x 1 , , a k , , x n )
for all a k A k . If t (0, ∞), then by Archimedean property of , there exists an n such that the point (t, 0) lies in the interior of circle with center at origin and radius n. Let t 1 = t + n 2 - t 2 i n T 1 and t 2 = t - n 2 - t 2 i n T 1 . We have t = t 1 + t 2 2 . Then
d k ( x 1 , , t a k , , x n ) = d k x 1 , , t 1 + t 2 2 a k , , x n = t 1 + t 2 2 d k ( x 1 , , a k , , x n )
for all a k A k . Let λ . Then λ = | λ | e i λ 1 and so
d k ( x 1 , , λ a k , , x n ) = | λ | e i λ 1 d k ( x 1 , , a k , , x n ) = λ d k ( x 1 , , a k , , x n )

for all a k A k . It follows that d k is -linear with respect to k-th variable.

By the same reasoning as above, we can show that the limit
D k ( x 1 , , x k , , x n ) : = lim m 2 - m T k ( x 1 , , 2 m x k , , x n )

exists for all x i A i ( i = 1 , 2 , , n ) and that D k is -linear with respect to k-th variable.

By te inequality Ψ k (a k ) ≤ 2L Ψ k (2-1a k ), we conclude that
lim m 2 - m Ψ k ( 2 m a k ) = 0

for all a k A k .

Now, by (2.2), we have
F k ( x 1 , , a k 2 , , x n ) - a k F k ( x 1 , , a k , , x n ) - F k ( x 1 , , a k , , x n ) a k Ψ k ( a k )
for all a k A k , x i A i ( i k ) . Replacing a k by 2 m a k in the above inequality, we obtain that
F k ( x 1 , , 2 2 m a k 2 , , x n ) - 2 m a k F K ( x 1 , , 2 m a k , , x n ) - F k ( x 1 , , 2 m a k , , x n ) 2 m a k Ψ k ( 2 m a k ) .
Then we have
2 - 2 m F k ( x 1 , , 2 2 m a k 2 , , x n ) - 2 - m a k F K ( x 1 , , 2 m a k , , x n ) - F k ( x 1 , , 2 m a k , , x n ) 2 - m a k 2 - 2 m Ψ k ( 2 m a k )
for all a k A k . Passing m, we obtain
d k ( x 1 , , a k 2 , , x n ) = a k d k ( x 1 , , a k , , x n ) + d k ( x 1 , , a k , , x n ) a k
for all a k A k and all x i A i ( i k ) . This shows that d k is a partial Jordan derivation. We have to show that D k is a partial generalized Jordan derivation related to d k . By (2.2), we have
T k ( x 1 , , a k 2 , , x n ) - a k T k ( x 1 , , a k , , x n ) - F k ( x 1 , , a k , , x n ) a k Ψ k ( a k )
for all a k A k , x i A i ( i k ) . Replacing a k by 2 m a k in the last inequality, we get
T k ( x 1 , , 2 2 m a k 2 , , x n ) - 2 m a k T K ( x 1 , , 2 m a k , , x n ) - F k ( x 1 , , 2 m a k , , x n ) 2 m a k Ψ k ( 2 m a k )
for all a k A k , x i A i ( i k ) . Then we have
2 - 2 m T k ( x 1 , , 2 2 m a k 2 , , x n ) - 2 - m a k T K ( x 1 , , 2 m a k , , x n ) - F k ( x 1 , , 2 m a k , , x n ) 2 - m a k 2 - 2 m Ψ k ( 2 m a k )
for all a k A k , x i A i ( i k ) . Passing m, we obtain that
D k ( x 1 , , a k 2 , , x n ) = a k D k ( x 1 , , a k , , x n ) + d k ( x 1 , , a k , , x n ) a k

for all a k A k and all x i A i ( i k ) . Hence D k is a partial generalized Jordan derivation related to d k .

Corollary 2.2. Let p (0, 1) and θ [0, ) be real numbers. Let T k , F k : A 1 × × A n X be mappings such that T k ( x 1 , , 0 k , , x n ) = F k ( x 1 , , 0 k , , x n ) = 0 X and that
2 S k x 1 , , λ a k + λ b k 2 , , x n - λ S k ( x 1 , , a k , , x n ) - λ S k ( x 1 , , b k , , x n ) θ ( a k p + b k p ) ,
max { F k ( x 1 , , a k 2 , , x n ) - a k F k ( x 1 , , a k , , x n ) - F k ( x 1 , , a k , , x n ) a k , T k ( x 1 , , a k 2 , , x n ) - T k ( x 1 , , a k , , x n ) a k - a k F k ( x 1 , , a k , , x n ) } θ ( a k p )
for S k {F k , T k } and for all λ T 1 1 n 0 and all a k , b k A k , x i A i ( i k ) . Then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable d k : A 1 × A 2 × × A n X and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d k ) D k : A 1 × A 2 × × A n X such that
max F k ( x 1 , x 2 , , x n ) - d k ( x 1 , x 2 , , x n ) , T k ( x 1 , x 2 , , x n ) - D k ( x 1 , x 2 , , x n ) 2 p 2 - 2 p θ x k p

for all x i A i ( i = 1 , 2 , , n ) .

Proof. It follows from Theorem 2.1 by putting Ψ k (a k ) = θ(||a k || p ), φ k (a k , b k ) = θ(|| a k || p + ||b k || p ) and L = 2p-1. □

Theorem 2.3. Let T k , F k : A 1 × × A n X be mappings with T k ( x 1 , , 0 k , , x n ) = F k ( x 1 , , 0 k , , x n ) = 0 X . Assume that there exist functions Ψ k : A k [ 0 , ) , φ k : A k 2 [ 0 , ) satisfying
2 S k x 1 , , λ a k + λ b k 2 , , x n - λ S k ( x 1 , , a k , , x n ) - λ S k ( x 1 , , b k , , x n ) , φ k ( a k , b k ) ,
max { F k ( x 1 , , a k 2 , , x n ) - a k F k ( x 1 , , a k , , x n ) - F k ( x 1 , , a k , , x n ) a k , T k ( x 1 , , a k 2 , , x n ) - T k ( x 1 , , a k , , x n ) a k - a k F k ( x 1 , , a k , , x n ) } Ψ k ( a k )
for S k {F k , T k } and for all λ T 1 1 n 0 and all a k , b k A k , x i A i ( i k ) . If there exists a constant 0 < L < 1 such that φ k (a k , b k ) ≤ 2-1 k (2a k , 2b k ), Ψ k (a k ) ≤ 2-1L Ψ k (2a k ) for all a k , b k A k , then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable d k : A 1 × A 2 × × A n X and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d k ) D k : A 1 × A 2 × × A n X such that
max { F k ( x 1 , x 2 , , x n ) - d k ( x 1 , x 2 , , x n ) , T k ( x 1 , x 2 , , x n ) - D k ( x 1 , x 2 , , x n ) } L 2 - 2 L φ k ( 2 x k , 0 )

for all x i A i ( i = 1 , 2 , , n ) .

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

Corollary 2.4. Let p (1, ) and θ [0, ) be real numbers. Let T k , F k : A 1 × × A n X be mappings such that T k ( x 1 , , 0 k , , x n ) = F k ( x 1 , , 0 k , , x n ) = 0 X and
2 S k x 1 , , λ a k + λ b k 2 , , x n - λ S k ( x 1 , , a k , , x n ) - λ S k ( x 1 , , b k , , x n ) θ ( a k p + b k p ) ,
max { F k ( x 1 , , a k 2 , , x n ) - a k F k ( x 1 , , a k , , x n ) - F k ( x 1 , , a k , , x n ) a k , T k ( x 1 , , a k 2 , , x n ) - T k ( x 1 , , a k , , x n ) a k - a k F k ( x 1 , , a k , , x n ) } θ ( a k p )
for S k {F k , T k } and for all λ T 1 1 n 0 and all a k , b k A k , x i A i ( i k ) . Then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable d k : A 1 × A 2 × × A n X and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d k ) D k : A 1 × A 2 × × A n X such that
max { F k ( x 1 , x 2 , , x n ) - d k ( x 1 , x 2 , , x n ) , T k ( x 1 , x 2 , , x n ) - D k ( x 1 , x 2 , , x n ) } θ 1 - 2 p - 1 x k p

for all x i A i ( i = 1 , 2 , , n ) .

Proof. It follows from Theorem 2.3 by putting Ψ k (a k )=θ(||a k || p ), φ k (a k , b k ) = θ(||a k || p + ||b k || p ) and L = 21-pfor each a k , b k A k . □

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

Corollary 2.5. Let p ( 0 , 1 2 ) and θ [0, ) be real numbers. Let T k , F k : A 1 × × A n X be mappings such that T k ( x 1 , , 0 k , , x n ) = F k ( x 1 , , 0 k , , x n ) = 0 X and
2 S k x 1 , , λ a k + λ b k 2 , , x n - λ S k ( x 1 , , a k , , x n ) - λ S k ( x 1 , , b k , , x n ) θ ( a k p b k p ) ,
max { F k ( x 1 , , a k 2 , , x n ) - a k F k ( x 1 , , a k , , x n ) - F k ( x 1 , , a k , , x n ) a k , T k ( x 1 , , a k 2 , , x n ) - T k ( x 1 , , a k , , x n ) a k - a k F k ( x 1 , , a k , , x n ) } θ ( a k p )

for S k {F k , T k }and for all λ T 1 1 n 0 and all a k , b k A k , x i A i ( i k ) . Then F k is a partial Jordan derivation of Jensen type with respect to k-th variable and T k is a partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to F k ).

Proof. It follows from Theorem 2.1 by putting Ψ k (a k )=θ(||a k || p ), φ k (a k , b k ) = θ(||a k || p + ||b k || p ), and L = 22p- 1. □

Declarations

Acknowledgements

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

Authors’ Affiliations

(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, Korea
(3)
Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea

References

  1. Ulam SM: A Collection of the Mathematical Problems. Interscience Publ., New York; 1960.Google Scholar
  2. Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
  3. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleMathSciNetMATHGoogle Scholar
  4. Gajda Z: On stability of additive mappings. Int J Math Math Sci 1991, 14: 431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
  5. Găvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl 1994, 184: 431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar
  6. Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. J Inequal Pure Appl Math 2003, 4: 7. no. 1, 7 (Article ID 4)MATHGoogle Scholar
  7. Kominek Z: On a local stability of the Jensen functional equation. Demonstratio Math 1989, 22: 499–507.MathSciNetMATHGoogle Scholar
  8. Czerwik S: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor, FL; 2003.Google Scholar
  9. Jung S: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press lnc, Palm Harbor, FL; 2001.Google Scholar
  10. Najati A, Kang J, Cho Y: Local stability of the pexiderized Cauchy and Jensen's equations in fuzzy spaces. J Inequal Appl 2011, 2011: 8. Article No. 78 10.1186/1029-242X-2011-8MathSciNetView ArticleMATHGoogle Scholar
  11. Rassias TM: On the stability of the quadratic functional equation and its applications. Studia Univ Babes-Bolyai 1998, XLIII: 89–124.MathSciNetMATHGoogle Scholar
  12. Rassias TM: The problem of S.M. Ulam for approximately multiplicative mappings. J Math Anal Appl 2000, 246: 352–378. 10.1006/jmaa.2000.6788MathSciNetView ArticleMATHGoogle Scholar
  13. Rassias TM: On the stability of functional equations in Banach spaces. J Math Anal Appl 2000, 251: 264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle Scholar
  14. Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000, 62: 23–130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar
  15. Rassias TM, Šemrl P: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc Am Math Soc 1992, 114: 989–993. 10.1090/S0002-9939-1992-1059634-1View ArticleMATHGoogle Scholar
  16. Ulam TM, Šemrl P: On the Hyers-Ulam stability of linear mappings. J Math Anal Appl 1993, 173: 325–338. 10.1006/jmaa.1993.1070MathSciNetView ArticleMATHGoogle Scholar
  17. Rassias TM, Shibata K: Variational problem of some quadratic functionals in complex analysis. J Math Anal Appl 1998, 228: 234–253. 10.1006/jmaa.1998.6129MathSciNetView ArticleMATHGoogle Scholar
  18. Mathieu M: Elementary Operators & Applications. World Scientific, NJ; 1992.View ArticleGoogle Scholar
  19. Feng W, Zhankui X: Generalized Jordan derivations on semiprime rings. Demonstratio Math 2007, 40: 789–798.MathSciNetMATHGoogle Scholar
  20. Hvala B: Generalized derivations. Commun Algebra 1998, 26: 1147–1166. 10.1080/00927879808826190MathSciNetView ArticleMATHGoogle Scholar
  21. Badora R: On approximate derivations. Math Inequal Appl 2006, 9: 167–173.MathSciNetMATHGoogle Scholar
  22. Eshaghi Gordji M, Moslehian MS: A trick for investigation of approximate derivations. Math Commun 2010, 15: 99–105.MathSciNetMATHGoogle Scholar
  23. Miura T, Hirasawa G, Takahasi SE: A perturbation of ring derivations on Banach algebras. J Math Anal Appl 2006, 319: 522–530. 10.1016/j.jmaa.2005.06.060MathSciNetView ArticleMATHGoogle Scholar
  24. Eshaghi Gordji M, Ghobadipour N: Nearly generalized Jordan derivations. Math Slovaca 2011, 61: 55–62. 10.2478/s12175-010-0059-xMathSciNetView ArticleMATHGoogle Scholar
  25. Chu H, Ku S, Park J: Partial stabilities and partial derivations of n -variable functions. Nonlinear Anal TMA 2010, 72: 1531–1541. 10.1016/j.na.2009.08.038MathSciNetView ArticleMATHGoogle Scholar
  26. Diaz JB, Margolis B: A fixed point theorem of the alternative for the contractions on generaliuzed complete metric space. Bull Am Math Soc 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar
  27. Rus IA: Principles and Applications of Fixed Point Theory. 1979.Google Scholar
  28. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel 1998.Google Scholar

Copyright

© Gordji et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement