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: 29 May 2012

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
(2)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University
(3)
Department of Mathematics, Daejin University

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