Open Access

Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras

  • Choonkil Park1,
  • Golamreza Zamani Eskandani2,
  • Hamid Vaezi2 and
  • Dong Yun Shin3Email author
Journal of Inequalities and Applications20122012:114

https://doi.org/10.1186/1029-242X-2012-114

Received: 1 December 2011

Accepted: 24 May 2012

Published: 24 May 2012

Abstract

Eskandani and Vaezi proved the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras associated with the following Pexiderized Jensen type functional equation

k f x + y k = f 0 ( x ) + f 1 ( y )

by using direct method. Using fixed point method, we prove the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras. Moreover, we investigate the Pexiderized Jensen type functional inequality in proper Jordan CQ*-algebras.

Mathematics Subject Classification 2010: Primary, 17B40; 39B52; 47N50; 47L60; 46B03; 47H10.

Keywords

Hyers-Ulam stabilityproper Jordan CQ*-algebraJordan derivationfixed point method

1. Introduction and preliminaries

In 1940, Ulam [1] asked the first question on the stability problem. In 1941, Hyers [2] solved the problem of Ulam. This result was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. In 1994, a generalization of Rassias' Theorem was obtained by Găvruta [5]. Since then, several stability problems for various functional equations have been investigated by numerous mathematicians (see [625], M Eshaghi Gordji, unpublished work).

The Jensen equation is 2 f x + y 2 = f ( x ) + f ( y ) , where f is a mapping between linear spaces. It is easy to see that a mapping f : XY between linear spaces with f(0) = 0 satisfies the Jensen equation if and only if it is additive [26]. Stability of the Jensen equation has been studied at first by Kominek [27].

We recall some basic facts concerning quasi *-algebras.

Definition 1.1. Let A be a linear space and let A0 be a *-algebra contained in A as a subspace. We say that A is a quasi *-algebra over A0 if

(i) the right and left multiplications of an element of A and an element of A0 are defined and linear;

(ii) x1(x2a) = (x1x2)a, (ax 1)x 2 = a(x1x2) and x1(ax2) = (x1a)x2 for all x1, x2 A0 and all a A;

(iii) an involution *, which extends the involution of A0, is defined in A with the property (ab)* = b*a* whenever the multiplication is defined.

Quasi *-algebras [28, 29] arise in natural way as completions of locally convex *-algebras whose multiplication is not jointly continuous; in this case one has to deal with topological quasi *-algebras.

A quasi *-algebra (A, A o ) is said to be a locally convex quasi *-algebra if in A a locally convex topology τ is defined such that

(i) the involution is continuous and the multiplications are separately continuous;

(ii) A o is dense in A[τ].

Throughout this article, we suppose that a locally convex quasi *-algebra (A, A0) is complete. For an overview on partial *-algebra and related topics we refer to [30].

In a series of articles [3135] many authors have considered a special class of quasi * algebras, called proper CQ*-algebras, which arise as completions of C*-algebras. They can be introduced in the following way:

Let A be a Banach module over the C*-algebra A0 with involution * and C*-norm || . ||0 such that A0 A. We say that (A, A0) is a proper CQ*-algebra if

(i) A0 is dense in A with respect to its norm || · ||;

(ii) (ab)* = b*a* whenever the multiplication is defined;

(iii) || y ||0= supa A, || a ||≤1|| ay || for all y A0.

Definition 1.2. A proper CQ*-algebra (A, A0), endowed with the Jordan product
z x = z x + x z 2

for all x A and all z A0, is called a proper Jordan CQ*-algebra.

Definition 1.3. Let (A, A0) be proper Jordan CQ*-algebras.

A -linear mapping δ: A0A is called a Jordan derivation if
δ ( x y ) = x δ ( y ) + δ ( x ) y

for all x, y A0.

Park and Rassias [36] investigated homomorphisms and derivations on proper JCQ*- triples.

Throughout this article, assume that k is a fixed positive integer.

Eskandani and Vaezi [37] proved the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras associated with the following Pexiderized Jensen type functional equation
k f x + y k = f 0 ( x ) + f 1 ( y )

by using direct method.

In this article, using fixed point method, we prove the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras.

Moreover, we investigate the Pexiderized Jensen type functional inequality in proper Jordan CQ*-algebras.

2. Derivations on proper Jordan CQ*-algebras

Throughout this section, assume that (A, A0) is a proper Jordan CQ*-algebra with C*-norm || · ||A0 and norm || · || A .

Theorem 2.1. Let φ : A0 × A0 → [0, + ∞) be a function such that
lim n 1 4 n φ ( 2 n x , 2 n y ) = 0
(2.1)
for all x, y A0 . Suppose that f, f0, f1 : A0A are mappings with f(0) = 0 and
| | μ f ( x ) - f 0 ( y ) - f 1 ( z ) | | A k f μ x + y + z k A
(2.2)
| | f ( x y ) + x f 1 ( y ) + f 0 ( x ) y | | A φ ( x , y )
(2.3)
for all μ T 1 : { μ : | μ | = 1 } and all x, y, z A0. Then the mapping f : A0A is a Jordan derivation. Moreover,
f ( x ) = f 0 ( 0 ) - f 0 ( x ) = f 1 ( 0 ) - f 1 ( x )

for all x A0.

Proof. Letting x = yz = 0 in (2.2), we get f0(0) + f1(0) = 0.

Letting µ = 1, y = -x and z = 0 in (2.2), we get
f ( x ) = f 0 ( - x ) + f 1 (0) = f 0 ( - x ) - f 0 (0)
(2.4)
for all x A0. Similarly, we have
f ( x ) = f 1 (- x ) + f 0 (0) = f 1 (- x ) - f 1 (0)
(2.5)
for all x A0. By (2.2), we have
| | f ( x + y ) - f ( x ) - f ( y ) | | A = | | f ( x + y ) - ( f 0 ( - x ) + f 1 ( 0 ) ) - ( f 1 ( - y ) + f 0 ( 0 ) ) | | A = | | f ( x + y ) - f 0 ( - x ) - f 1 ( - y ) | | A = 0
for all x, y A0. So the mapping f : A0A is additive. Letting y = -µx and z = 0 in (2.2), we get
μ f ( x ) = f 0 ( - μ x ) + f 1 ( 0 ) = f ( μ x )
for all x A0. By the same reasoning as in the proof of [[38], Theorem 2.1], the mapping f : A0A is -linear. By (2.1) and (2.3), we have
| | f ( x y ) - x f ( y ) - f ( x ) y | | A = lim n 1 4 n | | f ( 2 n x 2 n y ) - 2 n x ( f 1 ( - 2 n y ) - f 1 ( 0 ) ) - ( f 0 ( - 2 n x ) - f 0 ( 0 ) ) 2 n y | | A = lim n 1 4 n | | f ( 2 n x 2 n y ) - 2 n x f 1 ( - 2 n y ) - f 0 ( - 2 n x ) 2 n y | | A lim n φ ( - 2 n x , - 2 n y ) 4 n = 0
for all x, y A0. So
f ( x y ) = x f ( y ) + f ( x ) y

for all x, y A0. Therefore, the mapping f : A0A is a Jordan derivation.

Since f(-x) = -f(x) for all x A0, it follows from (2.4) that
f ( x ) = - f ( - x ) = - ( f 0 ( x ) - f 0 (0)) = f 0 (0) - f 0 ( x )
for all x A0. It follows from (2.5) that
f ( x ) = - f ( - x ) = - ( f 1 ( x ) - f 1 (0)) = f 1 (0) - f 1 ( x )

for all x A0. This completes the proof.

   □

Corollary 2.2. Let θ, r0, r1 be nonnegative real numbers with r0 + r1 < 2, and let f, f0, f1 : A0A be mappings satisfying f (0) = 0, (2.2) and
| | f ( x y ) + x f 1 ( y ) + f 0 ( x ) y | | A θ | | x | | A 0 r 0 | | y | | A 0 r 1
for all x, y A0. Then the mapping f : A0A is a Jordan derivation. Moreover,
f ( x ) = f 0 ( 0 ) - f 0 ( x ) = f 1 ( 0 ) - f 1 ( x )

for all x A0.

Proof. The proof follows from Theorem 2.1.

   □

Corollary 2.3. Let θ, r0, r1 be nonnegative real numbers with r < 2 and let f, f0, f1 : A0A be mappings satisfying f(0) = 0, (2.2) and
| | f ( x y ) + x f 1 ( y ) + f 0 ( x ) y | | A θ ( | | x | | A 0 r + | | y | | A 0 r )
for all x, y A0. Then the mapping f : A0A is a Jordan derivation. Moreover,
f ( x ) = f 0 ( 0 ) - f 0 ( x ) = f 1 ( 0 ) - f 1 ( x )

for all x A.

3. Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras

We now introduce one of fundamental results of fixed point theory. For the proof, refer to [39, 40]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [8].

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 [39].

Theorem 3.1. 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 ) for all y Λ.

Now we prove the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras by using fixed point method.

Theorem 3.2. Let f, f0, f1 : A0A be mappings with f(0) = 0 for which there exists a function φ : A 0 2 [ 0 , ) with φ(0, 0) = 0 such that
k f μ x + μ y k - μ f 0 ( x ) - μ f 1 ( y ) A φ ( x , y ) ,
(3.1)
k f x y k - x f 1 ( y ) - f 0 ( x ) y A φ ( x , y )
(3.2)
for all μ T 1 and all x, y A0. If there exists an L < 1 such that φ ( x , y ) 2 L φ ( x 2 , y 2 ) for all x, y A0, then there exists a unique Jordan derivation δ : A0A such that
| | f ( x ) - δ ( x ) | | A 1 2 k - 2 k L φ ( k x , k x ) , | | f 0 ( x ) - f 0 ( 0 ) - δ ( x ) | | A 1 2 - 2 L φ ( x , x )
(3.3)

for all x A0. Moreover, f0(x) - f0(0) = f1(x) - f1(0) for all x A0.

Proof. Letting x = y = 0 and µ = 1 in (3.1), we get f0(0) + f1(0) = 0.

Letting y = 0 and µ = 1 in (3.1), we get
k f x k = f 0 ( x ) + f 1 ( 0 ) = f 0 ( x ) - f 0 ( 0 )
(3.4)
for all x A0. Similarly, we get
k f y k = f 1 ( y ) + f 0 ( 0 ) = f 1 ( y ) - f 1 ( 0 )
(3.5)
for all y A0. Using (3.4) and (3.5), we get
f 0 ( x ) - f 0 ( 0 ) = f 1 ( x ) - f 1 ( 0 )

for all x A0.

Let H : A 0 → A be a mapping defined by
H ( x ) : = f 0 ( x ) - f 0 ( 0 ) = f 1 ( x ) - f 1 ( 0 ) = k f x k
for all x A0. Then we have
| | H ( μ x + μ y ) - μ H ( x ) - μ H ( y ) | | A φ ( x , y )
(3.6)

for all μ T 1 and x, y A0.

Consider the set
X : = { g : A 0 A }
and introduce the generalized metric on X:
d ( g , h ) = inf { C + : | | g ( x ) - h ( x ) | | A C φ ( x , x ) , x A 0 } .

It is easy to show that (X, d) is complete (see [[41], Lemma 2.1]).

Now we consider the linear mapping J : XX such that
J g ( x ) : = 1 2 g ( 2 x )

for all x A.

By [[41], Theorem 3.1],
d ( J g , J h ) L d ( g , h )

for all g, h X.

Letting µ = 1 and y = x in (3.6), we get
| | H ( 2 x ) - 2 H ( x ) | | φ ( x , x )
(3.7)
and so
| | H ( x ) - 1 2 H ( 2 x ) | | 1 2 φ ( x , x )

for all x A0. Hence d ( H , J H ) 1 2 .

By Theorem 3.1, there exists a mapping δ : A0A such that
  1. (1)
    δ is a fixed point of J, i.e.,
    δ ( 2 x ) = 2 δ ( x )
    (3.8)
     
for all x A0. The mapping δ is a unique fixed point of J in the set
Y = { g X : d ( f , g ) < } .
This implies that δ is a unique mapping satisfying (3.8) such that there exists C (0, ∞) satisfying
| | H ( x ) - δ ( x ) | | A C φ ( x , x )
for all x A0.
  1. (2)
    d(J n H, δ) → 0 as n → ∞. This implies the equality
    lim n H ( 2 n x ) 2 n = δ ( x )
    (3.9)
     
for all x A0.
  1. (3)
    d ( H , δ ) 1 1 - L d ( H , J H ) , which implies the inequality
    d ( H , δ ) 1 2 - 2 L .
     

This implies that the inequality (3.3) holds.

It follows from (3.6) and (3.9) that
| | δ ( μ x + μ y ) - μ δ ( x ) - μ δ ( y ) | | A = lim n 1 2 n | | H ( 2 n μ x + 2 n μ y ) - μ H ( 2 n x ) - μ H ( 2 n y ) | | A lim n 1 2 n φ ( 2 n x , 2 n y ) = 0
for μ T 1 and all x, y A0. So
δ ( μ x + μ y ) = μ δ ( x ) + μ δ ( y )

for μ T 1 and all x, y A0. By the same reasoning as in the proof of [[38], Theorem 2.1], the mapping δ : A0A is -linear.

It follows from φ ( x , y ) 2 L φ ( x 2 , y 2 ) that
lim n 1 4 n φ ( 2 n x , 2 n y ) 1 2 n φ ( 2 n x , 2 n y ) = 0
(3.10)

for all x, y A0.

It follows from (3.2) and (3.10) that
| | δ ( x y ) - x δ ( y ) - δ ( x ) y | | A lim n 1 4 n k f 4 n x y k - 2 n x ( f 1 ( 2 n y ) + f 0 ( 0 ) ) - ( f 0 ( 2 n x ) + f 1 ( 0 ) ) 2 n y A lim n 1 4 n k f 4 n x y k - 2 n x f 1 ( 2 n y ) - f 0 ( 2 n x ) 2 n y A lim n 1 4 n φ ( 2 n x , 2 n y ) = 0
for all x, y A0. Hence
δ ( x y ) = x δ ( y ) + δ ( x ) y

for all x, y A0. So δ : A0A is a Jordan derivation, as desired.

   □

Corollary 3.3. [[37], Theorem 3.1] Let be a nonnegative real number and r0, r1 positive real numbers with λ:= r0 + r1 < 1 and let f, f0, f1 : A0A be mappings with f (0) = 0 such that
k f μ x + μ y k - μ f 0 ( x ) - μ f 1 ( y ) A θ | | x | | A 0 r 0 | | y | | A 0 r 1 ,
(3.11)
k f x y k - x f 1 ( y ) - f 0 ( x ) y A θ | | x | | A 0 r 0 | | y | | A 0 r 1
(3.12)
for all μ T 1 and all x, y A0. Then there exists a unique Jordan derivation δ : A0A such that
| | f ( x ) - δ ( x ) | | A k λ - 1 θ 2 - 2 λ | | x | | A 0 λ , | | f 0 ( x ) - f 0 ( 0 ) - δ ( x ) | | A θ 2 - 2 λ | | x | | A 0 λ

for all x A0. Moreover, f0(x) - f0(0) = f1(x) - f1(0) for all x A0.

Proof. The proof follows from Theorem 3.2 by taking
φ ( x , y ) : = θ | | x | | A 0 r 0 | | y | | A 0 r 1

for all x, y A. Letting L = 2λ-1, we get the desired result.

   □

Corollary 3.4. [[37], Theorem 3.4] Let θ, r be a nonnegative real numbers with 0 < r < 1, and let f, f0, f1 : A0A be mappings with f (0) = 0 such that
k f μ x + μ y k - μ f 0 ( x ) - μ f 1 ( y ) A θ ( | | x | | A 0 r + | | y | | A 0 r ) ,
(3.13)
k f x o y k - x o f 1 ( y ) - f 0 ( x ) o y A θ ( | | x | | A 0 r + | | y | | A 0 r )
(3.14)
for all μ T 1 and all x, y A0. Then there exists a unique Jordan derivation δ : A0A such that
| | f ( x ) - δ ( x ) | | A 2 k r - 1 θ 2 - 2 r | | x | | A 0 r , | | f 0 ( x ) - f 0 ( 0 ) - δ ( x ) | | A 2 θ 2 - 2 r | | x | | A 0 r

for all x A0. Moreover, f0(x) - f0(0) = f1(x) - f1(0) for all x A0.

Proof. The proof follows from Theorem 3.2 by taking
φ ( x , y ) : = θ ( | | x | | A 0 r + | | y | | A 0 r )

for all x, y A. Letting L = 2r- 1, we get the desired result.

   □

Theorem 3.5. Let f, f0, f1 : A0A be mappings with f(0) = f0(0) = f1(0) = 0 for which there exists a function φ : A 0 2 [ 0 , ) satisfying (3.1) and (3.2). If there exists an L < 1 such that φ ( x , y ) L 4 φ ( 2 x , 2 y ) for all x, y A0, then there exists a unique Jordan derivationδ: A0A such that
| | f ( x ) - δ ( x ) | | A L 4 k - 4 k L φ ( k x , k x ) , | | f 0 ( x ) - δ ( x ) | | A L 4 - 4 L φ ( x , x )
(3.15)

for all x A0. Moreover, f0(x) = f1(x) for all x A0.

Proof. Let (X, d) be the generalized metric space defined in the proof of Theorem 3.2.

Now we consider the linear mapping J : XX such that
J g ( x ) : = 2 g x 2

for all x X.

Let H ( x ) : = f 0 ( x ) = f 1 ( x ) = k f ( x k ) for all x A0. It follows from (3.7) that
H ( x ) - 2 H x 2 φ x 2 , x 2 L 4 φ ( x , y )
for all x A0. Thus d ( H , J H ) L 4 . One can show that there exists a mapping δ : A0A such that
d ( H , δ ) L 4 - 4 L .

Hence we obtain the inequality (3.15).

It follows from φ ( x , y ) L 4 φ ( 2 x , 2 y ) that
lim n 4 n φ x 2 n , y 2 n = 0
for all x, y A0. So
| | δ ( x y ) - x δ ( y ) - δ ( x ) y | | A lim n 4 n k f 4 n x y 4 n k - x 2 n f 1 y 2 n - f 0 x 2 n y 2 n A lim n 4 n φ x 2 n , y 2 n = 0
for all x, y A0. Hence
δ ( x y ) = x δ ( y ) + δ ( x ) y

for all x, y A0. So δ : A0A is a Jordan derivation, as desired.

The rest of the proof is similar to the proof of Theorem 3.2.

   □

Corollary 3.6. [[37], Theorem 3.2] Let θ be a nonnegative real number and r0, r1 positive real numbers with λ:= r0 + r1 > 2 and let f, f0, f1 : A0A be mappings satisfying f(0) = f0(0) = f1(0) = 0, (3.11) and (3.12). Then there exists a unique Jordan derivation δ: A0A such that
| | f ( x ) - δ ( x ) | | A k λ - 1 θ 2 λ - 4 | | x | | A 0 λ | | f i ( x ) - δ ( x ) | | A θ 2 λ - 4 | | x | | A 0 λ

for all x A0. Moreover, f0(x) = f1(x) for all x A0.

Proof. The proof follows from Theorem 3.3 by taking
φ ( x , y ) : = θ | | x | | A 0 r 0 | | y | | A 0 r 1

for all x, y A. Letting L = 22-λ, we get the desired result.

   □

Corollary 3.7. [[37], Theorem 3.3] Let θ, r be nonnegative real numbers with r > 2, and let f, f0, f1 : A0A be mappings satisfying f (0) = f0(0) = f1(0) = 0, (3.13) and (3.14). Then there exists a unique Jordan derivation δ : A0A such that
| | f ( x ) - δ ( x ) | | A 2 k r - 1 θ 2 r - 4 | | x | | A 0 r , | | f 0 ( x ) - δ ( x ) | | A 2 θ 2 r - 4 | | x | | A 0 r

for all x A0. Moreover, f0(x) = f1(x) for all x A0.

Proof. The proof follows from Theorem 3.3 by taking
φ ( x , y ) : = θ ( | | x | | A 0 r + | | y | | A 0 r )

for all x, y A. Letting L = 22-r, we get the desired result.

   □

Declarations

Acknowledgements

Choonkil Park was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). Dong Yun Shin was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

Authors’ Affiliations

(1)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University
(2)
Faculty of Mathematical Sciences, University of Tabriz
(3)
Department of Mathematics, University of Seoul

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