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Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras
Journal of Inequalities and Applications volume 2012, Article number: 114 (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
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.
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for various functional equations have been investigated by numerous mathematicians (see [6–25], M Eshaghi Gordji, unpublished work).
The Jensen equation is , where f is a mapping between linear spaces. It is easy to see that a mapping f : X → Y 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 [31–35] 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
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 ℂ-linear mapping δ: A0 → A is called a Jordan derivation if
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
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
for all x, y ∈ A0 . Suppose that f, f0, f1 : A0 → A are mappings with f(0) = 0 and
for all and all x, y, z ∈ A0. Then the mapping f : A0 → A is a Jordan derivation. Moreover,
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
for all x ∈ A0. Similarly, we have
for all x ∈ A0. By (2.2), we have
for all x, y ∈ A0. So the mapping f : A0 → A is additive. Letting y = -µx and z = 0 in (2.2), we get
for all x ∈ A0. By the same reasoning as in the proof of [[38], Theorem 2.1], the mapping f : A0 → A is ℂ-linear. By (2.1) and (2.3), we have
for all x, y ∈ A0. So
for all x, y ∈ A0. Therefore, the mapping f : A0 → A is a Jordan derivation.
Since f(-x) = -f(x) for all x ∈ A0, it follows from (2.4) that
for all x ∈ A0. It follows from (2.5) that
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 : A0 → A be mappings satisfying f (0) = 0, (2.2) and
for all x, y ∈ A0. Then the mapping f : A0 → A is a Jordan derivation. Moreover,
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 : A0 → A be mappings satisfying f(0) = 0, (2.2) and
for all x, y ∈ A0. Then the mapping f : A0 → A is a Jordan derivation. Moreover,
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 : X → X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that
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
or other exists a natural number m 0 such 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
★ 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 : A0 → A be mappings with f(0) = 0 for which there exists a function with φ(0, 0) = 0 such that
for all and all x, y ∈ A0. If there exists an L < 1 such that for all x, y ∈ A0, then there exists a unique Jordan derivation δ : A0 → A such that
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
for all x ∈ A0. Similarly, we get
for all y ∈ A0. Using (3.4) and (3.5), we get
for all x ∈ A0.
Let H : A 0 → A be a mapping defined by
for all x ∈ A0. Then we have
for all and x, y ∈ A0.
Consider the set
and introduce the generalized metric on X:
It is easy to show that (X, d) is complete (see [[41], Lemma 2.1]).
Now we consider the linear mapping J : X → X such that
for all x ∈ A.
By [[41], Theorem 3.1],
for all g, h ∈ X.
Letting µ = 1 and y = x in (3.6), we get
and so
for all x ∈ A0. Hence .
By Theorem 3.1, there exists a mapping δ : A0 → A such that
-
(1)
δ is a fixed point of J, i.e.,
(3.8)
for all x ∈ A0. The mapping δ is a unique fixed point of J in the set
This implies that δ is a unique mapping satisfying (3.8) such that there exists C ∈ (0, ∞) satisfying
for all x ∈ A0.
-
(2)
d(JnH, δ) → 0 as n → ∞. This implies the equality
(3.9)
for all x ∈ A0.
-
(3)
, which implies the inequality
This implies that the inequality (3.3) holds.
It follows from (3.6) and (3.9) that
for and all x, y ∈ A0. So
for and all x, y ∈ A0. By the same reasoning as in the proof of [[38], Theorem 2.1], the mapping δ : A0 → A is ℂ-linear.
It follows from that
for all x, y ∈ A0.
It follows from (3.2) and (3.10) that
for all x, y ∈ A0. Hence
for all x, y ∈ A0. So δ : A0 → A 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 : A0 → A be mappings with f (0) = 0 such that
for all and all x, y ∈ A0. Then there exists a unique Jordan derivation δ : A0 → A such that
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
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 : A0 → A be mappings with f (0) = 0 such that
for all and all x, y ∈ A0. Then there exists a unique Jordan derivation δ : A0 → A such that
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
for all x, y ∈ A. Letting L = 2r- 1, we get the desired result.
□
Theorem 3.5. Let f, f0, f1 : A0 → A be mappings with f(0) = f0(0) = f1(0) = 0 for which there exists a function satisfying (3.1) and (3.2). If there exists an L < 1 such that for all x, y ∈ A0, then there exists a unique Jordan derivationδ: A0 → A such that
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 : X → X such that
for all x ∈ X.
Let for all x ∈ A0. It follows from (3.7) that
for all x ∈ A0. Thus . One can show that there exists a mapping δ : A0 → A such that
Hence we obtain the inequality (3.15).
It follows from that
for all x, y ∈ A0. So
for all x, y ∈ A0. Hence
for all x, y ∈ A0. So δ : A0 → A 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 : A0 → A be mappings satisfying f(0) = f0(0) = f1(0) = 0, (3.11) and (3.12). Then there exists a unique Jordan derivation δ: A0 → A such that
for all x ∈ A0. Moreover, f0(x) = f1(x) for all x ∈ A0.
Proof. The proof follows from Theorem 3.3 by taking
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 : A0 → A be mappings satisfying f (0) = f0(0) = f1(0) = 0, (3.13) and (3.14). Then there exists a unique Jordan derivation δ : A0 → A such that
for all x ∈ A0. Moreover, f0(x) = f1(x) for all x ∈ A0.
Proof. The proof follows from Theorem 3.3 by taking
for all x, y ∈ A. Letting L = 22-r, we get the desired result.
□
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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).
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Park, C., Eskandani, G.Z., Vaezi, H. et al. Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras. J Inequal Appl 2012, 114 (2012). https://doi.org/10.1186/1029-242X-2012-114
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DOI: https://doi.org/10.1186/1029-242X-2012-114