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

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.

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for various functional equations have been investigated by numerous mathematicians (see [6-25], M Eshaghi Gordji, unpublished work).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, (ax1)x2 = 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= sup_{a ∈ 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.

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.

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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.

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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.

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 m₀ such that

★ d(T^mx, T^m+1x) < ∞ for all m ≥ m₀;

★ the sequence {T^mx} 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 : A0 → 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.,

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

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.

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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 = 2^λ-1, we get the desired result.

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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 = 2^r-1, we get the desired result.

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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.

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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 = 2^2-λ, we get the desired result.

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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 = 2^2-r, we get the desired result.

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The authors declare that they have no competing interests.

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.

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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