Almost partial generalized Jordan derivations: a fixed point approach

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.

The following question posed by Ulam [1] in 1940: "When is it true that a mapping which approximately satisfies a functional equation must be somehow close to an exact solution of ?". Hyers [2] proved the problem for the Cauchy functional equation. In 1978, Rassias [3] proved the following theorem.

Theorem 1.1. Let f: E → E' be a mapping from a normed vector space E into a Banach space E' subject to the inequality

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: E → E' such that

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 ℝ into E' is continuous in real t for each x∈ E, then T is ℝ-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, , 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 [8-17] and references therein for more detailed information.

Let A, B be two Banach algebras. A ℂ-linear mapping d: A → B is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) δ: A → X 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 be normed algebras over the complex field ℂ and let be a Banach algebra over ℂ. A mapping is called a k-th partial derivation if

and there exists a mapping such that

for all and and all γ, μ ∈ ℂ.

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

Definition 1.2. Let be normed algebras over the complex field ℂ and let be a Banach module over and . Then

(i) A mapping is called a k-th partial Jordan derivation of Jensen type if

and

for all and and all γ ∈ ℂ.

(ii) A mapping is called a k-th partial generalized Jordan derivation of Jensen type if

and there exists a k-th partial Jordan derivation such that

for all and and all γ ∈ ℂ.

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: 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 [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

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

⋆.

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.

For n₀ ∈ ℕ, we define

and we denote by . Also, we suppose that are normed algebras over the complex field ℂ and is a Banach module over and . We denote that 0_k, are zero elements of , respectively.

Theorem 2.1. Let be mappings with . Assume that there exist functions , satisfying

for S_k ∈ {F_k, T_k} and for all and all , . If there exists a constant 0 < L < 1 such that φ_k(a_k, b_k) ≤ 2Lφ_k(2^-1a_k, 2^-1b_k), Ψ_k(a_k) ≤ 2LΨ_k(2^-1a_k) for all , then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d_k) such that

for all .

Proof. It follows from (2.1) that

for S_k ∈{F_k, T_k} and for all and all , .

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

for all . Put and define d: Ω × Ω → [0, ∞] by

It is easy to show that (Ω, d) is a complete generalized metric space. We define the mapping J: Ω → Ω by

for all . 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

for all . Hence we have

for all . So

for all G_k, H_k∈ Ω. It follows from (2.4) that

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

for all , and there exists α ∈ (0, ∞) such that

for all .

On the other hand, we have lim_m→∞ d(J^m(F_k), d_k) = 0. It follows that

for all . It follows from that that

This means that

for all . By the inequality φ_k(a_k, b_k) ≤ 2Lφ_k(2^-1a_k, 2^-1b_k), we conclude that

for all . In the inequality (2.3), replacing a_k, b_k by 2^ma_k, 2^mb_k, respectively, we obtain that

Passing the limit m → ∞, we obtain

for all and all . Now, we show that d_k is ℂ-linear with respect to k-th variable. First suppose that λ belongs to T¹. Then λ = e^iθ for some 0 ≤ θ ≤ 2π. We set . Then λ₁ belongs to and

for all . 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

for all . If t ∈ (0, ∞), then by Archimedean property of ℂ, there exists an n ∈ ℕ such that the point (t, 0) lies in the interior of circle with center at origin and radius n. Let and . We have . Then

for all . Let λ ∈ ℂ. Then and so

for all . It follows that d_k is ℂ-linear with respect to k-th variable.

By the same reasoning as above, we can show that the limit

exists for all and that D_k is ℂ-linear with respect to k-th variable.

By te inequality Ψ_k(a_k) ≤ 2LΨ_k(2^-1a_k), we conclude that

for all .

Now, by (2.2), we have

for all , . Replacing a_k by 2^ma_k in the above inequality, we obtain that

Then we have

for all . Passing m → ∞, we obtain

for all and all . 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

for all , . Replacing a_k by 2^ma_k in the last inequality, we get

for all , . Then we have

for all , . Passing m → ∞, we obtain that

for all and all . 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 be mappings such that and that

for S_k ∈ {F_k, T_k} and for all and all , . Then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d_k) such that

for all .

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 = 2^p-1. □

Theorem 2.3. Let be mappings with . Assume that there exist functions , satisfying

for S_k ∈ {F_k, T_k} and for all and all , . If there exists a constant 0 < L < 1 such that φ_k(a_k, b_k) ≤ 2^-1Lφ_k(2a_k, 2b_k), Ψ_k(a_k) ≤ 2^-1LΨ_k(2a_k) for all , then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d_k) such that

for all .

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 be mappings such that and

for S_k ∈ {F_k, T_k} and for all and all , . Then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d_k) such that

for all .

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 = 2^1-p for each . □

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

Corollary 2.5. Let and θ ∈ [0, ∞) be real numbers. Let be mappings such that and

for S_k ∈ {F_k, T_k}and for all and all , . 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 = 2^2p-1. □

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.

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

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