Almost partial generalized Jordan derivations: a fixed point approach

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.

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 $\mathcal{E}$ must be somehow close to an exact solution of $\mathcal{E}$?". 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

$$∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥\le \epsilon \left({∥x∥}^p+{∥y∥}^p\right)$$

(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: E → E' such that

$$∥f\left(x\right)-T\left(x\right)∥\le \frac{2\epsilon }{2-2^p}{∥x∥}^p$$

(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 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, $2f\left(\frac{x+y}{2}\right)=f\left(x\right)+f\left(y\right)$, 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 ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_n$ be normed algebras over the complex field ℂ and let $\mathcal{B}$ be a Banach algebra over ℂ. A mapping $d_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{B}$ is called a k-th partial derivation if

$$d_k\left(x_1,\dots ,\gamma a_k+\mu b_k,\dots ,x_n\right)=\gamma d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)+\mu d_k\left(x_1,\dots ,b_k,\dots ,x_n\right)$$

and there exists a mapping $f_k:{\mathcal{A}}_k\to \mathcal{B}$ such that

$$d_k\left(x_1,\dots ,a_k{b}_k,\dots ,x_n\right)=f_k\left(a_k\right)d_k\left(x_1,\dots ,b_k,\dots ,x_n\right)+d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)f_k\left(b_k\right)$$

for all $a_k,b_k\in {\mathcal{A}}_k$ and $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$ and all γ, μ ∈ ℂ.

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

Definition 1.2. Let ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_n$ be normed algebras over the complex field ℂ and let $\mathcal{X}$ be a Banach module over ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_{n-1}$ and ${\mathcal{A}}_n$. Then

• (i) A mapping $d_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ is called a k-th partial Jordan derivation of Jensen type if

  $$2d_k\left(x_1,\dots ,\frac{\gamma a_k+\gamma b_k}{2},\dots ,x_n\right)=\gamma d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)+\gamma d_k\left(x_1,\dots ,b_k,\dots ,x_n\right)$$

and

$$d_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)=a_k{d}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)+d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k$$

for all $a_k,b_k\in {\mathcal{A}}_k$ and $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$ and all γ ∈ ℂ.

• (ii) A mapping ${\delta }_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ is called a k-th partial generalized Jordan derivation of Jensen type if

  $$2{\delta }_k\left(x_1,\dots ,\frac{\gamma a_k+\gamma b_k}{2},\dots ,x_n\right)=\gamma {\delta }_k\left(x_1,\dots ,a_k,\dots ,x_n\right)+\gamma {\delta }_k\left(x_1,\dots ,b_k,\dots ,x_n\right)$$

and there exists a k-th partial Jordan derivation $d_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ such that

$${\delta }_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)={\delta }_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k+a_k{d}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)$$

for all $a_k,b_k\in {\mathcal{A}}_k$ and $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$ 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

$$d\left(Tx,Ty\right)\le Ld\left(x,y\right)$$

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

$$d\left(T^{m}x,T^{m+1}x\right)=\infty forallm\ge 0,$$

or other exists a natural number m ₀ 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

$$\Lambda =\left\{y\in \Omega :d\left(T^{m_0}x,y\right)<\infty \right\};$$

⋆$d\left(y,y^{*}\right)\le \frac{1}{1-L}d\left(y,Ty\right)forally\in \Lambda$.

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 ∈ ℕ, we define

$${{\mathcal{T}}^1}_{\frac{1}{n_O}}:=\left\{e^{i\theta }0\le \theta \le \frac{2\pi }{n_o}\right\}$$

and we denote ${\mathbb{T}}_{\frac{1}{1}}^1$ by ${\mathbb{T}}^1$. Also, we suppose that ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_n$ are normed algebras over the complex field ℂ and $\mathcal{X}$ is a Banach module over ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_{n-1}$ and ${\mathcal{A}}_n$. We denote that 0 _k , $0_{\mathcal{X}}$ are zero elements of ${\mathcal{A}}_k,\mathcal{X}$, respectively.

Theorem 2.1. Let $T_k,F_k:{\mathcal{A}}_1\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$be mappings with $T_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=F_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=0_{\mathcal{X}}$. Assume that there exist functions${\Psi }_k:{\mathcal{A}}_k\to \left[0,\infty \right)$, ${\phi }_k:{\mathcal{A}}_k^2\to \left[0,\infty \right)$satisfying

$$\begin{array}{cc}∥2S_k\left(x_1,\dots ,\frac{\lambda a_k+\lambda b_k}{2},\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,b_k,\dots ,x_n\right)∥\hfill & \le {\phi }_k\left(a_k,b_k\right),\hfill \end{array}$$

(2.1)

$$\begin{array}{cc}\hfill \text{max}\{& ∥F_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-F_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k∥,\hfill \\ ∥T_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-T_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)∥\}\hfill \\ \le {\Psi }_k\left(a_k\right)\hfill \end{array}$$

(2.2)

for S _k ∈ {F _k , T _k } and for all $\lambda \in {{\mathbb{T}}^1}_{\frac{1}{n_0}}$ and all $a_k,b_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. 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 ) ≤ 2L Ψ _k (2^-1a _k ) for all $a_k,b_k\in {\mathcal{A}}_k$, then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable $d_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d _k ) $D_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ such that

$$\begin{array}{cc}\hfill \text{max}\{∥F_k\left(x_1,x_2,\dots ,x_n\right)-d_k\left(x_1,x_2,\dots ,x_n\right)∥,& ∥T_k\left(x_1,x_2,\dots ,x_n\right)-D_k\left(x_1,x_2,\dots ,x_n\right)∥\}\hfill \\ \le \frac{L}{1-L}{\phi }_k\left(x_k,0\right)\hfill \end{array}$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$.

Proof. It follows from (2.1) that

$$\begin{array}{c}∥2S_k\left(x_1,\dots ,\frac{\lambda a_k+\lambda b_k}{2},\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,b_k,\dots ,x_n\right)∥\hfill \\ \le {\phi }_k\left(a_k,b_k\right),\hfill \end{array}$$

(2.3)

for S _k ∈{F _k , T _k } and for all $\lambda \in {{\mathbb{T}}^1}_{\frac{1}{n_0}}:=\left\{\lambda \in ℂ:\left|\lambda \right|=1\right\}$ and all $a_k,b_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$.

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

$$∥F_k\left(x_1,\dots ,x_k,\dots ,x_n\right)-2^{-1}{F}_k\left(x_1,\dots ,2x_k,\dots ,x_n\right)∥\le 2^{-1}{\phi }_k\left(2x_k,0\right)\le L\phi \left(x_k,0\right)$$

(2.4)

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$. Put $\Omega :=\left\{G_k|G_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}\right\}$ and define d: Ω × Ω → [0, ∞] by

$$\begin{array}{cc}\hfill d\left(H_k,G_k\right):& =\text{inf}\{\alpha \in {ℝ}^{+};∥G_k\left(x_1,\dots ,x_k,\dots ,x_n\right)-H_k\left(x_1,\dots ,x_k,\dots ,x_n\right)∥\hfill \\ \le \alpha {\phi }_k\left(x_k,0\right)\forall x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)\}.\hfill \end{array}$$

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

$$J\left(H_k\right)\left(x_1,\dots ,x_k,\dots ,x_n\right)=2^{-1}{H}_k\left(x_1,\dots ,2x_k,\dots ,x_n\right)$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$. 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

$$∥G_k\left(x_1,\dots ,x_k,\dots ,x_n\right)-H_k\left(x_1,\dots ,x_k,\dots ,x_n\right)∥\le \alpha {\phi }_k\left(x_k,0\right)$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$. Hence we have

$$\begin{array}{c}∥\left(J{G}_k\right)\left(x_1,\dots ,x_k,\dots ,x_n\right)-\left(J{H}_k\right)\left(x_1,\dots ,x_k,\dots ,x_n\right)∥\hfill \\ =2^{-1}∥G_k\left(x_1,\dots ,2x_k,\dots ,x_n\right)-H_k\left(x_1,\dots ,2x_k,\dots ,x_n\right)∥\hfill \\ \le 2^{-1}\alpha {\phi }_k\left(2x_k,0\right)\le \alpha L{\phi }_k\left(x_k,0\right)\hfill \end{array}$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$. So

$$d\left(J\left(G_k\right),J\left(H_k\right)\right)\le Ld\left(G_k,H_k\right)$$

for all G _k , H _k ∈ Ω. It follows from (2.4) that

$$d\left(F_k,J\left(F_k\right)\right)\le L.$$

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

$$d_k\left(x_1,\dots ,2x_k,\dots ,x_n\right)=2d_k\left(x_1,\dots ,x_k,\dots ,x_n\right)$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$, and there exists α ∈ (0, ∞) such that

$$∥d_k\left(x_1,\dots ,x_k,\dots ,x_n\right)-F_k\left(x_1,\dots ,x_k,\dots ,x_n\right)∥\le \alpha {\phi }_k\left(x_k,0\right)$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$.

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

$$\underset{m\to \infty }{\text{lim}}{2}^{-m}{F}_k\left(x_1,\dots ,2^m{x}_k,\dots ,x_n\right)=d_k\left(x_1,\dots ,x_k,\dots ,x_n\right)$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$. It follows from that $d\left(F_k,d_k\right)\le \frac{1}{1-L}d\left(F_k,J\left(F_k\right)\right)$ that

$$d\left(F_k,d_k\right)\le \frac{L}{1-L}.$$

This means that

$$∥F_k\left(x_1,x_2,\dots ,x_n\right)-d_k\left(x_1,x_2,\dots ,x_n\right)∥\le \frac{L}{1-L}{\phi }_k\left(x_k,0\right)$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$. By the inequality φ _k (a _k , b _k ) ≤ 2Lφ _k (2^-1a _k , 2^-1b _k ), we conclude that

$$\underset{m\to \infty }{\text{lim}}{2}^{-m}\phi \left(2^m{a}_k,2^m{b}_k\right)=0$$

for all $a_k,b_k\in {\mathcal{A}}_k$. In the inequality (2.3), replacing a _k , b _k by 2^ma _k , 2^mb _k , respectively, we obtain that

$$\begin{array}{c}{2}^{-m}∥2F_k\left(x_1,\dots ,\frac{\lambda 2^m{a}_k+\lambda 2^m{b}_k}{2},\dots ,x_n\right)-F_k\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)\\ -F_k\left(x_1,\dots ,2^m{b}_k,\dots ,x_n\right)∥\le 2^{-m}{\phi }_k\left(2^m{a}_k,2^m{b}_k\right).\end{array}$$

Passing the limit m → ∞, we obtain

$$2d_k\left(x_1,\dots ,\frac{\lambda a_k+\lambda b_k}{2},\dots ,x_n\right)=\lambda d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)+\lambda d_k\left(x_1,\dots ,b_k,\dots ,x_n\right)$$

for all $a_k,b_k\in {\mathcal{A}}_k$ and all $\lambda \in {{\mathbb{T}}^1}_{\frac{1}{n_0}}$. 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 ${\lambda }_1=e^{\frac{i\theta }{n_o}}$. Then λ₁ belongs to $T_{\frac{1}{n_o}}^1$ and

$$2d_k\left(x_1,\dots ,\frac{{\lambda }_1{a}_k+{\lambda }_1{b}_k}{2},\dots ,x_n\right)={\lambda }_1{d}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)+{\lambda }_1{d}_k\left(x_1,\dots ,b_k,\dots ,x_n\right)$$

for all $a_k,b_k\in {\mathcal{A}}_k$. 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

$$d_k\left(x_1,\dots ,\lambda a_k,\dots ,x_n\right)=\lambda d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)$$

for all $a_k\in {\mathcal{A}}_k$. 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 $t_1=t+\sqrt{n^2-t^2}i\in n{T}^1$ and $t_2=t-\sqrt{n^2-t^2}i\in n{T}^1$. We have $t=\frac{t_1+t_2}{2}$. Then

$$d_k\left(x_1,\dots ,t{a}_k,\dots ,x_n\right)=d_k\left(x_1,\dots ,\frac{t_1+t_2}{2}{a}_k,\dots ,x_n\right)=\frac{t_1+t_2}{2}{d}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)$$

for all $a_k\in {\mathcal{A}}_k$. Let λ ∈ ℂ. Then $\lambda =\left|\lambda \right|e^{i{\lambda }_1}$ and so

$$d_k\left(x_1,\dots ,\lambda a_k,\dots ,x_n\right)=\left|\lambda \right|e^{i{\lambda }_1}{d}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)=\lambda d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)$$

for all $a_k\in {\mathcal{A}}_k$. 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

$$D_k\left(x_1,\dots ,x_k,\dots ,x_n\right):=\underset{m\to \infty }{\text{lim}}{2}^{-m}{T}_k\left(x_1,\dots ,2^m{x}_k,\dots ,x_n\right)$$

exists for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$ and that D _k is ℂ-linear with respect to k-th variable.

By te inequality Ψ _k (a _k ) ≤ 2L Ψ _k (2^-1a _k ), we conclude that

$$\underset{m\to \infty }{\text{lim}}{2}^{-m}{\Psi }_k\left(2^m{a}_k\right)=0$$

for all $a_k\in {\mathcal{A}}_k$.

Now, by (2.2), we have

$$\begin{array}{c}∥F_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-F_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k∥\hfill \\ \le {\Psi }_k\left(a_k\right)\hfill \end{array}$$

for all $a_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. Replacing a _k by 2^ma _k in the above inequality, we obtain that

$$\begin{array}{c}∥F_k\left(x_1,\dots ,2^{2m}{a}_k^2,\dots ,x_n\right)-2^m{a}_k{F}_K\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)\hfill \\ -F_k\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)2^m{a}_k∥\le {\Psi }_k\left(2^m{a}_k\right).\hfill \end{array}$$

Then we have

$$\begin{array}{c}∥2^{-2m}{F}_k\left(x_1,\dots ,2^{2m}{a}_k^2,\dots ,x_n\right)-2^{-m}{a}_k{F}_K\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)\hfill \\ -F_k\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)2^{-m}{a}_k∥\le 2^{-2m}{\Psi }_k\left(2^m{a}_k\right)\hfill \end{array}$$

for all $a_k\in {\mathcal{A}}_k$. Passing m → ∞, we obtain

$$d_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)=a_k{d}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)+d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k$$

for all $a_k\in {\mathcal{A}}_k$ and all $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. 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

$$\begin{array}{c}∥T_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-a_k{T}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-F_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k∥\hfill \\ \le {\Psi }_k\left(a_k\right)\hfill \end{array}$$

for all $a_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. Replacing a _k by 2^ma _k in the last inequality, we get

$$\begin{array}{c}∥T_k\left(x_1,\dots ,2^{2m}{a}_k^2,\dots ,x_n\right)-2^m{a}_k{T}_K\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)\hfill \\ -F_k\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)2^m{a}_k∥\le {\Psi }_k\left(2^m{a}_k\right)\hfill \end{array}$$

for all $a_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. Then we have

$$\begin{array}{c}∥2^{-2m}{T}_k\left(x_1,\dots ,2^{2m}{a}_k^2,\dots ,x_n\right)-2^{-m}{a}_k{T}_K\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)\hfill \\ -F_k\left(x_1,\dots ,2^m{a}_k,\dots ,x_n\right)2^{-m}{a}_k∥\le 2^{-2m}{\Psi }_k\left(2^m{a}_k\right)\hfill \end{array}$$

for all $a_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. Passing m → ∞, we obtain that

$$D_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)=a_k{D}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)+d_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k$$

for all $a_k\in {\mathcal{A}}_k$ and all $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. 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 $T_k,F_k:{\mathcal{A}}_1\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$be mappings such that $T_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=F_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=0_{\mathcal{X}}$ and that

$$\begin{array}{c}∥2S_k\left(x_1,\dots ,\frac{\lambda a_k+\lambda b_k}{2},\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,b_k,\dots ,x_n\right)∥\\ \le \theta \left({∥a_k∥}^p+{∥b_k∥}^p\right),\end{array}$$

$$\begin{array}{c}\text{max}\{∥F_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-F_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k∥,\hfill \\ ∥T_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-T_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)∥\}\hfill \\ \le \theta \left({∥a_k∥}^p\right)\hfill \end{array}$$

for S _k ∈ {F _k , T _k } and for all $\lambda \in {{\mathbb{T}}^1}_{\frac{1}{n_0}}$and all $a_k,b_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. Then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable $d_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d _k ) $D_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ such that

$$\begin{array}{c}\text{max}\left\{∥F_k\left(x_1,x_2,\dots ,x_n\right)-d_k\left(x_1,x_2,\dots ,x_n\right)∥,∥T_k\left(x_1,x_2,\dots ,x_n\right)-D_k\left(x_1,x_2,\dots ,x_n\right)∥\right\}\\ \le \frac{2^p}{2-2^p}\theta {∥x_k∥}^p\end{array}$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$.

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 $T_k,F_k:{\mathcal{A}}_1\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$be mappings with $T_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=F_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=0_{\mathcal{X}}$. Assume that there exist functions ${\Psi }_k:{\mathcal{A}}_k\to \left[0,\infty \right)$, ${\phi }_k:{\mathcal{A}}_k^2\to \left[0,\infty \right)$satisfying

$$\begin{array}{c}∥2S_k\left(x_1,\dots ,\frac{\lambda a_k+\lambda b_k}{2},\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,b_k,\dots ,x_n\right)∥,\hfill \\ \le {\phi }_k\left(a_k,b_k\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \text{max}\{& ∥F_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-F_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k∥,\hfill \\ ∥T_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-T_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)∥\}\hfill \\ \le {\Psi }_k\left(a_k\right)\hfill \end{array}$$

for S _k ∈ {F _k , T _k } and for all $\lambda \in {{\mathbb{T}}^1}_{\frac{1}{n_0}}$and all $a_k,b_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. 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^-1L Ψ _k (2a _k ) for all $a_k,b_k\in {\mathcal{A}}_k$, then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable $d_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d _k ) $D_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ such that

$$\begin{array}{cc}\hfill \text{max}\{∥F_k\left(x_1,x_2,\dots ,x_n\right)-d_k\left(x_1,x_2,\dots ,x_n\right)∥,& ∥T_k\left(x_1,x_2,\dots ,x_n\right)-D_k\left(x_1,x_2,\dots ,x_n\right)∥\}\hfill \\ \le \frac{L}{2-2L}{\phi }_k\left(2x_k,0\right)\hfill \end{array}$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$.

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 $T_k,F_k:{\mathcal{A}}_1\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$be mappings such that $T_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=F_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=0_{\mathcal{X}}$ and

$$\begin{array}{c}∥2S_k\left(x_1,\dots ,\frac{\lambda a_k+\lambda b_k}{2},\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,b_k,\dots ,x_n\right)∥\\ \le \theta \left({∥a_k∥}^p+{∥b_k∥}^p\right),\end{array}$$

$$\begin{array}{cc}\hfill \text{max}& \{∥F_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-F_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k∥,\hfill \\ ∥T_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-T_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)∥\}\hfill \\ \le \theta \left({∥a_k∥}^p\right)\hfill \end{array}$$

for S _k ∈ {F _k , T _k } and for all $\lambda \in {{\mathbb{T}}^1}_{\frac{1}{n_0}}$and all $a_k,b_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. Then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable $d_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to d _k ) $D_k:{\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ such that

$$\begin{array}{cc}\hfill \text{max}\{∥F_k\left(x_1,x_2,\dots ,x_n\right)-d_k\left(x_1,x_2,\dots ,x_n\right)∥,& ∥T_k\left(x_1,x_2,\dots ,x_n\right)-D_k\left(x_1,x_2,\dots ,x_n\right)∥\}\hfill \\ \le \frac{\theta }{1-2^{p-1}}{∥x_k∥}^p\hfill \end{array}$$

for all $x_i\in {\mathcal{A}}_i\left(i=1,2,\dots ,n\right)$.

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-pfor each $a_k,b_k\in {\mathcal{A}}_k$. □

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

Corollary 2.5. Let $p\in \left(0,\frac{1}{2}\right)$and θ ∈ [0, ∞) be real numbers. Let $T_k,F_k:{\mathcal{A}}_1\times \cdots \times {\mathcal{A}}_n\to \mathcal{X}$ be mappings such that $T_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=F_k\left(x_1,\dots ,0_k,\dots ,x_n\right)=0_{\mathcal{X}}$ and

$$\begin{array}{c}∥2S_k\left(x_1,\dots ,\frac{\lambda a_k+\lambda b_k}{2},\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-\lambda S_k\left(x_1,\dots ,b_k,\dots ,x_n\right)∥\hfill \\ \le \theta \left({∥a_k∥}^p{∥b_k∥}^p\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \text{max}& \{∥F_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)-F_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k∥,\hfill \\ ∥T_k\left(x_1,\dots ,a_k^2,\dots ,x_n\right)-T_k\left(x_1,\dots ,a_k,\dots ,x_n\right)a_k-a_k{F}_k\left(x_1,\dots ,a_k,\dots ,x_n\right)∥\}\hfill \\ \le \theta \left({∥a_k∥}^p\right)\hfill \end{array}$$

for S _k ∈ {F _k , T _k }and for all $\lambda \in {{\mathbb{T}}^1}_{\frac{1}{n_0}}$and all $a_k,b_k\in {\mathcal{A}}_k$, $x_i\in {\mathcal{A}}_i\left(i\ne k\right)$. 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}. □