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Almost partial generalized Jordan derivations: a fixed point approach
Journal of Inequalities and Applications volume 2012, Article number: 119 (2012)
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 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 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
⋆.
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
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 ) ≤ 2L Ψ 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 limm→∞d(Jm(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 2ma k , 2mb 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 = eiθ 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 ) ≤ 2L Ψ k (2-1a k ), we conclude that
for all .
Now, by (2.2), we have
for all , . Replacing a k by 2ma 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 2ma 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 = 2p-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 ) ≤ 2-1L Ψ 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 = 21-pfor 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 = 22p- 1. □
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Acknowledgements
This work was supported by the Daejin University Research Grants in 2012.
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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.
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Gordji, M.E., Park, C. & Lee, J.R. Almost partial generalized Jordan derivations: a fixed point approach. J Inequal Appl 2012, 119 (2012). https://doi.org/10.1186/1029-242X-2012-119
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DOI: https://doi.org/10.1186/1029-242X-2012-119