Approximate Derivations with the Radical Ranges of Noncommutative Banach Algebras

and Applied Analysis 3 By (2), we find that 󵄩󵄩󵄩f (xy) − xf (y) − f (x) y 󵄩󵄩󵄩 = lim n→∞ ( u s ) 2n 󵄩󵄩󵄩󵄩󵄩󵄩󵄩 f(( s u ) 2n xy) − ( s u ) n xf(( s u ) n y)


Introduction and Preliminaries
Let A be an algebra over the real or complex field F. An additive mapping  : A → A is called a ring derivation if the functional equation () = () + () is valid for all ,  ∈ A. In addition, if the identity () = () holds for all  ∈ F and all  ∈ A, then  is said to be a linear derivation.
Singer and Wermer [1] obtained a fundamental result which started investigation into the ranges of linear derivations on Banach algebras.The result states that every continuous linear derivation on a commutative Banach algebra maps into the radical.They also made a very insightful conjecture that the assumption of continuity is unnecessary.This conjecture was proved by Thomas [2].So, in this paper, we will take into account the problems in [1,2] for the derivations on noncommutative Banach algebras.
On the other hand, the stability problem for ring derivations on Banach algebras was considered by Miura et al. in [3]: under suitable conditions every approximate ring derivation on Banach algebra is an exact ring derivation.Šemrl [4] obtained the first stability result concerning derivations between operator algebras.As just mentioned, the study of stability problem has originally been formulated by Ulam [5]: under what condition does there exist a homomorphism near an approximate homomorphism?Hyers [6] had answered affirmatively the question of Ulam under the assumption that the groups are Banach spaces.A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [7] and for approximately linear mappings was presented by Rassias [8] by considering an unbounded Cauchy difference.Since then, many interesting results of the stability problems to a number of functional equations and inequalities (or involving derivations) have been investigated (see, e.g., [9][10][11][12][13][14][15][16]).The reader is referred to the book [17] for more information on stability problem with a large variety of applications.
In this paper, we will establish the stability of functional inequalities with ring derivations and will deal with the problem for the radical range of these functional inequalities on Banach algebra by considering the base of noncommutative versions for the result of Singer and Wermer: Mathieu and Murphy [18] verify that every continuous centralizing linear derivation on a Banach algebra maps into the radical and Brešar [19] proved that every centralizing linear derivation on a semiprime Banach algebra maps into the intersection of the center and the radical.Moreover, Chaudhry and Thaheem [20] showed that two ring derivations on semiprime ring with suitable property map into the center.

Functional Inequalities for a Derivation and Its Applications
Theorem 1.Let A be a normed algebra.Assume that mappings Φ : A 3 → [0,∞) and  : A 2 → [0,∞) satisfy the assumptions for all , ,  ∈ A. Then  is a ring derivation.
Case II.Assume that lim for all ,  ∈ A. We get by ( 7) for all positive integers  and all  ∈ A. Therefore, one can obtain that for all  ∈ A. The remainder of the proof is similar to the proof of Case I.

Corollary 2.
Let A be a Banach algebra and let , , and  be fixed positive real numbers with  >  and ++ > 1. Suppose that mapping Φ : for some  > 0 and  > 0 and all ,  ∈ A, where [, ] =  − , then  maps A into its radical rad (A).
In view of (20), we have lim for all ,  ∈ A, which means that  is a centralizing mapping.With the help of Mathieu and Murphy's result [18], we arrive at the conclusion.

Corollary 3.
Let A be a Banach algebra with identity and let , , and  be fixed positive real numbers with  >  and  +  +  > 1. Suppose that mapping Φ : satisfies the assumption ( 1) and the second case of assumption (2) in Theorem 1 and suppose that  : A → A is a continuous centralizing mapping subjected to (18) and (19).Then  maps A into its radical rad (A).
Proof.We let  = 1 in (18).In the proof of Theorem 1, we find that  is an additive mapping.In this case,  is a mapping defined by () := lim  → ∞ (/)  ((/)  ) for all  ∈ A and  satisfies (7).As in the proof of the Corollary 2, the mapping  is linear.Since A contains the identity, Badora's result [10] implies that () = () + () for all ,  ∈ A. So  is a centralizing linear derivation.Based on the result of Mathieu and Murphy [18], we conclude that (A) ⊆ rad (A).

Approximate Derivations and Their Applications
Theorem 4. Let A be a Banach algebra.Assume that mappings Φ : A 3 → [0, ∞) and  : A 2 → [0, ∞) satisfy the following assumptions: where , , and  are fixed positive real numbers with  >  and  +  +  > 1. Suppose that  : A → A is a mapping subjected to the inequalities ( 1) and (2).Then, there exists a unique ring derivation L : A → A such that the inequality for all  ∈ A, where Φ (0, 0, 0)] . ( In addition, the equation holds for all  ∈ A.
In particular, by (2), we note that for all ,  ∈ A. Thus we get The conditions (35) and (39) guarantee that which implies that Therefore, we obtain (24) and L() = L() + L().Now, to show uniqueness of the mapping L, let us assume that  : A → A is another ring derivation satisfying (22).Then, we have by ( 22) and ( 35) for all  ∈ A, which means that L = .

Corollary 5.
Let A be a semiprime Banach algebra.Assume that the mappings Φ : A 3 → [0, ∞) and  : A 2 → [0, ∞) satisfy the assumptions of Theorem 4. Suppose that  : A → A is a mapping such that the inequality (18) holds for all , ,  ∈ A and all  ∈ U.Moreover, if a mapping  satisfies the conditions ( 2) and ( 20), then  maps A into the intersection of its center (A) and its radical rad (A).
Theorem 6.Let A be a Banach algebra.Assume that mappings Φ : where , , and  are fixed positive real numbers with  >  and  +  +  > 1. Suppose that  : A → A is a mapping subjected to inequalities ( 1) and (2).Then there exists a unique ring derivation L : A → A such that the inequality     L () −  ()     ≤  () for all  ∈ A. Moreover, (24) holds.
for all  ∈ A. The rest of proof can be carried out similarly as the corresponding part of Theorem 4.

Corollary 7.
Let A be a semiprime Banach algebra.Assume that a mapping Φ : A 3 → [0, ∞) satisfies the assumptions of Theorem 6 and that  : A → A is a centralizing mapping such that the inequality (18) holds for all , ,  ∈ A and all  ∈ U. Suppose that a mapping  is fulfilled with the inequality (19).Then  maps A into the intersection of its center (A) and its radical rad (A).
Proof.Employing the same method in the proof of Corollary 5, we find that  is linear derivation.According to Brešar's result [19], we get the result.
Proof.Employing the same argument in the proof of the previous corollaries, we see that  1 and  2 are linear derivations.