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Approximate Cubic ∗-Derivations on Banach ∗-Algebras
We study the stability of cubic ∗-derivations on Banach ∗-algebras. We also prove the superstability of cubic ∗-derivations on a Banach ∗-algebra A, which is left approximately unital.
In , Ulam proposed the stability problems for functional equations concerning the stability of group homomorphisms. In fact, a functional equation is called stable if any approximately solution to the functional equation is near a true solution of that functional equation and is superstable if every approximate solution is an exact solution to it. In , Hyers considered the case of approximate additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. Bourgin  was the second author to treat this problem for additive mappings (see also ). In , Rassias provided a generalization of Hyers Theorem, which allows the Cauchy difference to be unbounded. Gvruţa then generalized the Rassias’ result in  for the unbounded Cauchy difference. Subsequently, various approaches to the problem have been studied by a number of authors (see, e.g., [7–11]).
Recall that a Banach -algebra is a Banach algebra (complete normed algebra) which has an isometric involution. For a locally compact group , the algebraic group algebra is a Banach -algebra. The bounded operators on Hilbert space is also a Banach -algebra. In general, all -algebras are Banach -algebra. A left- (right-) bounded approximate identity for a normed algebra is a bounded net in such that () for each . A bounded approximate identity for is a bounded net , which is both a left- and a right-bounded approximate identity. Every group algebra and every -algebra has a bounded approximate identity.
The stability of functional equations of -derivations and of quadratic -derivations with the Cauchy functional equation and the Jensen functional equation on Banach -algebras is investigated in . The author also proved the superstability of -derivations and of quadratic -derivations on -algebras.
In 2003, Cdariu and Radu employed the fixed point method to the investigation of the Jensen functional equation. They presented a short and a simple proof (different from the “direct method,” initiated by Hyers in 1941) for the Cauchy functional equation  and for the quadratic functional equation  (see also [15–18]).
The functional equation which is called cubic functional equation. In addition, every solution of functional equation (1.1) is said to be a cubic mapping. It is easy to check that function is a solution of (1.1).
In , Bodaghi et al. proved the generalized Hyers-Ulam stability and the superstability for the functional equation (1.1) by using the alternative fixed point (Theorem 3.1) under certain conditions on Banach algebras. Also, the stability and the superstability of homomorphisms on -algebras by using the same fixed point method was proved in . The generalized Hyers-Ulam-Rassias stability of -homomorphisms between unital -algebras associated with the Trif functional equation and of linear -derivations on unital -algebras has earlier been proved by Park and Hou in .
In this paper, we prove the stability and the superstability of cubic -derivations on Banach -algebras. We also show that these functional equations, under some mild conditions, are superstable. We also establish the stability and the superstability of cubic -derivations on a Banach -algebra with a left-bounded approximate identity.
2. Stability of Cubic -Derivation
Throughout this paper, we assume that is a Banach -algebra. A mapping is a cubic derivation if is a cubic homogeneous mapping, that is, is cubic and for all and , and for all . In addition, if satisfies in condition for all , then it is called the cubic -derivation. An example of cubic derivations on Banach algebras is given in .
Let . For the given mapping , we consider for all .
Theorem 2.1. Suppose that is a mapping with for which there exists a function such that for all and all in which . Also, if for each fixed the mapping from to is continuous, then there exists a unique cubic -derivation on satisfying in which .
Proof. Putting and in (2.3), we have
for all in which . We can use induction to show that
for all and . On the other hand,
for all and . It follows from (2.2) and (2.7) that the sequence is a Cauchy sequence. Since is a Banach algebra, this sequence converges to the map , that is,
Thus the inequalities (2.2) and (2.8) show that (2.5) holds. Substituting by , respectively, in (2.3), we get
for all and . Since , the mapping is cubic. The equality implies that for all and . Now, let such that in which . We set , thus belongs to and for all . Now, suppose that is any continuous linear functional on and is a fixed element in . Define the mapping via for each . Obviously, is a cubic function. Under the hypothesis that is continuous in for each fixed , the function is the pointwise limit of the sequence of measurable functions in which , , . Hence, is a continuous function and has the form for all . Therefore,
Since is an arbitrary continuous linear functional on , for all and . Thus
for all and . Therefore, is a cubic homogeneous. If we replace by , respectively, and put in (2.4), we have
for all . Taking the limit as tends to infinity, we get , for all . Putting and substituting by in (2.4) and then dividing the both sides of the obtained inequality by , then we get
for all . Passing to the limit as in (2.14), we get for all . This shows that is a cubic -derivation.
Now, let be another cubic -derivation satisfying (2.5). Then we have which tends to zero as for all . So we can conclude that for all . This proves the uniqueness of .
We have the following theorem, which is analogous to Theorem 2.1. Since the proof is similar, it is omitted.
Theorem 2.2. Suppose that is a mapping with for which there exists a function satisfying (2.3), (2.4), and for all . Also, if for each fixed the mappings from to is continuous, then there exists a unique cubic -derivation on satisfying where .
Corollary 2.3. Let be positive real numbers with , and let be a mapping with such that for all and all . Then there exists a unique cubic -derivation on satisfying for all .
In the next theorem, we investigate the superstability of cubic -derivations of Banach -algebras with a left-bounded approximate identity.
Theorem 2.4. Suppose that is a Banach -algebra with a left-bounded approximate identity and . Let be a mapping for which there exists a function such that for all . Then is a cubic -derivation on .
Proof. First, we show that is cubic. For each , we have
Taking the limit from the right side as tends to infinity and using (2.21), we get for all . If is a left-bounded approximate identity for , then so is . Now, it follows from (2.26) that is cubic. For being cubic homogeneous of , we have Thus . By the same reasoning as in the above, is cubic homogeneous. For each , we have The above inequality and (2.21), (2.22), and (2.23) show that for all . Finally, we have for all . Note that in the last inequality we have used (2.22) and (2.24). This completes the proof.
Corollary 2.5. Let be the nonnegative real numbers with , and let be a Banach -algebra with a left bounded approximate identity. Suppose that is a mapping satisfying for all all . Then is a cubic -derivation on .
Proof. Using Theorem 2.4 with , we get the desired result.
3. A Fixed Point Approach
Before proceeding to the main results in this section, we bring the upcoming theorem, which is useful to our purpose (For an extension of the result see ).
Theorem 3.1 (The fixed point alternative ). Let be a complete generalized metric space and a mapping with Lipschitz constant . Then, for each element , either for all , or there exists a natural number such that:(i) for all ; (ii)the sequence is convergent to a fixed point of ;(iii) is the unique fixed point of in the set ; (iv) for all .
Theorem 3.2. Let be a continuous mapping with , and let be a continuous function such that for all and all . If there exists a constant such that for all , then there exists a unique cubic -derivation on satisfying in which .
Proof. First, we wish to provide the conditions of Theorem 3.1. We consider the set
and define the mapping on as follows:
if there exist such constant and , otherwise. It is easy to check that and , for all . For each , we have
Hence . If , then for every fixed , we have for all . This implies . Let be a -Cauchy sequence in . Then , and thus for all . Since is complete, then there exists such that in . Therefore, is a generalized metric on and the metric space is complete. Now, we define the mapping by
If such that , by definition of and , we have for all . By using (3.3), we get for all . The above inequality shows that for all . Hence, is a strictly contractive mapping on with a Lipschitz constant . To achieve inequality (3.4), we prove that . Putting and in (3.1), we obtain for all . Hence for all . We conclude from (3.12) that . It follows from Theorem 3.1 that for all , and thus in this theorem we have . Therefore, the parts (iii) and (iv) of Theorem 3.1 hold on the whole . Hence there exists a unique mapping such that is a fixed point of and that as . Thus for all , hence The above equalities show that (3.4) is true for all . It follows from (3.3) that Putting and substituting by , respectively, in (3.1), we get Taking the limit as tend to infinity, we obtain for all and all . Similar to the proof of Theorem 2.1, we have for all and . Since , we can show that for any rational number . The continuity of and imply that , for all and . Hence , for all and . Therefore, is a cubic homogeneous. If we put and replace by , respectively, in (3.1), we have for all . By letting in the preceding inequality, we find for all . Substituting by in (3.2) and then dividing the both sides of the obtained inequality by , we get for all . Passing to the limit as in (3.18) and applying (3.13), we conclude that for all . This shows that is a unique cubic -derivation.
Corollary 3.3. Let be positive real numbers with , and let be a mapping with such that for all and all . Then there exists a unique cubic -derivation on satisfying for all .
Proof. The result follows from Theorem 3.2 by letting
In the following corollary, we show the superstability for cubic -derivations.
Corollary 3.4. Let be nonnegative real numbers with , and let be a mapping such that for all and all . Then is a cubic -derivation on .
Proof. Putting in (3.22), we get . Now, if we put , in (3.22), then we have for all . It is easy to see by induction that , and thus for all and . It follows from Theorem 3.2 that is a cubic mapping. Now, by putting in Theorem 3.2, we can obtain the desired result.
The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments, and fruitful suggestions to improve the quality of the first draft of this paper. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299).
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