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Asymptotic Behavior of Almost Quartic ⁎-Derivations on Banach ⁎-Algebras
The purpose of this paper is to obtain the stability theorems of quartic ⁎-derivations associated with the quartic functional equation on Banach ⁎-algebras.
In 1940, Ulam  gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. In the next year, Hyers  gave a clear answer to this problem for additive mappings between Banach spaces. Then this theorem  was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. Since then, many mathematicians have come to deal with this problem and also there are many interesting results concerning this problem [5–8].
First, we recall definition of -derivation.
Definition 1. Let be a Banach -algebra and let be a Banach -subalgebra of . A -linear mapping is said to be derivation on if for all . Moreover, if satisfies the additional condition for all , then it is called a -derivation.
The stability of -derivations and of quadratic -derivations with the Cauchy functional equation and the Jensen functional equation on Banach -algebras was investigated in . Jang and Park  proved the superstability of -derivations and of quadratic -derivations on -algebras. The stability of -derivations on Banach -algebras by using fixed-point alternative was proven by Park and Bodaghi . Thereafter, Yang et al.  obtained the stability results of cubic -derivations on -algebras and superstability of cubic -derivations on Banach -algebras which are left approximately unital. Recently, Koh and Kang  proved the stability of a generalized cubic -derivations on Banach -algebras.
In 1999, Rassias  treated the stability of the following quartic equation:for a mapping where is a linear space and is a Banach space. Thereafter, Lee and Chung  studied the general solution and stability theorem of generalized quartic functional equations in the spaces of generalized functions between real vector spaces. Kang  has then extended the stability theorems of the following generalized quartic functional equation: where in quasi--normed spaces. Recently, Bodaghi  obtained the general solution of the generalized quartic functional equation for a fixed positive integer and proved the Hyers-Ulam stability for this quartic functional equation by the direct method and the fixed-point method on real Banach spaces and non-Archimedean spaces. For more information about the stability of quartic functional equations, we refer to [17–20].
Hyer’s direct method used in  has been widely applied for studying the generalized Hyers-Ulam stability of various functional equations. Nevertheless, there exist also other approaches proving the Hyers-Ulam stability of functional equations. The most popular technique of proving stability of functional equations except for direct method is the fixed-point method. Although fixed-point method was used for the first time by J.A. Baker , most authors follow the alternative fixed-point approach [22, 23] using a theorem of Diaz and Magolis .
In this paper, we deal with the following quartic functional equation:in Banach -algebras. First of all, we show that (4) is equivalent to (1) and then the mapping satisfying (4) on the punctured domain at zero is quartic. In the sequel, we investigate the stability of quartic -derivations associated with the given functional equation on Banach -algebras by using direct method and fixed-point method, respectively.
2. Approximate Quartic -Derivations
First of all, we find out the general solution of (4) in the class of mappings between vector spaces.
Throughout this section, let be a Banach -algebra and let be a Banach -subalgebra of . For a given mapping , we define for all and all .
The following proposition provides a solution of the functional equation (4) on the punctured domain at zero.
Proposition 3. If is a mapping satisfying the equality for all and , then for all and hence is quartic.
Proof. Since , it is trivial that . We obtain the equalities so for . And we get ; then for .
By using above properties, we can show that for . Thus, for all and so is quartic.
Definition 4. A mapping is called a quartic homogeneous mapping if satisfies (1) and for all and . A quartic homogeneous mapping is said to be a quartic derivation if for all . In addition, if for all , then it is called a quartic -derivation.
Now we present a main theorem, which is a stability of quartic functional equation (4) in Banach -algebras.
Theorem 5. Let be a mapping with and let be a function such thatfor all and all . Assume that the mapping from to is continuous for each fixed . Then there exists a unique quartic -derivation satisfyingfor all .
Proof. Taking , , and in inequality (8), we getfor all . By using induction, it is implied from inequality (11) thatfor and . By (7) and (12), the sequence () is a Cauchy sequence. So define a mapping byfor all . And letting in inequality (12), we getfor and . Hence (7) and (14) show that approximate inequality (10) holds.
Next, we have to show that the mapping is a quartic -derivation such that inequality (10) holds for all . Replacing by in (8), respectively, and putting , we have and so it follows from (7) and (13) that for all and . Thus, by the same argument in the proof of Theorem 3.2 in , and for all and , which implies that the mapping is quartic homogeneous by Lemma 2.
Next, replacing by in inequality (9), we get for all . By (7), we have for all . Letting and replacing by in inequality (8), we have for all . Also by (7), we have for all . Therefore is a quartic -derivation.
Lastly, we should show that is unique. Suppose that is another quartic -derivation satisfying approximate inequality (10). So which tends to zero as for all . Hence for all .
Corollary 6. Let be nonnegative real constants with either or and let be a mapping with such that for all and . Assume that the mapping from to is continuous for each fixed . Then there exists a unique quartic -derivation satisfying for all .
Proof. Letting and applying Theorem 5, we obtain the desired result.
Concerning the stability of quartic homogeneous function, the following example presents that the stability of functional equation in Corollary 6 with does not hold.
Example 7. Let be defined byConsider the function defined by for all , where . Then, using similar way to that in , satisfies for all and , but there do not exist a quartic mapping and a constant such that for all
However, the stability problem of and is open in Corollary 6 concerning the stability of quartic -derivations.
Corollary 8. Let be nonnegative real constants with either or and let be a mapping with such that for all and . Assume that the mapping from to is continuous for each fixed . Then there exists a unique quartic -derivation satisfying for all .
Proof. Letting , as well as applying Theorem 5, we obtain the desired result.
Now, we investigate the stability using the alternative fixed-point method. Before proceeding to the main result, we state the following definition and theorem which are useful for our purpose.
Definition 9. Let be a set. A function is called a generalized metric on if satisfies the following: (i), if and only if .(ii) for all .(iii) for all .
Theorem 10 (see ). Let be a complete generalized metric space and let be 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 11. Let be a continuous mapping with and let be a function such thatfor all and . If there exist constants such that for all , then there exists a unique quartic -derivation satisfyingfor all , where .
Proof. First, we consider a setand define a mapping on as follows:if there exists such constant and , if not. Then we can easily show that is a generalized metric on and the metric space is complete. We define a mapping bywhere and for all .
Now we remark that is a strictly contractive mapping on with the Lipschitz constant .
On the other hand, letting in inequality (27), we get for all . This implies that It follows from Theorem 10 that for all . So parts (iii) and (iv) of Theorem 10 hold on the whole . Therefore, there exists a unique mapping such that is a fixed point of and for all and So the mapping satisfies inequality (30) that holds for all .
Since , inequality (30) shows thatfor all . Replacing by and putting in inequality (27), we have and taking the limit as tends to infinity, we get for all and all . Also, by the same argument in the proof of Theorem 5, the mapping is quartic homogeneous. Next, replacing by in inequality (28), we get for all . By (39), we have for all . Letting and replacing by in inequality (27), we have for all . Also by (39), we have for all . Therefore the mapping is a quartic -derivation.
The rest of the proof is similar to the proof of Theorem 5.
Corollary 12. Let be nonnegative reals with either or and let be a continuous mapping with such that for all and . Then there exists a unique quartic -derivation on satisfying for all .
Proof. Letting and applying Theorem 11 with , we obtain the desired results.
Corollary 13. Let be nonnegative reals with either or and let be a continuous mapping with such that for all and . Then there exists a unique quartic -derivation on satisfying for all .
Proof. Letting and applying Theorem 11 with , we obtain the desired results.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A3B03930971).
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