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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 472531, 5 pages
Generalized Hyers-Ulam Stability of a Mixed Type Functional Equation
1Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea
2Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea
Received 20 April 2013; Accepted 28 May 2013
Academic Editor: Bing Xu
Copyright © 2013 Yang-Hi Lee and Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate the stability of a functional equation by applying the direct method in the sense of Hyers and Ulam.
In 1940, Ulam  gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms: Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?
The Ulam’s problem for the Cauchy additive functional equation was solved by Hyers under the assumption that and are Banach spaces. Indeed, Hyers  proved that every solution of the inequality (for all and ) can be approximated by an additive function. In this case, the Cauchy additive functional equation, , is said to satisfy the Hyers-Ulam stability.
Thereafter, Rassias  attempted to weaken the condition for the bound of norm of the Cauchy difference as follows: and he proved that Hyers’ theorem is also true for this case. Indeed, Rassias proved the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation between Banach spaces. We here remark that a paper of Aoki  was published concerning the generalized Hyers-Ulam stability of the Cauchy functional equation earlier than Rassias’ paper.
The stability concept that was introduced by Rassias’ theorem provided a large influence to a number of mathematicians to develop the notion of what is known today with the term generalized Hyers-Ulam stability of functional equations. Since then, the stability problems of several functional equations have been extensively investigated by several mathematicians (e.g., see [5–10] and the references therein).
Almost all subsequent proofs in this very active area have used the Hyers’ method presented in . Namely, starting from the given mapping that approximately satisfies a given functional equation, a solution of the functional equation is explicitly constructed by using the formula: which approximates the mapping . This method of Hyers is called the direct method.
We remark that another method for proving the Hyers-Ulam stability of various functional equations was introduced by Baker , which is called the fixed-point method. This method is very powerful technique of proving the stability of functional equations (see [12, 13]).
Now we consider the following functional equation: which is called the mixed type functional equation. The mapping is a solution of this functional equation, where are real constants. Every solution of (3) will be called a quadratic-additive mapping.
In 1998, Jung  proved the stability of (3) by decomposing into the odd and even parts. In his proof, using the direct method, an additive mapping and a quadratic mapping are separately constructed from the odd and even parts of , and then and are combined to provide a quadratic-additive mapping which is close to .
In this paper, we will prove the generalized Hyers-Ulam stability of (3) by making use of the direct method. In particular, we will approximate the given mapping by a solution of (3) without decomposing into its odd and even parts, while in the Jung’s paper  the mapping was decomposed into the odd and even parts, and each of them was separately approximated by the corresponding part of a solution of (3).
2. Main Results
Throughout this paper, let be a (real or complex) normed space and a Banach space. For an arbitrary , we define and .
For a given mapping , we use the following abbreviations: for all .
As we stated in the previous section, is called a quadratic-additive mapping provided that satisfies the functional equation for all .
Proposition 1. A mapping is a solution of (3) if and only if is a quadratic mapping and is an additive mapping.
Proof. Assume that is a solution of (3). Then we have
for all , that is, is a quadratic mapping and is an additive mapping.
Conversely, assume that is a quadratic mapping and is an additive mapping. Then we get for all ; that is, is a solution of (3).
We first prove the following lemma.
Lemma 2. If a mapping satisfies for all and , then is a quadratic-additive mapping.
Proof. Using the hypothesis, we have for all . Furthermore, by the last equality, we get for all . Since is invariant with respect to the permutation of , it holds that for all . It is also easy to show that , , , and for all as we desired.
In the following theorem, we can prove the generalized Hyers-Ulam stability of the functional equation (3) by making use of the direct method.
Theorem 3. If a mapping satisfies and for all with a real constant , then there exists a unique quadratic-additive mapping such that for all . Moreover, if , then itself is a quadratic-additive mapping.
Proof. Let us define , where . From the definitions of and , we have
for all and . It follows from (9) and (11) that
for all . So, it is easy to show that the sequence is a Cauchy sequence for all .
Since is complete and , the sequence converges for all . Hence, we can define a mapping by for all . Moreover, if we put and let in (12), we obtain the inequality (10).
From the definition of , we get for all . By Lemma 2, is a quadratic-additive mapping.
Now, we will show that is uniquely determined. Let be another quadratic-additive mapping satisfying (10). It is easy to show that for all quadratic-additive mapping . It follows from (11) that for all and . Since and are quadratic-additive, if we replace with in (10), then we have for all and . Taking the limit in the above inequality as , we can conclude that for all , which proves the uniqueness of .
Since for all and , if , then we conclude that for all by letting in the previous inequality. From the fact that , is a quadratic-additive mapping.
Theorem 4. If a mapping satisfies for all and for a nonnegative real constant , then there exists a unique quadratic-additive mapping such that for all .
we get for and for . From the definitions of and , we have
for all and , where is defined by and .
It follows from (18) and (21) that for all . So, it is easy to show that the sequence is a Cauchy sequence for all .
Since is complete, the sequence converges for all . Hence, we can define a mapping by for all . Moreover, putting and letting in (22), we get the inequality (19).
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012R1A1A4A01002971).
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