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On the stability of a mixed type quadratic-additive functional equation
Advances in Difference Equations volume 2013, Article number: 198 (2013)
Abstract
In this paper we establish the general solutions of the following mixed type quadratic-additive functional equation:
in the class of functions between real vector spaces. Moreover, we prove the generalized Hyers-Ulam-Rassias stability of this equation in Banach spaces.
MSC:39B82, 39B52.
1 Introduction
The stability problems of functional equations go back to 1940 when Ulam [1] proposed the following question: This question was solved affirmatively by Hyers [2] under the assumption that is a Banach space. He proved that if f is a mapping between Banach spaces satisfying for some fixed , then there exists a unique additive mapping A such that . In 1978, Rassias [3] generalized Hyers’ result to the unbounded Cauchy difference. Since then, the stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [4–9]).
Let f be a mapping from a group to a metric group with the metric such that
Then does there exist a group homomorphism and such that
for all ?
In particular, Kannappan [10] introduced the following mixed type quadratic-additive functional equation:
and proved that a function on a real vector space is a solution of (1.1) if and only if there exist a symmetric biadditive function B and an additive function A such that . In addition, Jung [11] investigated the Hyers-Ulam stability of (1.1) on restricted domains and applied the result to the study of an interesting asymptotic behavior of the quadratic functions. More generally, Jun and Kim [12] solved the general solutions and proved the stability of the following functional equation, which is a generalization of (1.1):
Najati and Moghimi [13] introduced another mixed type quadratic-additive functional equation
and investigated the generalized Hyers-Ulam-Rassias stability of this equation in quasi-Banach spaces.
In this paper, we introduce the following quadratic-additive functional equation:
to establish the general solutions and stability problems of this equation. For real vector spaces X and Y, we prove in Section 2 that a mapping satisfies (1.2) if and only if there exist a quadratic mapping satisfying
and an additive mapping satisfying
such that
for all . We refer to [14–24] for the stability results of other mixed type functional equations. In Section 3, we prove the generalized Hyers-Ulam-Rassias stability of (1.2) in Banach spaces.
2 General solutions of (1.2)
Throughout this section, X and Y will be real vector spaces. In order to solve the general solutions of (1.2), we need the following two lemmas.
Lemma 2.1 If an even mapping satisfies (1.2) for all , then f is quadratic.
Proof Putting in (1.2), we have . Putting in (1.2) yields
for all . Replacing y with x in (2.1) gives
for all . Putting in (2.1), we have
for all . Using (2.2) and (2.3), we can rewrite (1.2) as
for all . Putting in (2.4), we obtain
for all . □
Lemma 2.2 If an odd mapping satisfies (1.2) for all , then f is additive.
Proof Putting in (1.2), we have . Putting in (1.2) yields
for all . Replacing y by x in (2.5) gives
for all . Putting in (2.5), we have
for all . It follows from (2.5), (2.6) and (2.7) that
for all . Replacing x with and y with in (2.8), we obtain
for all . □
Now we are ready to establish the general solutions of (1.2).
Theorem 2.3 A function satisfies (1.2) for all if and only if there exist a symmetric biadditive mapping and an additive mapping such that
for all .
Proof (Necessity) We decompose f into the even part and the odd part by putting
for all . By Lemmas 2.1 and 2.2 we have the result.
(Sufficiency) This is obvious. □
3 Stability of (1.2)
In what follows, X and Y will be a real normed linear space and a real Banach space, respectively. For convenience, we define
for all . Let be a mapping satisfying one of the conditions (), (ℬ) and one of the conditions (), ():
for all . We note that the condition () implies (). Similarly, the condition (ℬ) implies (). One of the conditions (), (ℬ) will be needed to derive a quadratic mapping, and one of the conditions (), () will be required to derive an additive mapping in the following theorem.
Theorem 3.1 Suppose that a mapping satisfies
for all . Then there exist a unique quadratic mapping satisfying (1.3) and an additive mapping satisfying (1.4) such that
and
for all and for or 2, or 2. The mappings Q and A are given by
for all .
Proof We first consider the even part of f. Let be a function defined by for all . Then and
for all . Putting , in (3.2), we have
for all , where .
Case 1. Assume that φ satisfies the condition (). Replacing x by 3x in (3.3) and dividing by 9 yield
for all . Making use of an induction argument in (3.4) implies
for all and . From (3.5) we figure out
for all with and . The right-hand side of the inequality above tends to 0 as , the sequence is a Cauchy sequence for all and thus converges by the completeness of Y. Therefore, we can define a mapping by
for all . Note that , for all . It follows from the condition () and (3.2) that Q satisfies
for all . According to Lemma 2.1, the mapping Q satisfies (1.3). Letting in (3.5), we have
for all . Now we are going to prove the uniqueness of Q. Assume that is another quadratic function satisfying (1.3) and (3.6). Obviously, we have and for all . Then we figure out
for all and . Taking the limit as , we conclude that for all .
Case 2. If φ satisfies the condition (ℬ) (and hence implies ()), the proof is analogous to that of Case 1. By virtue of the condition (ℬ) and (3.1), we have and . An induction argument on (3.3) implies
for all and . Using a similar argument to that of Case 1, we see that the sequence is a Cauchy sequence for all . Thus we can define a mapping by
for all . Note that and for all . From the condition (ℬ) and (3.2), we see that Q satisfies
for all . By Lemma 2.1 the mapping Q satisfies (1.3). Taking the limit as in (3.7), we obtain
for all . The rest of the proof is similar to that of Case 1.
Next, we consider the odd part of f. Now, let be a function defined by for all . Then , and
for all . Putting , in (3.8) and dividing by 3 yield
for all .
Case 3. Assume that φ satisfies the condition () (and hence implies ()). Replacing x by 3x in (3.9) and dividing by 3, we have
for all . Making use of an induction argument in (3.10) implies
for all and . From (3.11) we can show that the sequence is a Cauchy sequence for all . Define a mapping by
for all . By the oddness of f, we see that for all . Also, from the condition () and (3.9), we verify that A satisfies
for all . According to Lemma 2.2, the mapping A satisfies (1.4). Letting in (3.11), we have
for all . Using a similar argument to that of Case 1, we can easily see the uniqueness of A.
Case 4. If φ satisfies the condition (), the proof is analogous to that of Case 3. An induction argument on (3.9) implies
for all and . It follows from (3.12) that is a Cauchy sequence for all . Thus we can define a mapping by
for all . Note that for all . Also, from the condition () and (3.8), we verify that A satisfies
for all . According to Lemma 2.2, the mapping A satisfies (1.4). Taking the limit as in (3.12), we have
for all . Similarly, we can show the uniqueness of A. □
From the theorem above, we have the following corollary immediately.
Corollary 3.2 Let , and be real numbers. Suppose that a mapping satisfies
for all ( if ). Then for each three cases , and , there exist a unique quadratic mapping satisfying (1.3) and an additive mapping satisfying (1.4) such that
and
for all ( if ), where if .
Proof Let for all . Then for all ( if ). If , the mapping φ satisfies (). Thus, we figure out
for all ( if ). If , the mapping φ satisfies (ℬ). Thus, we have
for all . If , the mapping φ satisfies (). Thus, we get
for all ( if ). If , the mapping φ satisfies (). Thus, we obtain
for all . Therefore, we have
for all ( if ). □
Corollary 3.3 Let be a real number. Suppose that a mapping satisfies
for all . Then there exist a unique quadratic mapping satisfying (1.3) and an additive mapping satisfying (1.4) such that
and
for all .
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The first author discovered the topic and the main ideas for the proof of the paper and made the actual writing. All authors discussed the paper together. The second and the third authors discovered some helpful ideas for the proof of this paper and checked the proof of the paper. All authors read and approved the final manuscript.
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Lee, YS., Na, J. & Woo, H. On the stability of a mixed type quadratic-additive functional equation. Adv Differ Equ 2013, 198 (2013). https://doi.org/10.1186/1687-1847-2013-198
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DOI: https://doi.org/10.1186/1687-1847-2013-198