`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 219435, 16 pageshttp://dx.doi.org/10.1155/2012/219435`
Research Article

Generalized Stability of Euler-Lagrange Quadratic Functional Equation

Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Republic of Korea

Received 7 May 2012; Accepted 15 July 2012

Copyright © 2012 Hark-Mahn Kim and Min-Young Kim. 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.

Abstract

The main goal of this paper is the investigation of the general solution and the generalized Hyers-Ulam stability theorem of the following Euler-Lagrange type quadratic functional equation , in -Banach space, where are fixed rational numbers such that and .

1. Introduction

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

Let be a group and let be a metric group with metric . Given , does there exist a such that if satisfies for all , then a homomorphism exists with for all ?

In 1941, the first result concerning the stability of functional equations was presented by Hyers [2]. He has answered the question of Ulam for the case where and are Banach spaces.

Let and be real vector spaces. A function is called a quadratic function if and only if is a solution function of the quadratic functional equation

It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all , where the mapping is given by . See [3, 4] for the details. The Hyers-Ulam stability of the quadratic functional equation (1.1) was first proved by Skof [5] for functions , where is a normed space and is a Banach space. Cholewa [6] demonstrated that Skof’s theorem is also valid if is replaced by an Abelian group . Assume that a function satisfies the inequality for some and for all . Then there exists a unique quadratic function such that for all . Czerwik [7] proved the Hyers-Ulam-Rassias stability of quadratic functional equation (1.1). Let and be a real normed space and a real Banach space, respectively, and let be a positive constant. If a function satisfies the inequality for some and for all , then there exists a unique quadratic function such that for all . Furthermore, according to the theorem of Borelli and Forti [8], we know the following generalization of stability theorem for quadratic functional equation. Let be an Abelian group and a Banach space, and let be a mapping with satisfying the inequality for all . Assume that one of the series then there exists a unique quadratic function such that for all . During the last three decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability of several functional equations, and there are many interesting results concerning this problem [913].

The notion of quasi--normed space was introduced by Rassias and Kim in [14]. This notion is a generalization of that of quasi-normed space. We consider some basic concepts concerning quasi--normed space. We fix a real number with and let denote either or . Let be a linear space over . A quasi--norm is a real-valued function on satisfying the following:(1) for all and if and only if ,(2) for all and all ,(3)there is a constant such that for all .

The pair is called a quasi--normed space if is a quasi--norm on . The smallest possible is called the modulus of concavity of . A quasi--Banach space is a complete quasi--normed space. A quasi--norm is called a -norm if for all . In this case, the quasi--Banach space is called a -Banach space. We observe that if are nonnegative real numbers, then where [15].

J. M. Rassias investigated the stability of Ulam for the Euler-Lagrange functional equation in the paper of [16]. Gordji and Khodaei investigated the generalized Hyers-Ulam stability of other Euler-Lagrange quadratic functional equations [17]. Jun et al. [18] introduced a new quadratic Euler-Lagrange functional equation for any fixed with , which was a modified and instrumental equation for [19], and solved the generalized stability of (1.12). Now, we improve the functional equation (1.12) to the following functional equations: for any fixed rational numbers with and , which are generalized versions of (1.12). In this paper, we establish the general solution of (1.13) and (1.14) and then prove the generalized Hyers-Ulam stability of (1.13) and (1.14). We remark that there are some interesting papers concerning the stability of functional equations in quasi-Banach spaces [15, 2023] and quasi--normed spaces [14, 24, 25].

2. General Solution of (1.13) and (1.14)

First, we present the general solution of (1.14) in the class of all functions between vector spaces.

Lemma 2.1. Let and be vector spaces over . Then a mapping is a solution of the functional equation (1.12) for any fixed rational number with if and only if is quadratic.

Proof. See the same proof in [18].

Lemma 2.2. Let and be vector spaces over . Then a mapping is a solution of the functional equation (1.13) if and only if is quadratic.

Proof. We assume that a mapping satisfies the functional equation (1.13). Letting in (1.13), then (1.13) is equivalent to (1.12). Then by Lemma 2.1, is quadratic. Conversely, if is quadratic, then it is obvious that satisfies (1.13).

Theorem 2.3. Let and be vector spaces over . Then a mapping with satisfies the functional equation (1.14) if and only if is quadratic. In this case, and hold for all .

Proof. We assume that a mapping with satisfies the functional equation (1.14). Then replacing in (1.14) by 0, we also get the equality for all . Now, we decompose into the even part and the odd part by setting for all . Then and satisfy the functional equation (1.14). Therefore, we may assume without loss of generality that is even and satisfies (1.14) for all . If we replace in (1.14) by 0, then we get for all . From this equality, we have for all . Therefore, (1.14) implies (1.13) for all . By Lemma 2.2,   is quadratic.
Now, we assume that is odd and satisfies (1.14) for all . For the case , we have for all . Setting by 0 in (2.3), one obtains . Let . Replacing by 0 in (1.14), we have for all . From (1.14) and (2.4), we get for all . Putting in (2.5), then we obtain for all . Replacing by in (2.6), we get for all . Since , (2.7) yields for all . Interchanging and in (2.8), we have by oddness of for all . Replacing by in (2.6), we get for all . Adding (2.9) and (2.10) side by side, this leads to for all . Therefore, is additive and so for all and for any odd function satisfying (1.14). Using the equality , we obtain for all . Therefore, is a quadratic mapping, as desired.
Conversely, if is quadratic, then it is obvious that satisfies (1.14).

We note that if and satisfies (1.14).

3. Generalized Stability of (1.14) for

For convenience, we use the following abbreviation: for any fixed rational numbers and with and , for all , which is called the approximate remainder of the functional equation (1.14) and acts as a perturbation of the equation.

From now on, let be a vector space, and let be a -Banach space unless we give any specific reference. We will investigate the generalized Hyers-Ulam stability problem for the functional equation (1.14). Thus, we find some conditions such that there exists a true quadratic function near an approximate solution of (1.14).

Theorem 3.1. Let be a function such that for all . Suppose that a function with satisfies for all . Then there exists a unique quadratic function satisfying for all . The function is given by for all .

Proof. Letting by 0 in (3.4), we get for all . Multiplying both sides by in (3.7), we have for all . Replacing by and multiplying both sides by in (3.8), we have for all . Next we show that the sequence is a Cauchy sequence. For any , , and , it follows from (3.9) that for all . It follows from (3.2) and (3.10) that the sequence is a Cauchy sequence in for all . Since is a -Banach space, the sequence converges for all . Therefore, we can define a mapping by for all . Taking and in (3.10), we have for all . Therefore, for all , that is, the mapping satisfies (3.5). It follows from (3.3) and (3.4) that for all . Therefore, satisfies (1.14), and so the function is quadratic.
To prove the uniqueness of the quadratic function , let us assume that there exists a quadratic function satisfying the inequality (3.5). Then we have for all and . Therefore, letting , one has for all , completing the proof of uniqueness.

In the following corollary, we get a stability result of (1.14).

Corollary 3.2. Let be a quasi--normed space for fixed real number with . Let be positive reals such that either (1)  , , and or (2)  , , and , for . Assume that a function with satisfies the inequality for all . Then there exists a unique quadratic function which satisfies the inequality for all . The function is given by for all .

Proof. Let . Then By Theorem 3.1, there exists a unique quadratic mapping such that for all .

Theorem 3.3. Let be a function such that for all . Suppose that a function with satisfies for all . Then there exists a unique quadratic function satisfying for all . The function Q is given by for all .

Proof. Letting by 0 in (3.24), we get for all . Replacing by in (3.27), we have for all . Replacing by and multiplying both sides by in (3.28), we have for all . Next we show that the sequence is a Cauchy sequence. For any , , and , it follows from (3.29) that It follows from (3.22) and (3.30) that the sequence is a Cauchy sequence in for all . Since is a -Banach space, the sequence converges for all . Therefore, we can define a mapping by for all . The rest of the proof is similar to the corresponding proof of Theorem 3.1.

Corollary 3.4. Let be a quasi--normed space for fixed real number with . Let be positive reals such that either (1)  , , and or (2)  , , and , for . Assume that a function with satisfies the inequality for all . Then there exists a unique quadratic function which satisfies the inequality for all . The function is given by for all .

Proof. Let . Then satisfies the conditions (3.22) and (3.23). Applying Theorem 3.3, we obtain the results, as desired.

4. Generalized Stability of (1.13)

For convenience, we use the following abbreviation: for any fixed rational numbers and with and , for all , which is called the approximate remainder of the functional equation (1.13) and acts as a perturbation of the equation.

We will investigate the generalized Hyers-Ulam stability problem for the functional equation (1.13).

Theorem 4.1. Let be a function such that for all . Suppose that a function with satisfies for all . Then there exists a unique quadratic function satisfying for all . The function Q is given by for all .

Proof. Replacing by in (4.4), we get for all . Letting be in (4.7), we have for all . Multiplying both sides by in (4.8), we have for all . Replacing by and multiplying both sides by in (4.9), we have for all . Next we show that the sequence is a Cauchy sequence. For any , , and , it follows from (4.10) that for all . It follows from (4.2) and (4.11) that the sequence is a Cauchy sequence in for all . Since is a -Banach space, the sequence converges for all . Therefore, we can define a mapping by for all . The rest of the proof is similar to the corresponding proof of Theorem 3.1.

In the following corollary, we get a stability result of (1.13).

Corollary 4.2. Let be a quasi--normed space for fixed real number with . Let be positive reals such that either (1)  , , and or (2)  , , and , for . Assume that a function with satisfies the inequality for all . Then there exists a unique quadratic function which satisfies the inequality for all . The function is given by for all .

Proof. Let . Then satisfies the conditions (4.2) and (4.3). By Theorem 4.1, there exists a unique quadratic mapping such that for all .

Theorem 4.3. Let be a function such that for all . Suppose that a function with satisfies for all . Then there exists a unique quadratic function satisfying for all . The function Q is given by for all .

Proof. Replacing by in (4.8), we have for all . The rest of the proof is similar to the corresponding proof of Theorem 3.3.

Corollary 4.4. Let be a quasi--normed space for fixed real number with . Let be positive reals such that either (1)   and , or (2)   and , , for . Assume that a function with satisfies the inequality for all . Then there exists a unique quadratic function which satisfies the inequality for all . The function is given by for all .

Proof. Let . Then satisfies the conditions (4.17). Applying Theorem 4.3, we obtain the results, as desired.

Acknowledgment

This study was supported by the Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (no. 2012R1A1A2008139).

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