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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 896032, 10 pages
http://dx.doi.org/10.1155/2012/896032
Research Article

Nearly Radical Quadratic Functional Equations in p-2-Normed Spaces

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran
3Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran
4Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 16 November 2011; Accepted 13 February 2012

Academic Editor: John Rassias

Copyright © 2012 M. Eshaghi Gordji et al. 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

We establish some stability results in 2-normed spaces for the radical quadratic functional equation and then use subadditive functions to prove its stability in -2-normed spaces.

1. Introduction and Preliminaries

The story of the stability of functional equations dates back to 1925 when a stability result appeared in the celebrated book by Póolya and Szeg [1]. In 1940, Ulam [2, 3] posed the famous Ulam stability problem which was partially solved by Hyers [4] in the framework of Banach spaces. Later Aoki [5] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [6] provided a generalization of Hyers’ theorem by proving the existence of unique linear mappings near approximate additive mappings. Găvruţa [7] obtained the generalized result of T. M. Rassias’ theorem which allows the Cauchy difference to be controlled by a general unbounded function. On the other hand, Rassias, Găvruţa, and several authors proved the Ulam-Gavruta-Rassias stability of several functional equations. For more details about the results concerning such problems, the reader is referred to [830].

Gajda and Ger [31] showed that one can get analogous stability results, for subadditive multifunctions. For further results see [3242], among others.

The most famous functional equation is the Cauchy equation any solution of which is called additive. It is easy to see that the function defined by with an arbitrary constant is a solution of the functional equation So, it is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known [43, 44] that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all . The for all .

We briefly recall some definitions and results used later on in the paper. For more details, the reader is referred to [4549]. The theory of 2-normed spaces was first developed by Gähler [46] in the mid-1960s, while that of 2-Banach spaces was studied later by Gähler and White [47, 49].

Definition 1.1 (see [45]). Let    be a real linear space over    with dim   and    a function.
Then   is called a linear 2-normed space if and if and only if and are linearly dependent,,, for any ,,for all . The function is called the 2-norm on .

Let be a linear 2-normed space. If and , for all , then . Moreover, for a linear 2-normed space , the functions are continuous functions of into for each fixed (see [48]).

A sequence in a linear 2-normed space is called a Cauchy sequence if there are two points such that and are linearly independent, and .

A sequence in a linear 2-normed space is called a convergent sequence if there is an such that , for all . If converges to , write as and call the limit of . In this case, we also write .

A linear 2-normed space in which every Cauchy sequence is a convergent sequence is called a 2-Banach space. For a convergent sequence in a 2-normed space , , for all (see [48]).

We fix a real number with , and let be a linear space. A -2-norm is a function on satisfying Definition 1.1; , , and ; the following: , for all and all . The pair is called a -2-normed space if is a -2-norm on . A -2-Banach space is a complete -2-normed space.

We recall that a subadditive function is a function , having a domain and a codomain that are both closed under addition, with the following property: for all . Let be fixed. If there exists a constant with such that a function satisfies for all , then we say that is contractively subadditive if , and is expansively superadditive if . It follows by the last inequality that satisfies the following properties: for all and integers .

Now, we consider the radical quadratic functional equation where is a fixed integer and prove generalized Ulam stability, in the spirit of Găvruta (see [7]), of this functional equation in 2-normed spaces. Moreover, in this paper, we investigate new results about the generalized Ulam stability by using subadditive functions in -2-normed spaces for the radical quadratic functional equation (1.5).

2. Main Results

In this section, let be a linear space, and let and denote the sets of real and positive real numbers, respectively. If a mapping satisfies the functional equation (1.5), by letting in (1.5), we get . Setting in (1.5) and using , we get for all . Putting in (1.5) and using , we get for all . It follows from (2.1) and (2.2) that for all and integers . Setting in (1.5) and then comparing it with (1.5), we obtain , for all . Letting in (1.5), we get for all . It follows from (2.4) and the evenness of that satisfies (1.1). So we have the following lemma.

Lemma 2.1. If a mapping satisfies the functional equation (1.5), then is quadratic.

Corollary 2.2. If a mapping satisfies the functional equation (1.5), then there exists a symmetric biadditive mapping such that , for all .

Hereafter, we will assume that is a 2-Banach space. First, using an idea of Găvruţa [7], we prove the stability of (1.5) in the spirit of Ulam, Hyers, and Rassias.

Let be a function from to . A mapping is called a -approximatively radical quadratic function if for all , where is a fixed integer.

Theorem 2.3. Let be fixed, and let be a - approximatively radical quadratic function with . If the function satisfies and , for all , then there exists a unique quadratic mapping , satisfies (1.5) and the inequality for all .

Proof. Letting in (2.5), we get for all . Setting in (2.5), we get for all . Replacing by in (2.8), we obtain for all . It follows from (2.9) and (2.10) that for all . Thus, for all . Hence, for all and integers . Thus, is a Cauchy sequence in the 2-Banach space . Hence, we can define a mapping by , for all . That is, for all . In addition, it is clear from (2.5) that the following inequality: holds for all , and so by Lemma 2.1, the mapping is quadratic. Taking the limit in (2.13) with , we find that the mapping is quadratic mapping satisfying the inequality (2.7) near the approximate mapping of (1.5). To prove the aforementioned uniqueness, we assume now that there is another quadratic mapping which satisfies (1.5) and the inequality (2.7). Since the mapping satisfies (1.5), then for all and integers . Thus, one proves by the last equality and (2.7) that for all and integers . Therefore, from , one establishes for all .

Corollary 2.4. Let be fixed. If there exist nonnegative real numbers with such that a mapping satisfies the inequality for all and some , then there exists a unique quadratic mapping , satisfies (1.5) and the inequality for all , where .

Corollary 2.5. Let be fixed. If there exist nonnegative real numbers with such that a mapping satisfies the inequality for all and some , then there exists a unique quadratic mapping satisfies (1.5) and the inequality for all .

Now, we are going to establish the modified Hyers-Ulam stability of (1.5).

Theorem 2.6. Let be fixed, let be a -2-Banach space, and, be a -approximatively radical quadratic function with . Assume that the map is contractively subadditive if and is expansively superadditive if with a constant satisfying , where , then there exists a unique quadratic mapping which satisfies (1.5) and the inequality for all , where

Proof. Using the same method as in the proof of Theorem 2.3, we have for all . Hence for all and integers . Thus, is a Cauchy sequence in the -2-Banach space . Hence, we can define a mapping by , for all . Also holds for all , and so by Lemma 2.1, the mapping is quadratic. Taking the limit in (2.25) with , we find that the mapping is quadratic mapping satisfying the inequality (2.22) near the approximate mapping of (1.5). The remaining assertion goes through in a similar way to the corresponding part of Theorem 2.3.

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