Abstract and Applied Analysis

Volume 2012 (2012), Article ID 896032, 10 pages

http://dx.doi.org/10.1155/2012/896032

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

^{1}Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran^{2}Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran^{3}Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran^{4}Department 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 [8–30].

Gajda and Ger [31] showed that one can get analogous stability results, for subadditive multifunctions. For further results see [32–42], 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 [45–49]. 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.

#### References

- G. Póolya and G. Szeg,
*Aufgaben und Lehrsätze aus der Analysis*, vol. 1, Springer, Berlin, Germany, 1925. - S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience, New York, NY, USA, 1960. View at Zentralblatt MATH - S. M. Ulam,
*Problems in Modern Mathematics*, Wiley, New York, NY, USA, 1964. View at Zentralblatt MATH - D. H. Hyers, “On the stability of the linear functional equation,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222–224, 1941. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Aoki, “On the stability of the linear transformation in Banach spaces,”
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64–66, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. M. Rassias, “On the stability of the linear mapping in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431–436, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. M. Rassias, “Solution of a problem of Ulam,”
*Journal of Approximation Theory*, vol. 57, no. 3, pp. 268–273, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Gavruta, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” in
*Advances in Equations and Inequalities*, Hadronic Mathematics, pp. 67–71, 1999. View at Google Scholar - J. M. Rassias and H.-M. Kim, “Generalized Hyers-Ulam stability for general additive functional equations in quasi-
*β*-normed spaces,”*Journal of Mathematical Analysis and Applications*, vol. 356, no. 1, pp. 302–309, 2009. View at Publisher · View at Google Scholar - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Journal of Functional Analysis*, vol. 46, no. 1, pp. 126–130, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Bulletin des Sciences Mathematiques*, vol. 108, no. 4, pp. 445–446, 1984. View at Google Scholar · View at Zentralblatt MATH - J. M. Rassias, “Solution of a problem of Ulam,”
*Journal of Approximation Theory*, vol. 57, no. 3, pp. 268–273, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Z. Eskandani, P. Gavruta, J. M. Rassias, and R. Zarghami, “Generalized Hyers-Ulam stability for a general mixed functional equation in quasi-
*β*-normed spaces,”*Mediterranean Journal of Mathematics*, vol. 8, no. 3, pp. 331–348, 2011. View at Publisher · View at Google Scholar - T. Z. Xu, J. M. Rassias, and W. X. Xu, “Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces,”
*Journal of Mathematical Physics*, vol. 51, no. 9, Article ID 093508, 19 pages, 2010. View at Publisher · View at Google Scholar - T. Z. Xu, J. M. Rassias, and W. X. Xu, “Intuitionistic fuzzy stability of a general mixed additive-cubic equation,”
*Journal of Mathematical Physics*, vol. 51, no. 6, Article ID 063519, 21 pages, 2010. View at Publisher · View at Google Scholar - T. Z. Xu, J. M. Rassias, and W. X. Xu, “On the stability of a general mixed additive-cubic functional equation in random normed spaces,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 328473, 16 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji, M. B. Ghaemi, Y. J. Cho, and H. Majani, “A general system of Euler-Lagrange-type quadratic functional equations in Menger probabilistic non-Archimedean 2-normed spaces,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 208163, 21 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji, “Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 393247, 12 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji and Z. Alizadeh, “Stability and superstability of ring homomorphisms on non-Archimedean Banach algebras,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 123656, 10 pages, 2011. View at Google Scholar · View at Zentralblatt MATH - M. B. Savadkouhi, M. E. Gordji, J. M. Rassias, and N. Ghobadipour, “Approximate ternary Jordan derivations on Banach ternary algebras,”
*Journal of Mathematical Physics*, vol. 50, no. 4, Article ID 042303, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Park and M. E. Gordji, “Approximate ternary Jordan derivations on Banach ternary algebras,”
*Journal of Mathematical Physics*, vol. 51, no. 4, Article ID 044102, 7 pages, 2010. View at Publisher · View at Google Scholar - M. Eshaghi Gordji, S. Abbaszadeh, and C. Park, “On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 153084, 26 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji, A. Ebadian, and S. Zolfaghari, “Stability of a functional equation deriving from cubic and quartic functions,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 801904, 17 pages, 2008. View at Google Scholar · View at Zentralblatt MATH - F. Rahbarnia, T. M. Rassias, R. Saadati, and G. Sadeghi, “Forti's approach in fixed point theory and the stability of a functional equation on metric and ultra metric spaces,”
*Journal of Computational Analysis and Applications*, vol. 13, no. 3, pp. 458–462, 2011. View at Google Scholar · View at Zentralblatt MATH - R. Saadati, S. M. Vaezpour, and C. Park, “The stability of the cubic functional equation in various spaces,”
*Mathematical Communications*, vol. 16, no. 1, pp. 131–145, 2011. View at Google Scholar · View at Zentralblatt MATH - R. Saadati, M. M. Zohdi, and S. M. Vaezpour, “Nonlinear
*ℒ*-random stability of an ACQ functional equation,”*Journal of Inequalities and Applications*, vol. 2011, Article ID 194394, 23 pages, 2011. View at Publisher · View at Google Scholar - R. Saadati and C. Park, “Non-Archimedian
*ℒ*-fuzzy normed spaces and stability of functional equations,”*Computers & Mathematics with Applications*, vol. 60, no. 8, pp. 2488–2496, 2010. View at Publisher · View at Google Scholar - M. M. Pourpasha, J. M. Rassias, R. Saadati, and S. M. Vaezpour, “A fixed point approach to the stability of Pexider quadratic functional equation with involution,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 839639, 18 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Gavruta and P. Gavruta, “On a problem of John M.Rassias concerning the stability in Ulam sense of Euler-Lagrange
equation,” in
*Functional Equations, Difference Inequalities*, pp. 47–53, Nova Sciences, 2010. View at Google Scholar - Z. Gajda and R. Ger, “Subadditive multifunctions and Hyers-Ulam stability,” in
*General Inequalities*, vol. 5 of*International Series of Numerical Mathematics*, Birkhäuser, Basel, Switzerland, 1987. View at Google Scholar · View at Zentralblatt MATH - Y. J. Cho, C. Park, and R. Saadati, “Functional inequalities in non-Archimedean Banach spaces,”
*Applied Mathematics Letters*, vol. 23, no. 10, pp. 1238–1242, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji and H. Khodaei,
*Stability of Functional Equations*, Lap Lambert Academic, 2010. - D. H. Hyers, G. Isac, and T. M. Rassias,
*Stability of Functional Equations in Several Variables*, Birkhäuser, Basel, Switzerland, 1998. - S.-M. Jung and P. K. Sahoo, “Stability of a functional equation for square root spirals,”
*Applied Mathematics Letters*, vol. 15, no. 4, pp. 435–438, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji, “Stability of a functional equation deriving from quartic and additive functions,”
*Bulletin of the Korean Mathematical Society*, vol. 47, no. 3, pp. 491–502, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji, “Stability of an additive-quadratic functional equation of two variables in F-spaces,”
*Journal of Nonlinear Science and its Applications*, vol. 2, no. 4, pp. 251–259, 2009. View at Google Scholar - M. Eshaghi Gordji, M. B. Ghaemi, and H. Majani, “Generalized Hyers-Ulam-Rassias theorem in Menger probabilistic normed spaces,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 162371, 11 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji, J. M. Rassias, and N. Ghobadipour, “Generalized Hyers-Ulam stability of generalized (
*n*,*k*)-derivations,”*Abstract and Applied Analysis*, vol. 2009, Article ID 437931, 8 pages, 2009. View at Google Scholar - M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, and S. Zolfaghari, “Solution and stability of a mixed type additive, quadratic, and cubic functional equation,”
*Advances in Difference Equations*, vol. 2009, Article ID 826130, 17 pages, 2009. View at Google Scholar · View at Zentralblatt MATH - M. E. Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi, “Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 417473, 14 pages, 2009. View at Google Scholar · View at Zentralblatt MATH - C. Park and A. Najati, “Homomorphisms and derivations in ${\text{C}}^{\ast}$-algebras,”
*Abstract and Applied Analysis*, vol. 2007, Article ID 80630, 12 pages, 2007. View at Google Scholar - J. Aczél and J. Dhombres,
*Functional Equations in Several Variables*, Cambridge University Press, Cambridge, UK, 1989. - P. Kannappan, “Quadratic functional equation and inner product spaces,”
*Results in Mathematics*, vol. 27, no. 3-4, pp. 368–372, 1995. View at Google Scholar · View at Zentralblatt MATH - Y. J. Cho, P. C. S. Lin, S. S. Kim, and A. Misiak,
*Theory of 2-Inner Product Spaces*, Nova Science, New York, NY, USA, 2001. - S. Gähler, “Lineare 2-normierte Räume,”
*Mathematische Nachrichten*, vol. 28, pp. 1–43, 1964. View at Publisher · View at Google Scholar - S. Gähler, “Über 2-banach-Räume,”
*Mathematische Nachrichten*, vol. 42, pp. 335–347, 1969. View at Publisher · View at Google Scholar - W.-G. Park, “Approximate additive mappings in 2-banach spaces and related topics,”
*Journal of Mathematical Analysis and Applications*, vol. 376, no. 1, pp. 193–202, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. G. White,, “2-banach spaces,”
*Mathematische Nachrichten*, vol. 42, pp. 43–60, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH