- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 415053, 8 pages

http://dx.doi.org/10.1155/2013/415053

## Solution and Stability of the Multiquadratic Functional Equation

^{1}Department of Mathematics, Zhejiang University, Hangzhou 310027, China^{2}Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan^{3}College of Mathematics and Information Science, Hebei Normal University and Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang 050024, China^{4}Department of Information Management, Yuan Ze University, Chungli 32003, Taiwan

Received 25 August 2013; Accepted 10 September 2013

Academic Editor: Chi-Ming Chen

Copyright © 2013 Xiaopeng Zhao 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 consider the multiquadratic functional equation. We establish its general solution and provide a characterization for this functional equation. Finally, we prove the Hyers-Ulam-Rassias stability of this functional equation.

#### 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. The stability of a functional equation originated from a question raised by Ulam: “when is it true that the solution of an equation differing slightly from a given one must of necessity be close to the solution of the given equation?” This question was solved by Hyers [2] in the case of the approximately additive functions between Banach spaces. In 1978, Rassias [3] provided a generalized version of Hyers’ result by allowing the Cauchy difference to be unbounded. The paper of Rassias [3] has provided a lot of influence in the development of the stability of functional equations, and this new concept is known as generalized Hyers-Ulam-Rassias stability or Hyers-Ulam-Rassias stability. Since then, the stability problems have been widely studied and extensively developed by many authors for a number of functional equations; see, for example, [4–10] and the books [11–14].

The functional equation is called the quadratic functional equation, and every solution of the quadratic functional equation is said to be a quadratic function. It is well known that a quadratic function between vector spaces can be expressed by a symmetric biadditive (i.e., additive for each fixed one variable) function . On the other hand, the stability problem for the quadratic functional equation has been studied by many mathematicians under various degrees of generality imposed on the equation or on the underlying space; see, for example, [15–20] and the references therein.

In [21], Park and Bae obtained the general solution and the generalized Hyers-Ulam-Rassias stability of the biquadratic functional equation. Let and be vector spaces. Recall from [21] that a mapping is called biquadratic if satisfies the system of equations for all ; that is, is quadratic for each fixed one variable.

A general version of the biquadratic functional equation is the multiquadratic functional equation. Recall from [22] that a mapping , where is a commutative group, is a linear space, and is an integer, is called multiquadratic if it is quadratic in each variable. On the other hand, for more details about the multiadditive (resp., the multi-Jensen mappings) (i.e., mappings satisfying Cauchy’s (resp., Jensen’s) functional equation in each variable) and the stability for them, one can see [23–28] and the references given there.

The stability of the multiquadratic functional equation was also studied by some authors. For example, Park [29] proved the stability of the multiquadratic functional equation in Banach spaces. Ciepliński [22] proved the stability of this functional equation in complete non-Archimedean spaces as well as in Banach spaces but using the fixed point method. However, to our knowledge, not many results are known about the solution of this functional equation.

In the present paper, we establish the general solution of the multiquadratic functional equation and provide a sufficient and necessary condition for a mapping to be multiquadratic. Finally, we prove its Hyers-Ulam-Rassias stability.

#### 2. General Solution

Throughout this section, let and be vector spaces, and let be a positive integer. We begin with the following useful proposition.

Proposition 1 (see [11]). *A function is quadratic if and only if there exists a unique symmetric biadditive function such that for any . The biadditive function is given by
*

In the following, we give the general solution of the multiquadratic functional equation.

Theorem 2. *A mapping is multiquadratic if and only if there exists a multiadditive mapping such that
**
for all , and satisfies the following symmetric condition
**
for all , where and . Moreover, the mapping is given by
**
where , , .*

*Proof. *We prove this theorem by using induction on . Clearly, Theorem 2 is true for thanks to Proposition 1. Now, we assume that the present theorem is true for some , and we consider the case for .

We first assume that there exists a multiadditive mapping such that
for all , and satisfies the following symmetric condition:
for all , where and . Then, for each , we have that

for all . Thus, is multiquadratic.

Conversely, we assume that is a multiquadratic function. We need to find the desired multiadditive function . For this, we give the following notations.

For each fixed , define the mapping by

Then is a multiquadratic mapping (as is multiquadratic). By induction, we let denote the corresponding multiadditive mapping for ; that is, satisfies the symmetric condition (5) and
for all . Moreover, the mapping is given by
for all , where and .

On the other hand, for any fixed elements , define by

for all . It can be verified that is a quadratic mapping. Thus, it follows from Proposition 1 that there exists a symmetric biadditive mapping such that

for all . The mapping is given by

for all .

Now, we define the mapping by
for all , , . In the following, we will show that is the desired function for . First, we show that is multiadditive. Indeed, by the definition of (see (16)) and noting that for any the function is multiadditive, one can obtain that for each ,
for all . Moreover, by the definition of in (16) and the notations we gave in (13) and (15), we have that
for all . Similarly, we can see that is additive in the other variables. Thus, we have shown that is multiadditive.

Furthermore, since is multiquadratic, we obtain that
for all . Thus, by the definition of in (16) and the notations we gave in (10) and (11), one has
for all .

Now, we verify the expression of the mapping . By the definition of again and the notations we gave in (10) and (12), also noting that is multiquadratic, one can obtain that
for all , , .

Finally, we check the symmetric property of . Fix any , where and . Since is multiquadratic, it follows that is an even mapping in each variable. Then by (21), it is easy to verify that
Moreover, due to the symmetric property of and and from the definition of (see (16)) we can get
for each . So the desired symmetric property of is proved. Thus, we have shown that is the desired multiadditive mapping for the multiquadratic mapping . The proof is complete.

#### 3. A Characterization for the Multiquadratic Functional Equation

The following theorem provides a sufficient and necessary condition for a mapping to be multiquadratic.

Theorem 3. *Let be a commutative semigroup with the identity element , and let be a linear space. A mapping is multiquadratic if and only if
**
for all , .*

*Proof. *Assume that satisfies (24). Putting
in (24) we get , and consequently, we have . Next, fix , , and put , where , for . Then, by (24),
and thus . Continuing in this fashion, we obtain that for any with at least one component which is equal to .

Now, fix , and put for in (24). Then
and thus
which proves that is multiquadratic.

Conversely, we assume that is multiquadratic, and we prove (24) by mathematical induction. If is quadratic, then for all . So, (24) holds for . It is easy to verify that (24) holds for . Indeed,
for all . Assume that (24) holds for some positive integer . Then,
Thus, (24) holds for , and this completes the proof.

#### 4. Stability

In this section, we give two results on the stability of the multiquadratic functional equation. Throughout this section, let be a commutative semigroup with the identity element , and let be a Banach space.

Theorem 4. *Assume that for every , is a mapping such that for any **
If is a function satisfying
**
for all , , then for every there exists a multiquadratic mapping such that for any one has
**
For every the function is given by
**
for all .*

*Proof. *Fix , (where denotes the set of the positive integers) and . Putting in (32), we get
Hence
Dividing both sides of the above inequality by and replacing by , we obtain
and consequently for any nonnegative integers and with , we obtain
Therefore, it follows from (31) that is a Cauchy sequence. Since the space is complete, this sequence is convergent, and we define by (34). Putting , letting in (38), and using (31), we see that (33) holds.

Finally, fix also , , and notice that according to (32) we have
Next, fix , , and assume that (the same arguments apply to the case where ). Then, it follows from (32) that
Letting in the above two inequalities and using (31), we see that the mapping is multiquadratic.

Theorem 5. *Assume that is a mapping such that
**
for all . If is a function satisfying
**
for all and letting for any with one component which is equal to , then there exists a unique multiquadratic mapping such that
**
for all . The function is given by
**
for all .*

*Proof. *Fix and . Putting for in (42), we get
Dividing both sides of the above inequality by and replacing by for , we see that
and consequently for any nonnegative integers and with we obtain
Therefore, it follows from (41) that is a Cauchy sequence. Since the space is complete, this sequence is convergent, and we define by (44). Putting , taking in (47), and using (41), we can see that the inequality (43) holds.

Next, fix also , and note that according to (42) we have
Letting in the above inequality and using (41), we see that satisfies (24). By Theorem 3, we obtain that is multiquadratic.

Finally, assume that is another multiquadratic mapping satisfying (43). Fix . Since and are multiquadratic mappings, it is easy to verify that
Then, using (41) and (43), we have
hence letting we obtain .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

- S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience Publishers, New York, NY, USA, 1960. View at MathSciNet - 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, no. 4, pp. 222–224, 1941. View at MathSciNet - Th. 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 MathSciNet - S.-M. Jung, “Hyers-Ulam-Rassias stability of Jensen's equation and its application,”
*Proceedings of the American Mathematical Society*, vol. 126, no. 11, pp. 3137–3143, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - K.-W. Jun and H.-M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,”
*Journal of Mathematical Analysis and Applications*, vol. 274, no. 2, pp. 867–878, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - K.-W. Jun and H.-M. Kim, “Stability problem of Ulam for generalized forms of Cauchy functional equation,”
*Journal of Mathematical Analysis and Applications*, vol. 312, no. 2, pp. 535–547, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - G. H. Kim, “On the stability of the Pexiderized trigonometric functional equation,”
*Applied Mathematics and Computation*, vol. 203, no. 1, pp. 99–105, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - S. H. Lee, S. M. Im, and I. S. Hwang, “Quartic functional equations,”
*Journal of Mathematical Analysis and Applications*, vol. 307, no. 2, pp. 387–394, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - A. Najati and Th. M. Rassias, “Stability of a mixed functional equation in several variables on Banach modules,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1755–1767, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - J. Sikorska, “Exponential functional equation on spheres,”
*Applied Mathematics Letters*, vol. 23, no. 2, pp. 156–160, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - J. Aczél and J. Dhombres,
*Functional Equations in Several Variables*, Cambridge University Press, Cambridge, UK, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - S. Czerwik,
*Functional Equations and Inequalities in Several Variables*, World Scientific Publishing, River Edge, NJ, USA, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - D. H. Hyers, G. Isac, and Th. M. Rassias,
*Stability of Functional Equations in Several Variables*, Birkhäauser, Boston, Mass, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - Pl. Kannappan,
*Functional Equations and Inequalities with Applications*, Springer, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - P. W. Cholewa, “Remarks on the stability of functional equations,”
*Aequationes Mathematicae*, vol. 27, no. 1-2, pp. 76–86, 1984. View at Publisher · View at Google Scholar · View at MathSciNet - S. Czerwik, “On the stability of the quadratic mapping in normed spaces,”
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*, vol. 62, no. 1, pp. 59–64, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - S.-M. Jung and P. K. Sahoo, “Hyers-Ulam stability of the quadratic equation of Pexider type,”
*Journal of the Korean Mathematical Society*, vol. 38, no. 3, pp. 645–656, 2001. View at MathSciNet - J. R. Lee, S.-Y. Jang, C. Park, and D. Y. Shin, “Fuzzy stability of quadratic functional equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 412160, 16 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - M. S. Moslehian, K. Nikodem, and D. Popa, “Asymptotic aspect of the quadratic functional equation in multi-normed spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 355, no. 2, pp. 717–724, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - F. Skof, “Local properties and approximation of operators,”
*Rendiconti del Seminario Matematico e Fisico di Milano*, vol. 53, no. 1, pp. 113–129, 1983. View at Publisher · View at Google Scholar · View at MathSciNet - W.-G. Park and J.-H. Bae, “On a bi-quadratic functional equation and its stability,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 62, no. 4, pp. 643–654, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - K. Ciepliński, “On the generalized Hyers-Ulam stability of multi-quadratic mappings,”
*Computers & Mathematics with Applications*, vol. 62, no. 9, pp. 3418–3426, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - K. Ciepliński, “On multi-Jensen functions and Jensen difference,”
*Bulletin of the Korean Mathematical Society*, vol. 45, no. 4, pp. 729–737, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - K. Ciepliński, “Stability of the multi-Jensen equation,”
*Journal of Mathematical Analysis and Applications*, vol. 363, no. 1, pp. 249–254, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - K. Ciepliński, “Generalized stability of multi-additive mappings,”
*Applied Mathematics Letters*, vol. 23, no. 10, pp. 1291–1294, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - K. Ciepliński, “Stability of multi-additive mappings in non-Archimedean normed spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 373, no. 2, pp. 376–383, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - W. Prager and J. Schwaiger, “Multi-affine and multi-Jensen functions and their connection with generalized polynomials,”
*Aequationes Mathematicae*, vol. 69, no. 1-2, pp. 41–57, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - W. Prager and J. Schwaiger, “Stability of the multi-Jensen equation,”
*Bulletin of the Korean Mathematical Society*, vol. 45, no. 1, pp. 133–142, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - C.-G. Park, “Multi-quadratic mappings in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 131, no. 8, pp. 2501–2504, 2003. View at Publisher · View at Google Scholar · View at MathSciNet