Stability of quadratic functional equations in tempered distributions
© Lee; licensee Springer 2012
Received: 12 September 2011
Accepted: 6 August 2012
Published: 16 August 2012
We reformulate the following quadratic functional equation:
as the equation for generalized functions. Using the fundamental solution of the heat equation, we solve the general solution of this equation and prove the Hyers-Ulam stability in the spaces of tempered distributions and Fourier hyperfunctions.
In 1940, Ulam  raised a question concerning the stability of group homomorphisms as follows: The case of approximately additive mappings was solved by Hyers  under the assumption that is a Banach space. In 1978, Rassias  generalized Hyers’ result to the unbounded Cauchy difference.
Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?
where k is a fixed positive integer. They proved the Hyers-Ulam-Rassias stability of this equation in Banach spaces. Wang  considered the intuitionistic fuzzy stability of (1.2) by using the fixed-point alternative. Saadati and Park  proved the Hyers-Ulam-Rassias stability of (1.2) in non-Archimedean -fuzzy normed spaces.
Here, ∘ denotes the pullback of generalized functions and the inequality in (1.4) means that for all test functions φ. We refer to  for pullbacks and to [2, 7–11] for more details of the spaces of generalized functions.
In this section, we introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the n-dimensional notations. If , where is the set of nonnegative integers, then , . For , we denote and .
2.1 Tempered distributions
We present a very useful space of test functions for the tempered distributions as follows.
for all . The set of all tempered distributions is denoted by .
for all , where is the Fourier transform of φ.
Note that tempered distributions are generalizations of -functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform.
2.2 Fourier hyperfunctions
Imposing the growth condition on in (2.1) Sato and Kawai introduced the new space of test functions for the Fourier hyperfunctions as follows.
Definition 2.2 ()
for some positive constants A, B depending only on φ. The strong dual of , denoted by , is called the Fourier hyperfunction.
for some .
3 General solution in generalized functions
of the heat kernel is very useful to convert Eq. (1.3) into the classical functional equation defined on upper-half plane. We also use the following famous result, the so-called heat kernel method, which is stated as follows.
Theorem 3.1 ()
- (i)There exist positive constants C, M, and N such that(3.1)
- (ii)as in the sense that for every ,
Conversely, every -solution of the heat equation satisfying the growth condition (3.1) can be uniquely expressed as for some .
Similarly, we can represent Fourier hyperfunctions as a special case of the results as in . In this case, the estimate (3.1) is replaced by the following:
Here, we need the following lemma to solve the general solution of (1.3).
for some .
for all , . □
As a direct consequence of the above lemma, we present the general solution of the quadratic functional equation (1.3) in the spaces of generalized functions.
for some .
as in (3.8). □
4 Stability in generalized functions
In this section, we are going to solve the stability problem of (1.4). For the case of in (1.4), the result is known as follows.
We here need the following lemma to solve the stability problem of (1.4).
for all , , . Taking the limit as , we conclude that for all , . □
We now state and prove the main theorem of this paper.
This completes the proof. □
- Aczél J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.View ArticleGoogle Scholar
- Baker JA: Distributional methods for functional equations. Aequ. Math. 2001, 62: 136–142. 10.1007/PL00000134View ArticleGoogle Scholar
- Borelli C, Forti GL: On a general Hyers-Ulam-stability result. Int. J. Math. Math. Sci. 1995, 18: 229–236. 10.1155/S0161171295000287MathSciNetView ArticleGoogle Scholar
- Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76–86. 10.1007/BF02192660MathSciNetView ArticleGoogle Scholar
- Chung J, Chung S-Y, Kim D:Une caractérisation de l’espace de Schwartz. C. R. Math. Acad. Sci. Paris 1993, 316: 23–25.MathSciNetGoogle Scholar
- Chung J, Chung S-Y, Kim D: A characterization for Fourier hyperfunctions. Publ. Res. Inst. Math. Sci. 1994, 30: 203–208. 10.2977/prims/1195166129MathSciNetView ArticleGoogle Scholar
- Chung J: Stability of functional equations in the spaces of distributions and hyperfunctions. J. Math. Anal. Appl. 2003, 286: 177–186. 10.1016/S0022-247X(03)00468-2MathSciNetView ArticleGoogle Scholar
- Chung J, Lee S: Some functional equations in the spaces of generalized functions. Aequ. Math. 2003, 65: 267–279. 10.1007/s00010-003-2657-yMathSciNetView ArticleGoogle Scholar
- Chung J, Chung S-Y, Kim D: The stability of Cauchy equations in the space of Schwartz distributions. J. Math. Anal. Appl. 2004, 295: 107–114. 10.1016/j.jmaa.2004.03.009MathSciNetView ArticleGoogle Scholar
- Chung J: A distributional version of functional equations and their stabilities. Nonlinear Anal. 2005, 62: 1037–1051. 10.1016/j.na.2005.04.016MathSciNetView ArticleGoogle Scholar
- Chung J: Stability of approximately quadratic Schwartz distributions. Nonlinear Anal. 2007, 67: 175–186. 10.1016/j.na.2006.05.005MathSciNetView ArticleGoogle Scholar
- Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 59–64. 10.1007/BF02941618MathSciNetView ArticleGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge; 2002.View ArticleGoogle Scholar
- Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleGoogle Scholar
- Hörmander L: The Analysis of Linear Partial Differential Operators I. Springer, Berlin; 1983.Google Scholar
- Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
- Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston; 1998.View ArticleGoogle Scholar
- Jun K-W, Lee Y-H: On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality. Math. Inequal. Appl. 2001, 4: 93–118.MathSciNetGoogle Scholar
- Kannappan P: Functional Equations and Inequalities with Applications. Springer, Berlin; 2009.View ArticleGoogle Scholar
- Kim KW, Chung S-Y, Kim D: Fourier hyperfunctions as the boundary values of smooth solutions of heat equations. Publ. Res. Inst. Math. Sci. 1993, 29: 289–300. 10.2977/prims/1195167274MathSciNetView ArticleGoogle Scholar
- Lee JR, An JS, Park C: On the stability of quadratic functional equations. Abstr. Appl. Anal. 2008., 2008: Article ID 628178Google Scholar
- Lee Y-S, Chung S-Y: Stability of cubic functional equation in the spaces of generalized functions. J. Inequal. Appl. 2007., 2007: Article ID 79893Google Scholar
- Matsuzawa T: A calculus approach to hyperfunctions III. Nagoya Math. J. 1990, 118: 133–153.MathSciNetGoogle Scholar
- Najati A, Eskandani GZ: A fixed point method to the generalized stability of a mixed additive and quadratic functional equation in Banach modules. J. Differ. Equ. Appl. 2010, 16: 773–788.MathSciNetView ArticleGoogle Scholar
- Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar
- Saadati R, Park C:Non-Archimedian -fuzzy normed spaces and stability of functional equations. Comput. Math. Appl. 2010, 60: 2488–2496. 10.1016/j.camwa.2010.08.055MathSciNetView ArticleGoogle Scholar
- Sahoo PK, Kannappan P: Introduction to Functional Equations. CRC Press, Boca Raton; 2011.Google Scholar
- Schwartz L: Théorie des distributions. Hermann, Paris; 1966.Google Scholar
- Skof F: Local properties and approximation of operators. Rend. Semin. Mat. Fis. Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar
- Trif T: On the stability of a general gamma-type functional equation. Publ. Math. (Debr.) 2002, 60: 47–61.MathSciNetGoogle Scholar
- Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.Google Scholar
- Wang L: Intuitionistic fuzzy stability of a quadratic functional equation. Fixed Point Theory Appl. 2010., 2010: Article ID 107182Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.