- Research Article
- Open Access

# Stability of Trigonometric Functional Equations in Generalized Functions

- Jeongwook Chang
^{1}Email author and - Jaeyoung Chung
^{2}

**2010**:801502

https://doi.org/10.1155/2010/801502

© J. Chang and J. Chung. 2010

**Received:**30 June 2009**Accepted:**10 January 2010**Published:**19 January 2010

## Abstract

We consider the Hyers-Ulam stability of a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions.

## Keywords

- Functional Equation
- Tensor Product
- Measurable Function
- Bounded Function
- Heat Kernel

## 1. Introduction

The Hyers-Ulam stability of functional equations go back to 1940 when Ulam proposed the following problem [1]:

*Let*
*be a mapping from a group*
*to a metric group*
*with metric*
*such that*

*Then does there exist a group homomorphism*
*and*
*such that*

*for all*
?

where . Following the formulations as in [6, 20–22], we generalize the classical stability problems of above functional equations to the spaces of generalized functions as

where , and denote the pullback and the tensor product of generalized functions, respectively, and denotes the space of bounded measurable functions on .

We prove as results that if generalized function satisfies (1.5), then satisfies one of the followings

(i) and is arbitrary;

(ii) and are bounded measurable functions;

(iii) ;

(iv) , ,

for some , and a bounded measurable function .

Also if generalized function satisfies (1.6), then satisfies one of the followings:

(i) and are bounded measurable functions,

(ii) , , .

## 2. Generalized Functions

For the spaces of tempered distributions , we refer the reader to [23–25]. Here we briefly introduce the spaces of Gelfand generalized functions and Fourier hyperfunctions. Here we use the following notations: , , , , and , for , , where is the set of nonnegative integers and .

Definition 2.1.

The topology on the space
is defined by the seminorms
in the left-hand side of (2.1) and the elements of the dual space
of
are called *Gelfand-Shilov generalized functions*. In particular, we denote
by
and call its elements *Fourier hyperfunctions*.

It is known that if and , the space consists of all infinitely differentiable functions on that can be continued to an entire function on satisfying

for some .

It is well known that the following topological inclusions hold:

We refer the reader to [24, chapter V-VI], for tensor products and pullbacks of generalized functions.

## 3. Stability of the Equations

In view of (2.2), it is easy to see that the -dimensional heat kernel

belongs to the Gelfand-Shilov space for each . Thus the convolution is well defined for all . It is well known that is a smooth solution of the heat equation in and in the sense of generalized functions, that is, for every ,

We call
*the Gauss transform* of
. Let
be a semigroup and
be the field of complex numbers. A function
is said to be *additive* provided
, and
is said to be *exponential* provided
.

We first discuss the solutions of the corresponding trigonometric functional equations in the space of Gelfand generalized functions. As a consequence of the result [6, 26], we have the following.

Lemma 3.1.

are equal, respectively, to the smooth solution of corresponding classical functional equations (1.3) and (1.4).

Remark 3.2.

We refer the reader to Aczél [27, page 180] and Aczél and Dhombres [28, pages 209–217] for the general solutions and measurable solutions of (1.3) and (1.4).

For the proof of the stability of (1.5), we need the following lemma.

Lemma 3.3.

for all .

Also the inequality (3.5) together with (3.4) implies one of the followings;

(i) , : arbitrary;

(ii) and are bounded functions;

(iii) and where , , with and is a bounded function.

Proof.

where , and .

Again considering (3.11) as a function of and for all fixed , we have .

for all . From (3.12) and (3.23), we have (iii). This completes the proof.

Remark 3.4.

for some and bounded solution of the heat equation.

Theorem 3.5.

Let satisfy (1.5). Then and satisfy one of the followings

(i) and is arbitrary;

(ii) and are bounded measurable functions;

(iii) , ;

(iv) , ,

for some , and a bounded measurable function .

Proof.

The nontrivial solutions of (3.27) are given by (iv) or which is included in case (iii). This completes the proof.

Now we prove the stability of (1.6).

Lemma 3.6.

for all .

Also the inequality (3.29) together with (3.28) implies one of the followings

(i) and are bounded functions;

(ii) is bounded; is an exponential;

(iii) for some bounded exponential ;

(iv) , , where , is a bounded exponential and is a bounded function.

Proof.

where and .

which gives (iv). This completes the proof.

Theorem 3.7.

Let satisfy (1.6). Then and satisfy one of the followings

(i) and are bounded measurable functions;

(ii) , ,

where .

Proof.

for all , where is a bounded exponential. Using the continuity of , it follows from (3.43) that is bounded in and so is , which implies that both and are bounded measurable functions. For the case (iv) since , are unbounded continuous, is unbounded continuous, and . On the other hand, it follows from (3.28) that , which occurs only when . Thus both and are bounded in and are bounded measurable functions.

By Lemma 3.1, the nonconstant solution of (3.44) is given by , for some . This completes the proof.

Remark 3.8.

Taking the growth of as into account, or only when for some . Thus the Theorems 3.5 and 3.7 are reduced to the followings.

Corollary 3.9.

Let or satisfy (1.5). Then and satisfy one of the followings

(i) and is arbitrary;

(ii) and are bounded measurable functions;

(iii) , only ,

for some , , and a bounded measurable function .

Corollary 3.10.

Let or satisfy (1.6). Then and are bounded measurable functions.

## Declarations

### Acknowledgments

The authors express their deep thanks to the referee for valuable comments on the paper, in particular, introducing the earlier results of Bourgin [3, 4] and Aoki [5]. This work was supported by the Korea Research Foundation Grant (KRF) Grant funded by the Korea Government (MEST) (no. 2009-0063887).

## Authors’ Affiliations

## References

- Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8*. Interscience, New York, NY, USA; 1960:xiii+150.Google Scholar - Hyers DH: On the stability of the linear functional equation.
*Proceedings of the National Academy of Sciences of the United States of America*1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar - Bourgin DG: Multiplicative transformations.
*Proceedings of the National Academy of Sciences of the United States of America*1950, 36: 564–570. 10.1073/pnas.36.10.564MathSciNetView ArticleMATHGoogle Scholar - Bourgin DG: Approximately isometric and multiplicative transformations on continuous function rings.
*Duke Mathematical Journal*1949, 16: 385–397. 10.1215/S0012-7094-49-01639-7MathSciNetView ArticleMATHGoogle Scholar - Aoki T: On the stability of the linear transformation in Banach spaces.
*Journal of the Mathematical Society of Japan*1950, 2: 64–66. 10.2969/jmsj/00210064MathSciNetView ArticleMATHGoogle Scholar - Baker JA: On a functional equation of Aczél and Chung.
*Aequationes Mathematicae*1993, 46(1–2):99–111. 10.1007/BF01834001MathSciNetView ArticleMATHGoogle Scholar - Baker JA: The stability of the cosine equation.
*Proceedings of the American Mathematical Society*1980, 80(3):411–416. 10.1090/S0002-9939-1980-0580995-3MathSciNetView ArticleMATHGoogle Scholar - Székelyhidi L: The stability of d'Alembert-type functional equations.
*Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum*1982, 44(3–4):313–320.MathSciNetMATHGoogle Scholar - Czerwik S:
*Stability of Functional Equations of Ulam-Hyers-Rassias Type*. Hadronic Press, Palm Harbor, Fla, USA; 2003.MATHGoogle Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar - Jun KW, Kim HM: Stability problem for Jensen-type functional equations of cubic mappings.
*Acta Mathematica Sinica, English Series*2006, 22(6):1781–1788. 10.1007/s10114-005-0736-9MathSciNetView ArticleMATHGoogle Scholar - Kim GH: The stability of d'Alembert and Jensen type functional equations.
*Journal of Mathematical Analysis and Applications*2007, 325(1):237–248. 10.1016/j.jmaa.2006.01.062MathSciNetView ArticleMATHGoogle Scholar - Kim GH: On the stability of the Pexiderized trigonometric functional equation.
*Applied Mathematics and Computation*2008, 203(1):99–105. 10.1016/j.amc.2008.04.011MathSciNetView ArticleMATHGoogle Scholar - Kim GH, Lee YW: Boundedness of approximate trigonometric functions.
*Applied Mathematics Letters*2009, 22(4):439–443. 10.1016/j.aml.2008.06.013MathSciNetView ArticleMATHGoogle Scholar - Park CG: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras.
*Bulletin des Sciences Mathématiques*2008, 132(2):87–96.View ArticleMATHMathSciNetGoogle Scholar - Rassias ThM: On the stability of functional equations in Banach spaces.
*Journal of Mathematical Analysis and Applications*2000, 251(1):264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM: On the stability of the linear mapping in Banach spaces.
*Proceedings of the American Mathematical Society*1978, 72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar - Székelyhidi L: The stability of the sine and cosine functional equations.
*Proceedings of the American Mathematical Society*1990, 110(1):109–115.MathSciNetView ArticleMATHGoogle Scholar - Tyrala I: The stability of d'Alembert's functional equation.
*Aequationes Mathematicae*2005, 69(3):250–256. 10.1007/s00010-004-2741-yMathSciNetView ArticleMATHGoogle Scholar - Chung J: A distributional version of functional equations and their stabilities.
*Nonlinear Analysis: Theory, Methods & Applications*2005, 62(6):1037–1051. 10.1016/j.na.2005.04.016MathSciNetView ArticleMATHGoogle Scholar - Chung J: Stability of functional equations in the spaces of distributions and hyperfunctions.
*Journal of Mathematical Analysis and Applications*2003, 286(1):177–186. 10.1016/S0022-247X(03)00468-2MathSciNetView ArticleMATHGoogle Scholar - Chung J: Distributional method for the d'Alembert equation.
*Archiv der Mathematik*2005, 85(2):156–160. 10.1007/s00013-005-1234-0MathSciNetView ArticleMATHGoogle Scholar - Gelfand IM, Shilov GE:
*Generalized Functions. II*. Academic Press, New York, NY, USA; 1968:x+261.Google Scholar - Hörmander L:
*The Analysis of Linear Partial Differential Operators. I, Grundlehren der Mathematischen Wissenschaften*.*Volume 256*. Springer, Berlin, Germany; 1983:ix+391.Google Scholar - Schwartz L:
*Théorie des Distributions*. Hermann, Paris, France; 1966:xiii+420.Google Scholar - Fenyö I: Über eine Lösungsmethode gewisser Funktionalgleichungen.
*Acta Mathematica Academiae Scientiarum Hungaricae*1956, 7: 383–396. 10.1007/BF02020533MathSciNetView ArticleMATHGoogle Scholar - Aczél J:
*Lectures on Functional Equations and Their Applications, Mathematics in Science and Engineering*.*Volume 19*. Academic Press, New York, NY, USA; 1966:xx+510.Google Scholar - Aczél J, Dhombres J:
*Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications*.*Volume 31*. Cambridge University Press, New York, NY, USA; 1989:xiv+462.View ArticleMATHGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications*.*Volume 34*. Birkhäuser, Boston, Mass, USA; 1998:vi+313.MATHGoogle Scholar - Widder DV:
*The Heat Equation, Pure and Applied Mathematics*.*Volume 6*. Academic Press, New York, NY, USA; 1975:xiv+267.Google Scholar

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