Stability of an additive functional equation in the spaces of generalized functions
© Lee; licensee Springer. 2011
Received: 24 July 2011
Accepted: 2 November 2011
Published: 2 November 2011
We reformulate the following additive functional equation with n-independent variables
as the equation for the spaces of generalized functions. Making use of the fundamental solution of the heat equation we solve the general solutions and the stability problems of this equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the regularizing functions, we extend these results to the space of distributions.
2000 MSC: 39B82; 46F05.
for all x, y ∈ ℝ. It is well-known that every measurable solution of (1.1) is of the form f(x) = ax for some constant a. In 1941, Hyers proved the stability theorem of (1.1) as follows:
for all x ∈ E 1.
The above stability theorem was motivated by Ulam . Forti  noticed that the theorem of Hyers is still true if E 1 is replaced by an arbitrary semigroup. In 1978, Rassias  generalized Hyers' result to the unbounded Cauchy difference. Thereafter, many authors studied the stability problems of (1.1) in various settings (see [5–7]).
was proposed by Nakmahachalasint , where n is a positive integer with n > 1. He proved that (1.2) is equivalent to (1.1). For that reason, we say that (1.2) is a generalization of the Cauchy functional equation. The stability theorem of (1.2) was also proved.
Here, ○ denotes the pullback of generalized functions and the inequality ||v|| ≤ ε in (1.4) means that for all test functions φ.
where and μ is a bounded measurable function such that . Subsequently, in Section 3, these results are extended to the space .
2. Stability in
We first introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the m-dimensional notations, |α| = α 1 + ⋯ + α m , α! = α 1! ⋯ α m !, and , for ζ = (ζ 1, ..., ζ m ) ∈ ℝ m , , where ℕ0 is the set of non-negative integers and .
for all . The set of all tempered distributions is denoted by .
Note that tempered distributions are generalizations of L p -functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform. Imposing the growth condition on ||·|| α,β in (2.1) a new space of test functions has emerged as follows:
for some positive constants A, B depending only on φ. The strong dual of , denoted by , is called the Fourier hyperfunction.
of the heat kernel is very useful to convert Equation (1.3) into the classical functional equation defined on upper-half plane. We also use the following famous result, so-called heat kernel method, which states as follows:
Conversely, every C ∞ -solution U(x, t) of the heat equation satisfying the growth condition (2.4) 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 (2.4) is replaced by the following:
We are now going to solve the general solutions and the stability problems of (1.2) in the spaces of and . Here, we need the following lemma.
where , .
where , . □
From the above lemma, we can find the general solutions of (1.2) in the spaces of and . Taking the inclusions of (2.3) into account, it suffices to consider the space .
This completes the proof. □
From the above theorem, we have the general solution of (1.1) in the spaces of and immediately.
We are going to prove the stability theorem of (1.2) in the spaces of and as follows:
Now inequality (2.18) implies that u - a · x belongs to (L 1)' = L ∞. Thus, all the solution u in can be written uniquely in the form u = a · x + h(x), where . □
From the above theorem, we shall prove the stability theorem of (1.1) in the spaces of and as follows:
3. Stability in
Making use of the regularizing functions we can find the general solution of (1.2) in the space as follows:
This completes the proof. □
In a similar manner, we have the following corollary immediately.
Using the regularizing functions, Chung  extended the stability theorem of the Cauchy functional equation (1.1) to the space . Similarly, we shall extend the stability theorem of (1.2) mentioned in the previous section to the space .
Inequality (3.13) implies that h(x) : = u - g(x) belongs to (L 1)' = L ∞. Thus, we conclude that . □
From the above theorem, we have the following corollary.
- Hyers DH: On the stability of the linear functional equation. Proc Nat Acad Sci USA 1941, 27: 222-224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
- Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.Google Scholar
- Forti GL: The stability of homomorphisms and amenability, with applications to functional equations. Abh Math Sem Univ Hamburg 1987, 57: 215-226. 10.1007/BF02941612MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: 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 ArticleMathSciNetMATHGoogle Scholar
- Gajda Z: On stability of additive mappings. Int J Math Math Sci 1991, 14: 431-434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
- Isac G, Rassias ThM: On the Hyers-Ulam stability of ψ -additive mappings. J Approx Theory 1993, 72: 131-137. 10.1006/jath.1993.1010MathSciNetView ArticleMATHGoogle Scholar
- Skof F: On the approximation of locally δ -additive mappings (Italian). Atti Accad Sci Torino Cl Sci Fis Mat Natur 1983, 117: 377-389.MathSciNetMATHGoogle Scholar
- Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston; 1998.View ArticleGoogle Scholar
- Kannappan Pl: Functional Equations and Inequalities with Applications. Springer, New York; 2009.View ArticleGoogle Scholar
- Aczél J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.View ArticleGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Co., Inc., River Edge; 2002.Google 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 ArticleMATHGoogle Scholar
- Jun K-W, Kim H-M: On the stability of an n -dimensional quadratic and additive functional equation. Math Inequal Appl 2006, 9: 153-165.MathSciNetMATHGoogle Scholar
- Nakmahachalasint P: Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities of an additive functional equation in several variables. Int J Math Math Sci 2007, 2007: 6. Art. ID 13437MathSciNetMATHGoogle Scholar
- Chung J, Lee S: Some functional equations in the spaces of generalized functions. Aequationes Math 2003, 65: 267-279. 10.1007/s00010-003-2657-yMathSciNetView ArticleMATHGoogle 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 ArticleMATHGoogle 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 ArticleMATHGoogle Scholar
- Lee Y-S, Chung S-Y: The stability of a general quadratic functional equation in distributions. Publ Math Debrecen 2009, 74: 293-306.MathSciNetMATHGoogle Scholar
- Lee Y-S, Chung S-Y: Stability of quartic functional equations in the spaces of generalized functions. Adv Difference Equ 2009, 2009: 16. Art. ID 838347MathSciNetView ArticleMATHGoogle Scholar
- Hörmander L: The Analysis of Linear Partial Differential Operators I. Springer, Berlin; 1983.Google Scholar
- Schwartz L: Théorie des Distributions. Hermann, Paris; 1966.Google 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 ArticleMATHGoogle Scholar
- Matsuzawa T: A calculus approach to hyperfunctions III. Nagoya Math J 1990, 118: 133-153.MathSciNetMATHGoogle 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 ArticleMATHGoogle Scholar
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