- Open Access
On the stability of a mixed type functional equation in generalized functions
© Lee; licensee Springer. 2012
- Received: 18 November 2011
- Accepted: 16 February 2012
- Published: 16 February 2012
We reformulate the following mixed type quadratic and additive functional equation with n-independent variables
as the equation for the spaces of generalized functions. Using the fundamental solution of the heat equation, we solve the general solution and prove the Hyers-Ulam stability of this equation in the spaces of tempered distributions and Fourier hyperfunctions.
Mathematics Subject Classification 2000: 39B82; 39B52.
- quadratic functional equation
- additive functional equation
- heat kernel
- Gauss transform
In 1940, Ulam  raised a question concerning the stability of group homomorphisms as follows:
Let G1 be a group and let G2 be a metric group with the metric d(·,·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1, then there exists a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x ∈ G1?
for all x ∈ X. For this reason, equation (1.1) is called the mixed type quadratic and additive functional equation. We refer to [9–14] for the stability results of other mixed type functional equations.
Here ○ denotes the pullback of generalized functions and the inequality ||v|| ≤ ε in (1.3) means that for all test functions φ.
where µ is a bounded measurable function such that
In this section, we introduce the spaces of tempered distributions and Fourier hyperfunctions. We first consider the space of rapidly decreasing functions which is a test function space of tempered distributions.
for all nonnegative integers α, β.
In other words, φ(x) as well as its derivatives of all orders vanish at infinity faster than the reciprocal of any polynomial. For that reason, we call the element of as the rapidly decreasing function. It can be easily shown that the function φ(x) = exp(−ax2), a > 0, belongs to , but ψ(x) = (1 + x2)−1 is not a member of . Next we consider the space of tempered distributions which is a dual space of .
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, but not all distributions have one. Imposing the growth condition on || · ||α,βin (2.1) a new space of test functions has emerged as follows.
for some positive constants A, B depending only on φ.
for some constants h, k > 0.
Definition 2.4.  The strong dual space of is called the Fourier hyperfunctions. We denote the Fourier hyperfunctions by .
holds for convolution. It is useful to convert equation (1.2) into the classical functional equation defined on upper-half plane. We also use the following famous result 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 initial values of solutions of the heat equation as a special case of the results as in . In this case, the condition (i) in the above theorem is replaced by the following:
for all x1, . . . , x n ∈ ℝ, t1, . . . , t n > 0. We here need the following lemma which will be crucial role in the proof of main theorem.
for all x ∈ ℝ, t > 0.
where c = c1 + c2.
Conversely, if f (x, t) = ax2 + bx + c for some a, b, c ∈ ℂ, then it is obvious that f satisfies equation (3.1). □
According to the above lemma, we solve the general solution of (1.2) in the space of (or , resp.) as follows.
for some a, b ∈ ℂ.
for some a, b, c ∈ ℂ. Letting t → 0+ in (3.8), we finally obtain the general solution of (1.2). □
In this section, we are going to state and prove the Hyers-Ulam stability of (1.3) in the space of (or , resp.).
for all x ∈ ℝ, t > 0.
for all x ∈ ℝ, t > 0, where
for all x ∈ ℝ, t > 0.
for some b, c2 ∈ ℂ.
for all x ∈ ℝ, t > 0, where c = c1 + c2. □
From the above lemma we immediately prove the Hyers-Ulam stability of (1.3) in the space of (or , resp.) as follows.
for all x ∈ ℝ, t > 0. Letting t → 0+ in (4.18) finally we have the stability result (4.17). □
where µ is a bounded measurable function such that .
- Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.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 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
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Co., Inc., River Edge; 2002.Google Scholar
- Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston; 1998.View ArticleGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer Optimization and Its Applications. Springer, New York; 2011.View ArticleGoogle Scholar
- Kannappan Pl: Functional Equations and Inequalities with Applications. Springer, New York; 2009.View ArticleGoogle Scholar
- Towanlong W, Nakmahachalasint P: An n -dimensional mixed-type additive and quadratic functional equation and its stability. ScienceAsia 2009, 35: 381–385. 10.2306/scienceasia1513-1874.2009.35.381View ArticleMATHGoogle Scholar
- Eshaghi Gordji M, Savadkouhi MB: Stability of mixed type cubic and quartic functional equations in random normed spaces. J Inequal Appl 2009, 2009: 9. Article ID 527462View ArticleMathSciNetMATHGoogle Scholar
- Eshaghi Gordji M, Kaboli Gharetapeh S, Moslehian MS, Zolfaghari S: Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation. In Nonlinear Analysis and Variational Problems. Volume 35. Springer Optimization and Its Applications. Springer, New York; 2010:65–80. 10.1007/978-1-4419-0158-3_6View ArticleGoogle 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
- Kannappan Pl, Sahoo PK: On generalizations of the Pompeiu functional equation. Int J Math Math Sci 1998, 21: 117–124. 10.1155/S0161171298000155MathSciNetView ArticleMATHGoogle 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 Diff Equ Appl 2010, 16: 773–788.MathSciNetView ArticleMATHGoogle Scholar
- Wang L, Liu B, Bai R: Stability of a mixed type functional equation on multi-Banach spaces: a fixed point approach. Fixed Point Theory Appl 2010, 2010: 9. Article ID 283827MathSciNetMATHGoogle 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 ArticleMATHGoogle 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
- Lee Y-S: Stability of a quadratic functional equation in the spaces of generalized functions. J Inequal Appl 2008, 2008: 12. Article ID 210615MathSciNetMATHGoogle 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 Diff 2009, 2009: 16. Article ID 838347MathSciNetMATHGoogle 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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