fference EquationsVolume 2008, Article ID 147979, 11 pages doi:10.1155/2008/147979 Research Article A Functional Equation of Acz ´el and Chung in Generalized Functions Jae-Young Chung Dep
Trang 1fference Equations
Volume 2008, Article ID 147979, 11 pages
doi:10.1155/2008/147979
Research Article
A Functional Equation of Acz ´el and Chung in
Generalized Functions
Jae-Young Chung
Department of Mathematics, Kunsan National University,
Kunsan 573-701, South Korea
Correspondence should be addressed to Jae-Young Chung,jychung@kunsan.ac.kr
Received 1 October 2008; Revised 22 December 2008; Accepted 25 December 2008
Recommended by Patricia J Y Wong
We consider an n-dimensional version of the functional equations of Acz´el and Chung in the spaces
of generalized functions such as the Schwartz distributions and Gelfand generalized functions As
a result, we prove that the solutions of the distributional version of the equation coincide with those of classical functional equation
Copyrightq 2008 Jae-Young Chung 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
1 Introduction
In1, Acz´el and Chung introduced the following functional equation:
l
j1
f j
α j x β j y
m
k1
where f j , g k , h k : R → C and α j , β j ∈ R for j 1, , l, k 1, , m Under the
natural assumptions that{g1, , g m } and {h1, , h m } are linearly independent, and α j β j / 0,
α i β j / α j β i for all i / j, i, j 1, , l, it was shown that the locally integrable solutions of 1.1
are exponential polynomials, that is, the functions of the form
q
k1
where r k ∈ C and p k ’s are polynomials for all k 1, 2, , q.
Trang 2In this paper, we introduce the following n-dimensional version of the functional
equation1.1 in generalized functions:
l
j1
u j ◦ T jm
k1
where u j , v k , w k ∈ DRn resp., S1/2
1/2Rn, and ◦ denotes the pullback, ⊗ denotes the
tensor product of generalized functions, and T j x, y α j x β j y, α j α j,1 , , α j,n,
β j β j,1 , , β j,n , x x1, , x n , y y1, , y n , α j x α j,1 x1, , α j,n x n , β j y
β j,1 y1, , β j,n y n , j 1, , l As in 1, we assume that α j,p β j,p / 0 and α i,p β j,p / α j,p β i,p for
all p 1, , n, i / j, i, j 1, , l.
In2, Baker previously treated 1.3 By making use of differentiation of distributions which is one of the most powerful advantages of the Schwartz theory, and reducing1.3 to
a system of differential equations, he showed that, for the dimension n 1, the solutions
of 1.3 are exponential polynomials We refer the reader to 2 6 for more results using this method of reducing given functional equations to differential equations
In this paper, by employing tensor products of regularizing functions as in7,8, we consider the regularity of the solutions of1.3 and prove in an elementary way that 1.3 can
be reduced to the classical equation1.1 of smooth functions This method can be applied
to prove the Hyers-Ulam stability problem for functional equation in Schwartz distribution
7, 8 In the last section, we consider the Hyers-Ulam stability of some related functional equations For some elegant results on the classical Hyers-Ulam stability of functional equations, we refer the reader to6,9 21
2 Generalized functions
In this section, we briefly introduce the spaces of generalized functions such as the Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions Here we use the following notations:|x| x2
1 · · · x2
n,|α| α1 · · · α n , α! α1!, , α n !, x α x α1
1 , , x α n
n ,
and ∂ α ∂ α1
1 , , ∂ α n
n , for x x1, , x n ∈ Rn , α α1, , α n ∈ Nn
0, whereN0 is the set of
nonnegative integers and ∂ j ∂/∂x j
Definition 2.1 A distribution u is a linear functional on C∞c Rn of infinitely differentiable functions onRn with compact supports such that for every compact set K ⊂ Rn there exist
constants C and k satisfying
|α|≤k
for all ϕ ∈ C∞
c Rn with supports contained in K One denotes by DRn the space of the Schwartz distributions onRn
Definition 2.2 For given r, s ≥ 0, one denotes by Ss
rRn the space of all infinitely differentiable functions ϕx on Rn such that there exist positive constants h and k satisfying
x∈R n , α,β∈N n
x α ∂ β ϕ x
Trang 3The topology on the spaceSs
ris defined by the seminorms h,kin the left-hand side of2.2, and the elements of the dual spaceSs
r are called Gelfand-Shilov generalized functions In
particular, one denotesS1
1byFand calls its elements Fourier hyperfunctions.
It is known that if r > 0 and 0 ≤ s < 1, the space S s
rRn consists of all infinitely differentiable functions ϕx on Rnthat can be continued to an entire function onCnsatisfying
|ϕx iy| ≤ C exp− a|x| 1/r b|y| 1/1−s
2.3
for some a, b > 0.
It is well known that the following topological inclusions hold:
S1/2 1/2 → F, F→ S1/2
We briefly introduce some basic operations on the spaces of the generalized functions
Definition 2.3 Let u∈ DRn Then, the kth partial derivative ∂ k u of u is defined by
∂ k u, ϕ
−u, ∂ k ϕ
2.5
for k 1, , n Let f ∈ C∞Rn Then the multiplication fu is defined by
2.6
Definition 2.4 Let u j∈ DRn j , j 1, 2 Then, the tensor product u1⊗u2of u1and u2is defined by
u1⊗ u2, ϕ
x1, x2
u1,
u2, ϕ
x1, x2
, ϕ
x1, x2
∈ C∞
c
Rn1× Rn2
The tensor product u1⊗ u2belongs toDRn1× Rn2
Definition 2.5 Let u j ∈ DRn j , j 1, 2, and let f : R n1 → Rn2 be a smooth function such
that for each x∈ Rn1the derivative fx is surjective Then there exists a unique continuous linear map f∗ :DRn2 → DRn1 such that f∗u u ◦ f, when u is a continuous function One calls f∗u the pullback of u by f and simply is denoted by u ◦ f.
The differentiations, pullbacks, and tensor products of Fourier hyperfunctions and Gelfand generalized functions are defined in the same way as distributions For more details
of tensor product and pullback of generalized functions, we refer the reader to9,22
Trang 43 Main result
We employ a function ψ ∈ C∞Rn such that
ψ x ≥ 0 ∀x ∈ R n , supp ψ⊂ x∈ Rn :|x| ≤ 1 ,
Rn
ψ x dx 1.
3.1
Let u ∈ DRn and ψ t x : t −n ψ x/t, t > 0 Then, for each t > 0, u ∗ ψ t x :
sinceu ∗ ψ t x is a smooth function of x satisfying u ∗ ψ t x → u as t → 0in the sense
of distributions, that is, for every ϕ ∈ C∞
c Rn,
t→ 0
u ∗ ψ t
Theorem 3.1 Let u j , v k , w k ∈ DRn , j 1, , l, k 1, , m, be a solution of 1.3, and both {v1, , v m } and {w1, , w m } are linearly independent Then, u j f j , v k g k , w k h k ,
j 1, , l, k 1, , m, where f j , g k , h k:Rn → C, j 1, , l, k 1, , m, a smooth solution of
1.1.
Proof By convolving the tensor product ψ t xψ s y in each side of 1.3, we have, for j
1, , l,
u j ◦ T j
∗ψ t xψ s y ξ, η u j ◦ T j , ψ t ξ − xψ s η − y
u j , α
j−1ψ t
α−1j
α j ξ −xyβ j−1ψ s
β−1j
β j η −ydy
u j ,
ψ t,α j
α j ξ − x yψ s,β j
β j η − ydy
u j ,
ψ t,α j ∗ ψ s,β j
α j ξ β j η − x
u j ∗ ψ t,α j ∗ ψ s,β j
α j ξ β j η
,
3.3 where|α j | α j,1 , , α j,n , α−1j α−1
j,1 , , α−1j,n , ψ t,α j x |α j|−1ψ t α−1
j x Similarly we have for
k 1, , m,
v k ⊗ w k
∗ψ t xψ s y ξ, η v k ∗ ψ t
ξw k ∗ ψ s
Thus1.3 is converted to the following functional equation:
l
j1
F j x, y, t, s m
k1
Trang 5F j x, y, t, s u j ∗ ψ t,α j ∗ ψ s,β j
α j x β j y
,
G k x, t v k ∗ ψ t
x, H k y, s w k ∗ ψ s
for j 1, , l, k 1, , m We first prove that lim t→ 0G k x, t are smooth functions and equal to v k for all k 1, , m Let
F x, y, t, s l
j1
Then,
lim
t→ 0 F x, y, t, s l
j1
u j ∗ ψ s,β j
α j x β j y
3.8
is a smooth function of x for each y ∈ Rn , s > 0, and {H1, , H m} is linearly independent
We may choose y m ∈ Rn , s m > 0 such that H m y m , s m : b0m / 0 Then, it follows from 3.5 that
G m x, t b m0
−1
F
x, y m , t, s m
−m−1
k1
b0k G k x, t
where b k0 H k y m , s m , k 1, , m − 1 Putting 3.9 in 3.5, we have
F1x, y, t, s m−1
k1
where
F1x, y, t, s Fx, y, t, s − b0m
−1
F
x, y m , t, s m
H k1y, s H k y, s − b0m
−1
b0k H m y, s, k 1, , m − 1. 3.12
Since limt→ 0F x, y, t, s is a smooth function of x for each y ∈ R n , s > 0, it follows from3.11 that
lim
is a smooth function of x for each y ∈ Rn , s > 0 Also, since {H1, , H m} is linearly independent, it follows from3.12 that
Trang 6
is linearly independent Thus we can choose y m−1∈ Rn , s m−1 > 0 such that H m−11 y m−1 , s m−1 :
b1m−1 / 0 Then, it follows from 3.10 that
G m−1 x, t b1m−1−1
F1
x, y m−1 , t, s m−1
−m−2
k1
b1k G k x, t
where b k1 H k1y m−1 , s m−1 , k 1, , m − 2 Putting 3.15 in 3.10, we have
F2x, y, t, s m−2
k1
where
F2x, y, t, s F1x, y, t, s − b m−11 −1F1
x, y m−1 , t, s m−1
H m−11 y, s,
H k2y, s H k1y, s − b1m−1−1b1k H m−11 y, s, k 1, , m − 2.
3.17
By continuing this process, we obtain the following equations:
F p x, y, t, s m−p
k1
for all p 0, 1, , m − 1, where F0 F, H k0 H k , k 1, , m,
G m−p x, t b p m−p−1
F p
x, y m−p , t, s m−p
−m−p−1
k1
b k p G k x, t
for all p 0, 1, , m − 2, and
G1x, t b m−11 −1
F m−1
x, y1, t, s1
By the induction argument, we have for each p 0, 1, , m − 1,
lim
is a smooth function of x for each y∈ Rn , s > 0 Thus, in view of3.20,
g1x : lim
Trang 7is a smooth function Furthermore, G1x, t converges to g1x locally uniformly, which implies that v1 g1in the sense of distributions, that is, for every ϕx ∈ C∞
c Rn,
v1, ϕ
lim
t→ 0
G1x, tϕx dx
g1xϕx dx.
3.23
In view of3.19 and the induction argument, for each k 2, , m, we have
g k x : lim
is a smooth function and v k g k for all k 2, 3, , m Changing the roles of G k and H kfor
k 1, 2, , m, we obtain, for each k 1, 2, , m,
h k x : lim
is a smooth function and w k h k Finally, we show that for each j 1, 2, , l, u jis equal to a
smooth function Letting s → 0in3.5, we have
l
j1
u j ∗ ψ t,α j
α j x β j y
m
k1
For each fixed i, 1 ≤ i ≤ l, replacing x by α−1
i x − β i y , multiplying ψ s y and integrating with respect to y, we have
u i ∗ ψ t,α i
x −
j / i
u j ∗ ψ t,α j ∗ ψ s,γ j
x m
k1
G k
α−1i x − α−1
i β i y, t
h k yψ s y dy, 3.27
where γ j α−1
i β i α j − α i β j for all 1 ≤ j ≤ l, j / i Letting t → 0in3.27, we have
u i −
j / i
u j ∗ ψ s,γ j
x m
k1
g k
α−1i x − α−1
i β i y
h k yψ s y dy : f i x. 3.28
It is obvious that f i is a smooth function Also it follows from3.27 that each u i ∗
ψ t x, i 1, , l, converges locally and uniformly to the function f i x as t → 0, which implies that the equality3.28 holds in the sense of distributions Finally, letting s → 0and
t → 0 in3.5 we see that f j , g k , h k , j 1, , l, k 1, , m are smooth solutions of 1.1 This completes the proof
Combined with the result of Acz´el and Chung1, we have the following corollary as
a consequence of the above result
Trang 8Corollary 3.2 Every solution u j , v k , w k ∈ DR, j 1, , l, k 1, , m, of 1.3 for the dimension n 1 has the form of exponential polynomials.
The result ofTheorem 3.1holds for u j , v k , w k ∈ S1/2
1/2Rn , j 1, , l, k 1, , m Using the following n-dimensional heat kernel,
E t x 4πt −n/2exp
−|x|2
4t
Applying the proof ofTheorem 3.1, we get the result for the space of Gelfand generalized functions
4 Hyers-Ulam stability of related functional equations
The well-known Cauchy equation, Pexider equation, Jensen equation, quadratic functional equation, and d’Alembert functional equation are typical examples of the form1.1 For the distributional version of these equations and their stabilities, we refer the reader to7, 8
In this section, as well-known examples of1.1, we introduce the following trigonometric differences:
T1f, g : fx y − fxgy − gxfy,
T2f, g : gx y − gxgy fxfy,
T3f, g : fx − y − fxgy gxfy,
T4f, g : gx − y − gxgy − fxfy,
4.1
where f, g : Rn → C In 1990, Sz´ekelyhidi 23 has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with stability questions for the sine and cosine functional equations As the results, he proved that if
T j f, g, j 1, 2, 3, 4, is a bounded function on R 2n , then either there exist λ, μ ∈ C, not
both zero, such that λf − μg is a bounded function on R n , or else T j f, g 0, j 1, 2, 3, 4,
respectively For some other elegant Hyers-Ulam stability theorems, we refer the reader to
6,9 21
By generalizing the differences 4.1, we consider the differences
G1u, v : u ◦ A − u ⊗ v − v ⊗ u,
G2u, v : v ◦ A − v ⊗ v u ⊗ u,
G3u, v : u ◦ S − u ⊗ v v ⊗ u,
G4u, v : v ◦ S − v ⊗ v − u ⊗ u,
4.2
and investigate the behavior of u, v ∈ S1/2
1/2Rn
j
each j 1, 2, 3, 4, where Ax, y x y, Sx, y x − y, x, y ∈ R n,◦ denotes the pullback,
⊗ denotes the tensor product of generalized functions as inTheorem 3.1, and j
1/2Rn
Trang 9As a result, we obtain the following theorems.
Theorem 4.1 Let u, v ∈ S 1/2
1/2 satisfy 1 items:
i u 0, v: arbitrary,
ii u and v are bounded measurable functions,
iii u c · xe ia · x Bx, v e ia · x ,
iv u λe c · x − Bx, v 1/2e c · x Bx,
v u λe b · x − e c · x , v 1/2e b · x e c · x ,
vi u b · xe c · x , v e c · x ,
where a∈ Rn , b, c∈ Cn , λ ∈ C, and B is a bounded measurable function.
Theorem 4.2 Let u, v ∈ S 1/2
1/2 satisfy 2 items:
i u and v are bounded measurable functions,
ii v e c · x and u is a bounded measurable function,
iii v c · xe ia · x Bx, u ±1 − c · xe ia · x − Bx,
iv v e c · x λBx/1 − λ2, u λe c · x Bx/1 − λ2,
v v 1 − b · xe c· x , u ±b · xe c · x ,
vi v e b · x cosc · x λ sinc · x, u √λ2 1 e b · x sinc · x,
where a∈ Rn , b, c∈ Cn , λ ∈ C, and B is a bounded measurable function.
Theorem 4.3 Let u, v ∈ S 1/2
1/2 satisfy 3 items:
i u ≡ 0 and v is arbitrary,
ii u and v are bounded measurable functions,
iii u c · x rx, v λc · x rx 1,
iv u λ sinc · x, v cosc · x λ sinc · x,
for some c∈ Cn , λ ∈ C and a bounded measurable function rx.
Theorem 4.4 Let u, v ∈ S 1/2
1/2 satisfy 4
items:
i u and v are bounded measurable functions,
ii u cosc · x, v sinc · x, c ∈ C n
For the proof of the theorems, we employ the n-dimensional heat kernel
E t x 4πt −n/2exp
−|x|2
4t
Trang 10
In view of2.3, it is easy to see that for each t > 0, E tbelongs to the Gelfand-Shilov space
S1/2
1/2Rn Thus the convolution u ∗ E t x : u y , E t
solution of the heat equation∂/∂ t − ΔU 0 in {x, t : x ∈ R n , t > 0 } and u ∗ E t x →
u as t → 0in the sense of generalized functions for all u∈ S1/2
1/2 Similarly as in the proof ofTheorem 3.1, convolving the tensor product E t xE s y of
heat kernels and using the semigroup property
E t ∗ E s
of the heat kernels, we can convert the inequalities j
Hyers-Ulam stability problems, respectively,
U x y, t s − Ux, tV y, s − V x, tUy, s ≤ M,
V x y, t s − V x, tV y, s Ux, tUy, s ≤ M,
U x − y, t s − Ux, tV y, s V x, tUy, s ≤ M,
V x − y, t s − V x, tV y, s − Ux, tUy, s ≤ M,
4.5
for the smooth functions Ux, t u ∗ E t x, V x, t v ∗ E t x Proving the Hyers-Ulam
stability problems for the inequalities4.5 and taking the initial values of U and V as t → 0,
we get the results For the complete proofs of the result, we refer the reader to24
Remark 4.5 The referee of the paper has recommended the author to consider the
Hyers-Ulam stability of the equations, which will be one of the most interesting problems in this field However, the author has no idea of solving this question yet Instead, Baker25 proved the Hyers-Ulam stability of the equation
l
j1
f j
α j x β j y
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... well-known Cauchy equation, Pexider equation, Jensen equation, quadratic functional equation, and d’Alembert functional equation are typical examples of the form1.1 For the distributional version of. .. distributional analog of a sum form functionalequation, ” Acta Mathematica Hungarica, vol 78, no 4, pp 333–344, 1998.
5 E Deeba and S Xie, “Distributional analog of a functional. .. of Acz´el and Chung 1, we have the following corollary as
a consequence of the above result
Trang 8Corollary