Volume 2008, Article ID 147979,11pages 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,[email protected] Received 1 October 2008; Revised 22 December 2008; Accepted 25 December 2008 Recommended by Patricia J. Y. Wong
We consider ann-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.
Copyrightq2008 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
fj
αjxβjy m
k1
gkxhky, 1.1
where fj, gk, hk : R → C and αj, βj ∈ R for j 1, . . . , l, k 1, . . . , m. Under the natural assumptions that{g1, . . . , gm}and{h1, . . . , hm}are linearly independent, andαjβj/0, αiβj/αjβifor alli /j,i, j 1, . . . , l, it was shown that the locally integrable solutions of1.1 are exponential polynomials, that is, the functions of the form
q k1
erkxpkx, 1.2
whererk∈Candpk’s are polynomials for allk1,2, . . . , q.
In this paper, we introduce the following n-dimensional version of the functional equation1.1in generalized functions:
l j1
uj◦Tjm
k1
vk⊗wk, 1.3
where uj, vk, wk ∈ DRn resp., S1/21/2Rn, and ◦ denotes the pullback, ⊗ denotes the tensor product of generalized functions, and Tjx, y αjx βjy, αj αj,1, . . . , αj,n, βj βj,1, . . . , βj,n, x x1, . . . , xn, y y1, . . . , yn, αjx αj,1x1, . . . , αj,nxn, βjy βj,1y1, . . . , βj,nyn,j 1, . . . , l. As in 1, we assume thatαj,pβj,p/0 andαi,pβj,p/αj,pβi,p for allp1, . . . , n,i /j,i, j1, . . . , l.
In2, Baker previously treated1.3. By making use of differentiation of distributions which is one of the most powerful advantages of the Schwartz theory, and reducing1.3to a system of differential equations, he showed that, for the dimensionn 1, the solutions of 1.3are exponential polynomials. We refer the reader to 2–6for 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 of1.3and prove in an elementary way that1.3can be reduced to the classical equation1.1of 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|
x21· · ·x2n,|α|α1· · ·αn,α!α1!, . . . , αn!,xαx1α1, . . . , xαnn, and∂α ∂α11, . . . , ∂αnn, forx x1, . . . , xn ∈Rn,α α1, . . . , αn∈ Nn0, whereN0 is the set of nonnegative integers and∂j ∂/∂xj.
Definition 2.1. A distribution uis a linear functional onC∞c Rnof infinitely differentiable functions onRnwith compact supports such that for every compact setK ⊂ Rn there exist constantsCandksatisfying
| u, ϕ| ≤C
|α|≤k
sup∂αϕ 2.1
for allϕ ∈ Cc∞Rnwith supports contained in K. One denotes byDRnthe space of the Schwartz distributions onRn.
Definition 2.2. For given r, s ≥ 0, one denotes by Ssr or SrsRn the space of all infinitely differentiable functionsϕxonRnsuch that there exist positive constantshandksatisfying
ϕh,k: sup
x∈Rn, α,β∈Nn0
xα∂βϕx
h|α|k|β|α!rβ!s <∞. 2.2
The topology on the spaceSsris defined by the seminorms·h,kin the left-hand side of2.2, and the elements of the dual spaceSsr ofSrsare called Gelfand-Shilov generalized functions. In particular, one denotesS11byFand calls its elements Fourier hyperfunctions.
It is known that ifr > 0 and 0 ≤ s < 1, the spaceSrsRn consists of all infinitely differentiable functionsϕxonRnthat can be continued to an entire function onCnsatisfying
|ϕxiy| ≤Cexp
−a|x|1/rb|y|1/1−s
2.3
for somea, b >0.
It is well known that the following topological inclusions hold:
S1/21/2→ F, F→ S1/21/2. 2.4
We briefly introduce some basic operations on the spaces of the generalized functions.
Definition 2.3. Letu∈ DRn. Then, thekth partial derivative∂kuofuis defined by ∂ku, ϕ
− u, ∂kϕ
2.5
fork1, . . . , n. Letf∈C∞Rn. Then the multiplicationfuis defined by
fu, ϕ u, fϕ. 2.6
Definition 2.4. Letuj∈ DRnj,j1,2. Then, the tensor productu1⊗u2ofu1andu2is defined by
u1⊗u2, ϕ x1, x2
u1,
u2, ϕ
x1, x2 , ϕ
x1, x2
∈C∞c
Rn1×Rn2
. 2.7
The tensor productu1⊗u2belongs toDRn1×Rn2.
Definition 2.5. Letuj ∈ DRnj,j 1,2, and letf : Rn1 → Rn2 be a smooth function such that for eachx∈Rn1the derivativefxis surjective. Then there exists a unique continuous linear mapf∗ :DRn2 → DRn1such thatf∗u u◦f, whenuis a continuous function.
One callsf∗uthe pullback ofubyfand simply is denoted byu◦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.
3. Main result
We employ a functionψ∈C∞Rnsuch that
ψx≥0 ∀x∈Rn, suppψ ⊂ x∈Rn :|x| ≤1
,
Rnψxdx1.
3.1
Let u ∈ DRn and ψtx : t−nψx/t,t > 0. Then, for eacht > 0,u∗ψtx : uy, ψtx−yis well defined. We callu∗ψtxa regularizing function of the distributionu, sinceu∗ψtxis a smooth function ofxsatisfyingu∗ψtx → u as t → 0in the sense of distributions, that is, for everyϕ∈C∞c Rn,
u, ϕ lim
t→0
u∗ψt
xϕxdx. 3.2
Theorem 3.1. Let uj, vk, wk ∈ DRn,j 1, . . . , l, k 1, . . . , m, be a solution of 1.3, and both{v1, . . . , vm}and {w1, . . . , wm}are linearly independent. Then,uj fj,vk gk,wk hk, j1, . . . , l,k1, . . . , m, wherefj, gk, hk:Rn → C,j1, . . . , l,k1, . . . , m, a smooth solution of 1.1.
Proof. By convolving the tensor productψtxψsyin each side of 1.3, we have, for j 1, . . . , l,
uj◦Tj
∗
ψtxψsy
ξ, η
uj◦Tj, ψtξ−xψsη−y
uj,αj−1ψt
α−1j
αjξ−xyβj−1ψs
β−1j
βjη−y dy
uj,
ψt,αj
αjξ−xy ψs,βj
βjη−y dy
uj,
ψt,αj∗ψs,βj
αjξβjη−x
uj∗ψt,αj∗ψs,βj
αjξβjη ,
3.3 where|αj|αj,1, . . . , αj,n,α−1j α−1j,1, . . . , α−1j,n,ψt,αjx |αj|−1ψtα−1j x. Similarly we have for k1, . . . , m,
vk⊗wk
∗
ψtxψsy
ξ, η vk∗ψt
ξ
wk∗ψs
η. 3.4
Thus1.3is converted to the following functional equation:
l j1
Fjx, y, t, s m
k1
Gkx, tHky, s, 3.5
where
Fjx, y, t, s
uj∗ψt,αj∗ψs,βj
αjxβjy , Gkx, t
vk∗ψt
x, Hky, s wk∗ψs
y, 3.6
forj 1, . . . , l,k 1, . . . , m. We first prove that limt→0Gkx, tare smooth functions and equal tovkfor allk1, . . . , m. Let
Fx, y, t, s l
j1
Fjx, y, t, s. 3.7
Then,
tlim→0Fx, y, t, s l
j1
uj∗ψs,βj
αjxβjy
3.8
is a smooth function ofxfor eachy ∈ Rn,s >0, and{H1, . . . , Hm}is linearly independent.
We may chooseym ∈Rn,sm >0 such thatHmym, sm:b0m /0. Then, it follows from3.5 that
Gmx, t bm0
−1 F
x, ym, t, sm
−m−1
k1
b0k Gkx, t
, 3.9
wherebk0Hkym, sm,k1, . . . , m−1. Putting3.9in3.5, we have
F1x, y, t, s m−1
k1
Gkx, tHk1y, s, 3.10
where
F1x, y, t, s Fx, y, t, s−b0m−1F
x, ym, t, sm
Hmy, s, 3.11
Hk1y, s Hky, s−b0m −1b0k Hmy, s, k1, . . . , m−1. 3.12 Since limt→0Fx, y, t, sis a smooth function ofxfor eachy∈Rn,s >0, it follows from3.11 that
tlim→0F1x, y, t, s 3.13
is a smooth function of x for each y ∈ Rn, s > 0. Also, since {H1, . . . , Hm} is linearly independent, it follows from3.12that
H11, . . . , Hm−11
3.14
is linearly independent. Thus we can chooseym−1∈Rn,sm−1>0 such thatHm−11 ym−1, sm−1: b1m−1/0. Then, it follows from3.10that
Gm−1x, t b1m−1−1
F1
x, ym−1, t, sm−1
−m−2
k1
b1k Gkx, t
, 3.15
wherebk1Hk1ym−1, sm−1,k1, . . . , m−2. Putting3.15in3.10, we have
F2x, y, t, s m−2
k1
Gkx, tHk2y, s, 3.16
where
F2x, y, t, s F1x, y, t, s−bm−11 −1F1
x, ym−1, t, sm−1
Hm−11 y, s, Hk2y, s Hk1y, s−b1m−1−1b1k Hm−11 y, s, k1, . . . , m−2.
3.17
By continuing this process, we obtain the following equations:
Fpx, y, t, s m−p
k1
Gkx, tHkpy, s, 3.18
for allp0,1, . . . , m−1, whereF0F,Hk0Hk,k1, . . . , m,
Gm−px, t bpm−p−1
Fp
x, ym−p, t, sm−p
−m−p−1
k1
bkpGkx, t
, 3.19
for allp0,1, . . . , m−2, and
G1x, t
bm−11 −1
Fm−1
x, y1, t, s1
. 3.20
By the induction argument, we have for eachp0,1, . . . , m−1,
t→lim0Fpx, y, t, s 3.21
is a smooth function ofxfor eachy∈Rn,s >0. Thus, in view of3.20, g1x: lim
t→0G1x, t 3.22
is a smooth function. Furthermore, G1x, t converges to g1x locally uniformly, which implies thatv1g1in the sense of distributions, that is, for everyϕx∈C∞c Rn,
v1, ϕ lim
t→0
G1x, tϕxdx
g1xϕxdx.
3.23
In view of3.19and the induction argument, for eachk2, . . . , m, we have gkx: lim
t→0Gkx, t 3.24
is a smooth function andvkgkfor allk 2,3, . . . , m. Changing the roles ofGkandHkfor k1,2, . . . , m, we obtain, for eachk1,2, . . . , m,
hkx: lim
t→0Hkx, t 3.25
is a smooth function andwkhk. Finally, we show that for eachj1,2, . . . , l,ujis equal to a smooth function. Lettings → 0in3.5, we have
l j1
uj∗ψt,αj
αjxβjy m
k1
Gkx, thky. 3.26
For each fixedi, 1≤i≤l, replacingxbyα−1i x−βiy, multiplyingψsyand integrating with respect toy, we have
ui∗ψt,αi
x −
j /i
uj∗ψt,αj∗ψs,γj
x m
k1
Gk
α−1i x−α−1i βiy, t
hkyψsydy, 3.27
whereγj α−1i βiαj−αiβjfor all 1≤j≤l,j /i. Lettingt → 0in3.27, we have
ui −
j /i
uj∗ψs,γj
x m
k1
gk
α−1i x−α−1i βiy
hkyψsydy:fix. 3.28
It is obvious thatfi is a smooth function. Also it follows from3.27that eachui ∗ ψtx,i 1, . . . , l, converges locally and uniformly to the functionfixast → 0, which implies that the equality3.28holds in the sense of distributions. Finally, lettings → 0and t → 0 in3.5we see thatfj, gk, hk, j 1, . . . , l,k 1, . . . , mare smooth solutions of1.1.
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.
Corollary 3.2. Every solution uj, vk, wk ∈ DR, j 1, . . . , l, k 1, . . . , m, of 1.3 for the dimensionn1 has the form of exponential polynomials.
The result ofTheorem 3.1holds foruj, vk, wk ∈ S1/21/2Rn,j 1, . . . , l,k 1, . . . , m.
Using the followingn-dimensional heat kernel,
Etx 4πt−n/2 exp −|x|2
4t
, t >0. 3.29
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:fxy−fxgy−gxfy, T2f, g:gxy−gxgy fxfy, T3f, g:fx−y−fxgy gxfy, T4f, g:gx−y−gxgy−fxfy,
4.1
wheref, g : Rn → C. In 1990, Sz´ekelyhidi23has 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 Tjf, g,j 1,2,3,4, is a bounded function onR2n, then either there exist λ, μ ∈ C, not both zero, such thatλf −μgis a bounded function onRn, or elseTjf, 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 differences4.1, we consider the differences G1u, v:u◦A−u⊗v−v⊗u,
G2u, v:v◦A−v⊗vu⊗u, G3u, v:u◦S−u⊗vv⊗u, G4u, v:v◦S−v⊗v−u⊗u,
4.2
and investigate the behavior ofu, v∈ S1/21/2Rnsatisfying the inequalityGju, v ≤Mfor eachj 1,2,3,4, whereAx, y xy,Sx, y x−y,x, y ∈Rn,◦denotes the pullback,
⊗denotes the tensor product of generalized functions as inTheorem 3.1, andGju, v ≤M means that| Gju, v, ϕ| ≤ ϕL1for allϕ∈S1/21/2Rn.
As a result, we obtain the following theorems.
Theorem 4.1. Letu, v∈ S1/21/2satisfyG1u, v ≤M. Then,uandvsatisfy one of the following items:
iu0,v: arbitrary,
iiuandvare bounded measurable functions, iiiuc·xeia·xBx,veia·x,
ivuλec·x−Bx,v 1/2ec·xBx, vuλeb·x−ec·x,v 1/2eb·xec·x, viub·xec·x,vec·x,
wherea∈Rn,b, c∈Cn,λ∈C, andBis a bounded measurable function.
Theorem 4.2. Letu, v∈ S1/21/2satisfyG2u, v ≤M. Then,uandvsatisfy one of the following items:
iuandvare bounded measurable functions, iivec·xanduis a bounded measurable function, iiivc·xeia·xBx,u±1−c·xeia·x−Bx, ivv ec·xλBx/1−λ2,u λec·xBx/1−λ2,
vv 1−b·xec·x,u±b·xec·x, viveb·xcosc·x λ sinc·x,u√
λ21eb·x sinc·x, wherea∈Rn,b, c∈Cn,λ∈C, andBis a bounded measurable function.
Theorem 4.3. Letu, v∈ S1/21/2satisfyG3u, v ≤M. Then,uandvsatisfy one of the following items:
iu≡0 andvis arbitrary,
iiuandvare bounded measurable functions, iiiuc·xrx,vλc·xrx 1, ivuλ sinc·x,vcosc·x λ sinc·x,
for somec∈Cn,λ∈Cand a bounded measurable functionrx.
Theorem 4.4. Letu, v∈ S1/21/2satisfyG4u, v ≤M. Then,uandvsatisfy one of the following items:
iuandvare bounded measurable functions, iiucosc·x,vsinc·x,c∈Cn.
For the proof of the theorems, we employ then-dimensional heat kernel
Etx 4πt−n/2exp −|x|2
4t
, t >0. 4.3
In view of2.3, it is easy to see that for eacht > 0,Etbelongs to the Gelfand-Shilov space S1/21/2Rn. Thus the convolutionu∗Etx: uy, Etx−yis well defined and is a smooth solution of the heat equation∂/∂t−ΔU 0 in{x, t : x ∈ Rn, t > 0}andu∗Etx → uast → 0in the sense of generalized functions for allu∈ S1/21/2.
Similarly as in the proof ofTheorem 3.1, convolving the tensor productEtxEsyof heat kernels and using the semigroup property
Et∗Es
x Etsx 4.4
of the heat kernels, we can convert the inequalitiesGju, v ≤M,j 1,2,3,4,to the classical Hyers-Ulam stability problems, respectively,
Uxy, ts−Ux, tVy, s−Vx, tUy, s≤M, Vxy, ts−Vx, tVy, s Ux, tUy, s≤M, Ux−y, ts−Ux, tVy, s Vx, tUy, s≤M, Vx−y, ts−Vx, tVy, s−Ux, tUy, s≤M,
4.5
for the smooth functionsUx, t u∗Etx,Vx, t v∗Etx. Proving the Hyers-Ulam stability problems for the inequalities4.5and taking the initial values ofUandVast → 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, Baker25proved the Hyers-Ulam stability of the equation
l j1
fj
αjxβjy
0. 4.6
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