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
areexponential polynomials, that is, the functions of the form
q
k1
erkxpkx, 1.2
In this paper, we introduce the following n-dimensional version of the functional equation1.1in generalized functions:
l
j1
uj◦Tj m
k1
vk⊗wk, 1.3
where uj, vk, wk ∈ DRn resp., S11//22Rn, 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 to2–6for 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|
x2
1· · ·x2n,|α|α1· · ·αn,α!α1!, . . . , αn!,xαx1α1, . . . , x αn n ,
and∂α ∂α1 1 , . . . , ∂
αn
n , 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 Ss
r or SrsRn the space of all infinitely
differentiable functionsϕxonRnsuch that there exist positive constantshandksatisfying
ϕh,k: sup x∈Rn, α,β∈Nn
0
xα∂βϕx
The topology on the spaceSs
ris defined by the seminorms·h,kin the left-hand side of2.2,
and the elements of the dual spaceSsr ofSs
r are calledGelfand-Shilov generalized functions. In
particular, one denotesS11byFand calls its elementsFourier hyperfunctions.
It is known that ifr > 0 and 0 ≤ s < 1, the spaceSs
rRn 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:
S1/2
1/2→ F, F
→ S11//22. 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
, ϕx1, 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.
3. Main result
Thus1.3is converted to the following functional equation:
where
independent, it follows from3.12that
is linearly independent. Thus we can chooseym−1∈Rn,sm−1>0 such thatHm1−1ym−1, sm−1:
bm1−1/0. Then, it follows from3.10that
Gm−1x, t bm1−1 −1
F1x, ym−1, t, sm−1
−m−2
k1
bk1Gkx, 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−b1
m−1 −1
F1x, y
m−1, t, sm−1
Hm1−1y, s,
Hk2y, s Hk1y, s−bm1−1−1bk1Hm1−1y, 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,H0
k Hk,k1, . . . , m,
Gm−px, t bmp−p
−1
Fpx, y
m−p, t, sm−p
−
m−p−1
k1
bkpGkx, t
, 3.19
for allp0,1, . . . , m−2, and
G1x, t
b1m−1−1Fm−1x, y1, t, s1
. 3.20
By the induction argument, we have for eachp0,1, . . . , m−1,
lim
t→0F
px, y, t, s 3.21
is a smooth function ofxfor eachy∈Rn,s >0. Thus, in view of3.20,
g1x: lim
is a smooth function. Furthermore, G1x, t converges to g1x locally uniformly, which
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.
Corollary 3.2. Every solution uj, vk, wk ∈ DR, j 1, . . . , l, k 1, . . . , m, of 1.3 for the dimensionn1has the form of exponential polynomials.
The result ofTheorem 3.1holds foruj, vk, wk ∈ S11//22Rn,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∈ S11//22Rnsatisfying 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
As a result, we obtain the following theorems.
Theorem 4.1. Letu, v∈ S11//22satisfyG1u, 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∈ S11//22satisfyG2u, 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∈ S11//22satisfyG3u, v ≤M. Then,uandvsatisfy one of the following items:
iu≡0andvis 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∈ S11//22satisfyG4u, 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
In view of2.3, it is easy to see that for eacht > 0,Etbelongs to the Gelfand-Shilov space
S1/2
1/2Rn. 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∈ S11//22.
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
References
1 J. Acz´el and J. K. Chung, “Integrable solutions of functional equations of a general type,”Studia Scientiarum Mathematicarum Hungarica, vol. 17, no. 1–4, pp. 51–67, 1982.
2 J. A. Baker, “On a functional equation of Acz´el and Chung,”Aequationes Mathematicae, vol. 46, no. 1-2, pp. 99–111, 1993.
3 J. A. Baker, “Distributional methods for functional equations,”Aequationes Mathematicae, vol. 62, no. 1-2, pp. 136–142, 2001.
4 E. Deeba, E. L. Koh, P. K. Sahoo, and S. Xie, “On a distributional analog of a sum form functional equation,”Acta Mathematica Hungarica, vol. 78, no. 4, pp. 333–344, 1998.
5 E. Deeba and S. Xie, “Distributional analog of a functional equation,”Applied Mathematics Letters, vol. 16, no. 5, pp. 669–673, 2003.
6 E. Deeba, P. K. Sahoo, and S. Xie, “On a class of functional equations in distribution,” Journal of Mathematical Analysis and Applications, vol. 223, no. 1, pp. 334–346, 1998.
7 J. Chung, “Stability of approximately quadratic Schwartz distributions,”Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 1, pp. 175–186, 2007.
9 L. H¨aormander,The Analysis of Linear Partial Differential Operators I, Springer, Berlin, Germany, 1983.
10 P. Gavruta, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” inAdvances in Equations and Inequalities, Hadronic Mathematics, pp. 67–71, Hadronic Press, Palm Harbor, Fla, USA, 1999.
11 S.-M. Jung and J. M. Rassias, “Stability of general Newton functional equations for logarithmic spirals,”Advances in Difference Equations, vol. 2008, Article ID 143053, 5 pages, 2008.
12 H.-M. Kim, J. M. Rassias, and Y.-S. Cho, “Stability problem of Ulam for Euler-Lagrange quadratic mappings,”Journal of Inequalities and Applications, vol. 2007, Article ID 10725, 15 pages, 2007.
13 Y.-S. Lee and S.-Y. Chung, “Stability of Euler-Lagrange-Rassias equation in the spaces of generalized functions,”Applied Mathematics Letters, vol. 21, no. 7, pp. 694–700, 2008.
14 P. Nakmahachalasint, “On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations,”International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 63239, 10 pages, 2007.
15 A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,”Demonstratio Mathematica, vol. 39, no. 3, pp. 523–530, 2006.
16 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.
17 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.
18 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Bulletin des Sciences Math´ematiques. 2e S´erie, vol. 108, no. 4, pp. 445–446, 1984.
19 J. M. Rassias, “Solution of a problem of Ulam,”Journal of Approximation Theory, vol. 57, no. 3, pp. 268–273, 1989.
20 J. M. Rassias, “Solution of a stability problem of Ulam,”Discussiones Mathematicae, vol. 12, pp. 95–103, 1992.
21 J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185–190, 1992.
22 I. M. Gel’fand and G. E. Shilov,Generalized Functions. Vol. 2. Spaces of Fundamental and Generalized Functions, Academic Press, New York, NY, USA, 1968.
23 L. Sz´ekelyhidi, “The stability of the sine and cosine functional equations,”Proceedings of the American Mathematical Society, vol. 110, no. 1, pp. 109–115, 1990.
24 J. Chang and J. Chung, “The stability of the sine and cosine functional equations in Schwartz distributions,”Bulletin of the Korean Mathematical Society, vol. 46, no. 1, pp. 87–97, 2009.
