Stability of a Quadratic Functional Equation in the Spaces of Generalized Functions

Journal of Inequalities and Applications Volume 2008, Article ID 210615,12pages doi:10.1155/2008/210615

Research Article

Stability of a Quadratic Functional Equation in

the Spaces of Generalized Functions

Young-Su Lee

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea

Correspondence should be addressed to Young-Su Lee,[email protected]

Received 30 June 2008; Accepted 20 August 2008

Recommended by L´aszl ´o Losonczi

Making use of the pullbacks, we reformulate the following quadratic functional equation:fxy z fx fy fz fxy fyz fzxin the spaces of generalized functions. Also, using the fundamental solution of the heat equation, we obtain the general solution and prove the Hyers-Ulam stability of this equation in the spaces of generalized functions such as tempered distributions and Fourier hyperfunctions.

Copyrightq2008 Young-Su Lee. 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

Functional equations can be solved by reducing them to differential equations. In this case, we need to assume differentiability up to a certain order of the unknown functions, which is not required in direct methods. From this point of view, there have been several works dealing with functional equations based on distribution theory. In the space of distributions, one can differentiate freely the underlying unknown functions. This can avoid the question of regularity. Actually using distributional operators, it was shown that some functional equations in distributions reduce to the classical ones when the solutions are locally integrable functions1–4.

Another approach to distributional analogue for functional equations is via the use of the regularization of distributions5,6. More exactly, this method gives essentially the same formulation as in1–4, but it can be applied to the Hyers-Ulam stability7–10for functional equations in distributions11–14.

In accordance with the notions in11–14, we reformulate the following quadratic functional equation:

in the spaces of generalized functions. Also, we obtain the general solution and prove the Hyers-Ulam stability of 1.1 in the spaces of generalized functions such as SRn _of

tempered distributions andFRn_{of Fourier hyperfunctions.}

The functional equation1.1was first solved by Kannappan15. In fact, he proved that a function on a real vector space is a solution of1.1if and only if there exist a symmetric biadditive functionBand an additive functionAsuch thatfx Bx, xAx. In addition, Jung16investigated Hyers-Ulam stability of1.1on restricted domains, and applied the result to the study of an interesting asymptotic behavior of the quadratic functions.

As a matter of fact, we reformulate 1.1 and related inequality in the spaces of generalized functions as follows. Foru∈ SRn_{oru∈ FR}n_,

u◦Au◦P1u◦P2u◦P3 u◦B1u◦B2u◦B3, 1.2

u◦Au◦P1u◦P2u◦P3−u◦B1−u◦B2−u◦B3≤, 1.3

whereA, B1, B2, B3, P1, P2, andP3are the functions defined by

Ax, y, z xyz,

P1x, y, z x, P2x, y, z y, P3x, y, z z, B1x, y, z xy, B2x, y, z yz, B3x, y, z zx.

1.4

Here, ◦ denotes the pullbacks of generalized functions, and v ≤ in 1.3 means that |v, ϕ| ≤ϕL1for all test functionsϕ.

As a consequence, we prove that every solutionuof inequality1.3can be written uniquely in the form

ux ux1, . . . , xn

1≤i≤j≤n

aijxixj

1≤i≤n

bixiμ, 1.5

whereμis a bounded measurable function such thatμL∞ ≤13/3.

2. Preliminaries

We first introduce briefly spaces of some generalized functions such as tempered distribu-tions and Fourier hyperfuncdistribu-tions. Here, we use the multi-index notadistribu-tions,|α|α1· · ·αn,

α! α1!· · ·αn!, xα x1α1· · ·x

αn

n , and ∂α ∂α11· · ·∂

αn

n , for x x1, . . . , xn ∈ Rn and

α α1, . . . , αn∈N₀n, whereN0is the set of nonnegative integers and∂j ∂/∂xj.

Definition 2.1 see 17, 18. One denotes by SRn _{the Schwartz space of all infinitely}

differentiable functionsϕinRn_satisfying

ϕα,βsup x∈Rn

for all α, β ∈ Nn₀, equipped with the topology defined by the seminorms ·α,β. A linear

functionaluonSRn_{is said to betempered distribution if there are a constantC ≥ 0 and a}

nonnegative integerNsuch that

_{u, ϕ≤C}

|α|,|β|≤N

sup

x∈Rn

_xα_∂β_{ϕ 2.2}

for allϕ∈ SRn_{. The set of all tempered distributions is denoted bySR}n_.

Imposing the growth condition on ·α,β in 2.1, a new space of test functions has

emerged as follows.

Definition 2.2see19. One denotes byFRn_{the Sato space of all infinitely differentiable}

functionsϕinRnsuch that

ϕA,Bsup x,α,β

|xα_∂β_ϕx|

A|α|_B|β|_α!β! <∞ 2.3

for some positive constants A, B depending only on ϕ. One says that ϕj→0 as j→ ∞ if

ϕjA,B→0 asj→ ∞for some A, B > 0, and denotes byFRn the strong dual of FRn

and calls its elementsFourier hyperfunctions.

It can be verified that the seminorms2.3are equivalent to

ϕh,k sup x∈Rn_,α∈Nn

0

|∂α_ϕx|expk|x|

h|α|_α! <∞ 2.4

for some constantsh, k >0. It is easy to see the following topological inclusions:

FRn_{→ SR}n_{, SR}n_{→ FR}n_{. 2.5}

From the above inclusions, it suffices to say that one considers1.2and 1.3in the space F_Rn_.

In order to obtain the general solution and prove the Hyers-Ulam stability of 1.1

in the space FRn,one employs the n-dimensional heat kernel, that is, the fundamental solution of the heat operator∂t−ΔxinRnx×Rt given by

Etx

⎧ ⎪ ⎨ ⎪ ⎩

4πt−n/2exp

− |x|2 4t

, t >0,

0, t≤0.

In view of2.1, one sees thatEt·belongs toSRnfor eacht >0. Thus, itsGauss transform

ux, t u∗Et

x uy, Etx−y

, x∈Rn, t >0, 2.7

is well defined for eachu∈ FRn_{. In relation to the Gauss transform, it is well known that}

thesemigroup propertyof the heat kernel

Et∗Es

x Etsx 2.8

holds for convolution. Moreover, the following result holds20.

Letu∈ SRn_{. Then, its Gauss transformux, tis aC}∞_{-solution of the heat equation}

∂ ∂t−Δ

ux, t 0 2.9

satisfying what follows.

iThere exist positive constantsC, M, andNsuch that

ux, t≤Ct−M_1|x|N _{in R}n_{×0, δ. 2.10}

iiux, t→uast→0in the sense that for everyϕ∈ SRn_,

u, ϕ lim

t→0

ux, tϕxdx. 2.11

Conversely, everyC∞-solutionUx, tof the heat equation satisfying the growth condition

2.10can be uniquely expressed asUx, t ux, tfor someu∈ SRn_.

Analogously, we can represent Fourier hyperfunctions as initial values of solutions of the heat equation as a special case of the results21. In this case, the estimate2.10is replaced by what follows.

For every >0, there exists a positive constantCsuch that

ux, t≤C exp

|x|1

t

3. General solution and stability inFRn

We will now consider the general solution and the Hyers-Ulam stability of1.1in the space F_Rn_{. Convolving the tensor productE}

tξEsηErζofn-dimensional heat kernels in both

sides of1.2, we have

u◦A∗EtξEsηErζ

x, y, z u◦A, Etx−ξEsy−ηErz−ζ

uξ,

Etx−ξηζEsy−ηErz−ζdη dζ

uξ,

Etxyz−ξ−η−ζEsηErζdη dζ

uξ,

Et∗Es∗Er

xyz−ξ

uξ,

Etsr

xyz−ξ

uxyz, tsr,

3.1

and similarly we obtain

u◦P1

∗EtξEsηErζ

x, y, z ux, t,

u◦P2

∗EtξEsηErζ

x, y, z uy, s,

u◦P3

∗EtξEsηErζ

x, y, z uz, r,

u◦B1

∗EtξEsηErζ

x, y, z uxy, ts,

u◦B2

∗EtξEsηErζ

x, y, z uyz, sr,

u◦B3

∗EtξEsηErζ

x, y, z uzx, rt,

3.2

where u is the Gauss transform of u. Thus, 1.2 is converted into the classical functional equation

uxyz, tsrux, tuy, suz, r uxy, ts uyz, sr uzx, rt

3.3

for allx, y, z∈Rn_{andt, s, r >0. For that reason, we first prove the following lemma which is}

essential to prove the main result.

Lemma 3.1. Suppose that a functionf:Rn_{×0,∞→Csatisfies}

fxyz, tsr fx, t fy, s fz, r fxy, ts fyz, sr fzx, rt

for allx, y, z∈Rn_{andt, s, r >0. Also, assume thatfx, tis continuous and2-times differentiable}

with respect toxandt, respectively. Then, there exist constantsaij, bi, ci, d, e∈Csuch that

fx, t 1≤i≤j≤n

aijxixj

1≤i≤n

bixit

1≤i≤n

cixidt2et 3.5

for allx x1, . . . , xn∈Rnandt >0.

Proof. In view of3.4,fx,0:limt→0fx, texists for eachx∈Rn. Lettingtsr→0 in3.4, we see thatfx,0satisfies1.1. By the result as that in15, there exist a symmetric biadditive functionBand an additive functionAsuch that

fx,0Bx, x Ax 3.6

for allx∈Rn_{. From the hypothesis that fx, tis continuous with respect tox, we have}

fx,0 1≤i≤j≤n

aijxixj

1≤i≤n

bixi 3.7

for someaij, bi∈C. We now define a functionhashx, t:fx, t−fx,0−f0, tfor all

x∈Rn_{andt >0. Puttingxyz0 andtsr→0in3.4, we havef0,0 0. From} the definition ofhandf0,0 0, we see thathsatisfiesh0, t 0, hx,0 0, and

hxyz, tsr hx, t hy, s hz, r hxy, ts hyz, sr hzx, rt

3.8

for allx, y, z∈Rn_{andt, s, r >0. Puttingyz0 in3.8, we get}

hx, tsr hx, t hx, ts hx, rt. 3.9

Now lettingt→0in3.9yields

hx, sr hx, s hx, r. 3.10

Given the continuity,hx, tcan be written as

hx, t hx,1t 3.11

for allx∈Rn_{andt >0. Settingx0, t1, andsr→0in3.8, we obtain}

hyz,1 hy,1 hz,1 3.12

for ally, z∈Rn_{. This shows thathx,1is additive. Thus,hx, tcan be written in the form}

hx, t t 1≤i≤n

for someci∈C. Now we are going to find the general solution off0, t. Puttingxyz0

in3.4, we obtain

f0, tsr f0, t f0, s f0, r f0, ts f0, sr f0, rt. 3.14

Differentiating3.14with respect tot, we have

f0, tsr f0, t f0, ts f0, rt 3.15

for allt, s, r >0. Similarly, differentiation of3.15with respect tosyields

f0, tsr f0, ts 3.16

which shows thatf0, tis a constant function. By virtue off0,0 0, f0, tcan be written as

f0, t dt2_{et 3.17}

for somed, e∈C. Combining3.7,3.13, and3.17,fx, tcan be written in the form

fx, t fx,0 hx, t f0, t 1≤i≤j≤n

aijxixj

1≤i≤n

bixit

1≤i≤n

cixidt2et _3.18

for someaij, bi, ci, d, e∈C. This completes the proof.

As an immediate consequence ofLemma 3.1, we establish the general solution of1.1

in the spaceFRn_.

Theorem 3.2. Every solutionuinFRn_of

u◦Au◦P1u◦P2u◦P3u◦B1u◦B2u◦B3 3.19

has the form

ux ux1, . . . , xn

1≤i≤j≤n

aijxixj

1≤i≤n

bixi 3.20

for someaij, bi∈C.

Proof. As we see above, if we convolve the tensor productEtξEsηErζofn-dimensional

heat kernels in both sides of 3.19, then 3.19 is converted into the classical functional equation

uxyz, tsr ux, t uy, s uz, r uxy, ts uyz, sr uzx, rt

for allx, y, z∈Rn_{andt, s, r >0, whereuis the Gauss transform ofu. According toLemma 3.1,}

ux, tis of the form

ux, t 1≤i≤j≤n

aijxixj

1≤i≤n

bixit

1≤i≤n

cixidt2et 3.22

for some constantsaij, bi, ci, d, e∈C. Now lettingt→0, we have

u 1≤i≤j≤n

aijxixj

1≤i≤n

bixi 3.23

which completes the proof.

We now in a position to state and prove the main result of this paper.

Theorem 3.3. Suppose thatuinFRn_{satisfies the inequality}

_{u◦Au◦P1u◦P2u◦P3−u◦B1−u◦B2−u◦B3≤ε. 3.24}

Then, there exists a functionT defined by

Tx 1≤i≤j≤n

aijxixj

1≤i≤n

bixi, aij, bi∈C, 3.25

such that

u−Tx≤ 13

3 ε. 3.26

Proof. Convolving the tensor productEtξEsηErζofn-dimensional heat kernels in both

sides of3.24, we have the classical functional inequality

uxyz, tsrux, tuy, suz, r−uxy, ts−uyz, sr−uzx, rt≤

3.27

for allx, y, z ∈ Rn and t, s, r > 0, where u is the Gauss transform ofu. Define a function fe :Rn×0,∞→Cbyfex, t : 1/2ux, t u−x, t−u0, tfor allx ∈Rn andt >0.

Then,fe−x, t fex, t, fe0, t 0, and

fexyz, tsrfex, tfey, sfez, r−fexy, ts−feyz, sr−fezx, rt≤2

3.28

for allx, y, z∈Rn_{andt, s, r >0. Replacingzby−yin3.28, we have}

Puttingyz0 in3.28yields

_{fex, tsr fex, t−fex, ts−fex, rt≤2. 3.30}

Taking3.29into3.30, we obtain

fexy, ts fex−y, rtfex, ts−fex, rt−fey, s−fey, r≤4. 3.31

Lettingt→0and switchingrbys, we have

_{fexy, s fex−y, s−2fex, s−2fey, s≤4. 3.32}

Substitutingy, sbyx, t, respectively, and then dividing by 4, we lead to

fe2x, t

4 −fex, t

≤. 3.33

Making use of an induction argument, we obtain

4−k_f

e2kx, t−fex, t≤

4

3 3.34

for allk ∈N, x ∈ Rn_{, andt >0. Exchangingxby 2}l_{xin3.34and then dividing the result}

by 4l, we can see that{4−kfe2kx, t}is a Cauchy sequence which converges uniformly. Let

gx, t limk→ ∞4−kfe2kx, tfor allx∈ Rn andt >0. It follows from3.28and3.34that

gx, tis the unique function satisfying

gxyz, tsrgx, tgy, sgz, rgxy, ts gyz, sr gzx, rt,

_{fex, t−gx, t≤} 4 3

3.35

for allx, y, z∈Rn_{andt, s, r >0. By virtue ofLemma 3.1,gis of the form}

gx, t 1≤i≤j≤n

aijxixj

1≤i≤n

bixit

1≤i≤n

cixidt2et 3.36

for some constantsaij, bi, ci, d, e∈ C. Sincefe−x, t fex, tandfe0, t 0 for allx ∈Rn

andt >0, we have

gx, t 1≤i≤j≤n

On the other hand, letfo:Rn×0,∞→Cbe the function defined byfox, t: 1/2ux, t−

u−x, tfor allx∈Rn_{andt >0. Then,f}

o−x, t −fox, t, fo0, t 0, and

foxyz, tsrfox, tfoy, sfoz, r−foxy, ts−foyz, sr−fozx, rt≤

3.38

for allx, y, z∈Rn_{andt, s, r >0. Replacingzby−yin3.38, we have}

fox, tsr fox, t foy, s−foy, r−foxy, ts−fox−y, rt≤. 3.39

Settingyz0 in3.38yields

_{fox, tsr fox, t−fox, ts−fox, rt≤. 3.40}

Adding3.39to3.40, we obtain

foxy, ts fox−y, rt−fox, ts−fox, rt−foy, s foy, r≤2. 3.41

Lettingt→0and replacingrbys, we have

_{foxy, s fox−y, s−2fox, s≤2. 3.42}

Substitutingy, sbyx, t, respectively, and then dividing by 2, we lead to

fo2x, t

2 −fox, t

≤. 3.43

Using the iterative method, we obtain

2−kfo2kx, t−fox, t≤2 3.44

for allk∈N, x∈Rn_{, andt >0. From3.38and3.44, we verify thathis the unique function}

satisfying

hxyz, tsrhx, thy, shz, rhxy, ts hyz, sr hzx, rt,

fox, t−hx, t≤2

3.45

for allx, y, z∈Rn_{andt, s, r > 0. According toLemma 3.1, there exista}

ij, bi, ci, d, e∈Csuch

that

hx, t 1≤i≤j≤n

aijxixj

1≤i≤n

bixit

1≤i≤n

On account offo−x, t fox, tandfo0, t 0 for allx∈Rnandt >0, we have

hx, t 1≤i≤n

bixit

1≤i≤n

cixi. 3.47

In turn, sinceux, t fex, t fox, t u0, t, we figure out

ux, t−gx, t−hx, t≤fex, t−gx, tfox, t−hx, tu0, t

≤ 10

3 u0, t

3.48

In view of3.27, it is easy to see thatc:lim sup_t→0f0, texists. Lettingxyz0 and tsr→0in3.27, we have|c| ≤. Finally, takingt→0in3.48, we have

u−

1≤i≤j≤n

aijxixj

1≤i≤n

bixi

≤

13

3 3.49

which completes the proof.

Remark 3.4. The above norm inequality3.49implies thatu−Txbelongs toL1 _L∞_. Thus, all the solutionuinFRn_{can be written uniquely in the form}

uTx μ, 3.50

whereμis a bounded measurable function such that||μ||_L∞ ≤13/3.

Acknowledgment

This work was supported by the second stage of Brain Korea 21 project, the Development Project of Human Resources in Mathematics, KAIST, 2008.

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