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:

fxyz fx fy fz fxy fyz fzx 1.1

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 andFRⁿof 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∈ SRⁿoru∈ FRⁿ,

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 u

x₁, . . . , x_n

1≤i≤j≤n

a_ijx_ix_j

1≤i≤n

b_ix_iμ, 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 hyperfunctions. Here, we use the multi-index notations,|α|α₁· · ·α_n, α! α₁!· · ·α_n!, x^α x₁^α1· · ·x^α_nⁿ, and ∂^α ∂^α₁¹· · ·∂^α_nⁿ, for x x1, . . . , x_n ∈ Rⁿ and α α1, . . . , αn∈Nⁿ₀, 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 functionsϕinRⁿsatisfying

ϕα,βsup

x∈Rⁿ

x^α∂^βϕx<∞ 2.1

for all α, β ∈ Nⁿ₀, equipped with the topology defined by the seminorms ·α,β. A linear functionaluonSRⁿis said to be tempered distribution if there are a constantC ≥ 0 and a nonnegative integerNsuch that

u, ϕ≤C

|α|,|β|≤N

sup

x∈Rⁿ

x^α∂^βϕ 2.2

for allϕ∈ SRⁿ. The set of all tempered distributions is denoted bySRⁿ.

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 byFRⁿthe Sato space of all infinitely differentiable functionsϕinRⁿsuch that

ϕA,Bsup

x,α,β

|x^α∂^βϕx|

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 elements Fourier hyperfunctions .

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

ϕh,k sup

x∈Rⁿ,α∈Nⁿ₀

|∂^αϕx|expk|x|

h^|α|α! <∞ 2.4

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

FRⁿ→ SRⁿ, SRⁿ→ FRⁿ. 2.5

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

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−ΔxinRⁿ_x×R_t given by

E_tx

⎧⎪

⎨

⎪⎩

4πt^−n/2exp

− |x|² 4t

, t >0,

0, t≤0.

2.6

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

ux, t u∗Et

x

u_y, E_tx−y

, x∈Rⁿ, t >0, 2.7

is well defined for eachu∈ FRⁿ. In relation to the Gauss transform, it is well known that the semigroup property of the heat kernel

Et∗Es

x E_tsx 2.8

holds for convolution. Moreover, the following result holds20.

Letu∈ SRⁿ. Then, its Gauss transformux, t is 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 Rⁿ×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, t for 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

inRⁿ×0, δ. 2.12

3. General solution and stability inFRⁿ

We will now consider the general solution and the Hyers-Ulam stability of1.1in the space FRⁿ. Convolving the tensor productEtξEsηErζofn-dimensional heat kernels in both sides of1.2, we have

u◦A∗

E_tξEsηErζ

x, y, z

u◦A, E_tx−ξEsy−ηErz−ζ

u_ξ,

E_tx−ξηζEsy−ηErz−ζdη dζ

u_ξ,

E_txyz−ξ−η−ζEsηErζdη dζ

uξ,

Et∗Es∗Er

xyz−ξ

u_ξ, E_tsr

xyz−ξ ux yz, tsr,

3.1

and similarly we obtain u◦P₁

∗

E_tξEsηErζ

x, y, z ux, t, u◦P₂

∗

E_tξEsηErζ

x, y, z uy, s, u◦P3

∗

EtξEsηErζ

x, y, z uz, r, u◦B₁

∗

E_tξEsηErζ

x, y, z ux y, ts, u◦B2

∗

EtξEsηErζ

x, y, z uy z, sr, u◦B3

∗

EtξEsηErζ

x, y, z uz x, rt,

3.2

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

uxyz, tsrux, t uy, suz, r ux y, ts uy z, sr uz x, rt 3.3

for allx, y, z∈Rⁿandt, 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 3.4

for allx, y, z∈Rⁿandt, s, r >0. Also, assume thatfx, tis continuous and 2-times differentiable with respect toxandt, respectively. Then, there exist constantsa_ij, b_i, c_i, d, e∈Csuch that

fx, t

1≤i≤j≤n

a_ijx_ix_j

1≤i≤n

b_ix_it

1≤i≤n

c_ix_idt²et 3.5

for allx x1, . . . , xn∈Rⁿandt >0.

Proof. In view of3.4,fx,0:lim_t→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

f x,0

Bx, 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∈Rⁿandt >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∈Rⁿandt, 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∈Rⁿandt >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

cixi 3.13

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 dt²et 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

a_ijx_ix_j

1≤i≤n

b_ix_it

1≤i≤n

c_ix_idt²et 3.18

for somea_ij, b_i, c_i, 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 solutionuinFRⁿof

u◦Au◦P₁u◦P₂u◦P₃u◦B₁u◦B₂u◦B₃ 3.19

has the form

ux u

x1, . . . , 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 productE_tξEsηErζofn-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 ux y, ts uy z, sr uz x, rt 3.21

for allx, y, z∈Rⁿandt, 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 constantsa_ij, b_i, c_i, d, e∈C. Now lettingt→0, we have

u

1≤i≤j≤n

a_ijx_ix_j

1≤i≤n

b_ix_i 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 thatuinFRⁿsatisfies 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, t uy, s uz, r− uxy, ts −uy z, sr−uz x, rt≤ 3.27 for allx, y, z ∈ Rⁿ and t, s, r > 0, where u is the Gauss transform ofu. Define a function f_e :Rⁿ×0,∞→Cbyf_ex, t : 1/2ux, t u−x, t −u0, t for allx ∈Rⁿ andt >0.

Then,fe−x, t fex, t, fe0, t 0, and

f_exyz, tsrfex, tf_ey, sf_ez, r−fexy, ts−feyz, sr−fezx, rt≤2 3.28 for allx, y, z∈Rⁿandt, s, r >0. Replacingzby−yin3.28, we have

fex, tsr fex, t fey, s fey, r−fexy, ts−fex−y, rt≤2. 3.29

Puttingyz0 in3.28yields

fex, tsr fex, t−fex, ts−fex, rt≤2. 3.30

Taking3.29into3.30, we obtain

f_exy, ts f_ex−y, rtfex, ts−f_ex, rt−f_ey, s−f_ey, 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 f_e2x, t

4 −f_ex, t

≤. 3.33

Making use of an induction argument, we obtain

4^−kf_e2^kx, t−f_ex, t≤ 4

3 3.34

for allk ∈N, x ∈ Rⁿ, andt >0. Exchangingxby 2^lxin3.34and then dividing the result by 4^l, we can see that{4^−kfe2^kx, t}is a Cauchy sequence which converges uniformly. Let gx, t limk→ ∞4^−kfe2^kx, 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∈Rⁿandt, 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 constantsa_ij, b_i, c_i, d, e∈ C. Sincef_e−x, t f_ex, tandf_e0, t 0 for allx ∈Rⁿ andt >0, we have

gx, t

1≤i≤j≤n

a_ijx_ix_j. 3.37

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

u−x, tfor allx∈Rⁿandt >0. Then,f_o−x, t −fox, t, fo0, t 0, and

f_oxyz, tsrfox, tf_oy, sf_oz, r−foxy, ts−foyz, sr−fozx, rt≤ 3.38

for allx, y, z∈Rⁿandt, s, r >0. Replacingzby−yin3.38, we have

f_ox, tsr f_ox, t f_oy, s−f_oy, r−f_oxy, ts−f_ox−y, rt≤. 3.39

Settingyz0 in3.38yields

fox, tsr fox, t−fox, ts−fox, rt≤. 3.40

Adding3.39to3.40, we obtain

f_oxy, ts f_ox−y, rt−f_ox, ts−f_ox, rt−f_oy, s f_oy, 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 f_o2x, t

2 −f_ox, t

≤. 3.43

Using the iterative method, we obtain

2^−kf_o2^kx, t−f_ox, 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, f_ox, t−hx, t≤2

3.45 for allx, y, z∈Rⁿandt, s, r > 0. According toLemma 3.1, there existaij, bi, ci, d, e∈Csuch that

hx, t

1≤i≤j≤n

a_ijx_ix_j

1≤i≤n

b_ix_it

1≤i≤n

c_ix_idt²et. 3.46

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

hx, t

1≤i≤n

b_ix_it

1≤i≤n

c_ix_i. 3.47

In turn, sinceux, t f_ex, t f_ox, 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 solutionuinFRⁿcan 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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