Stability of Quartic Functional Equations in the Spaces of Generalized Functions

doi:10.1155/2009/838347

Research Article

Stability of Quartic Functional Equations in

the Spaces of Generalized Functions

Young-Su Lee

_{and Soon-Yeong Chung}

1_{Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology,}

Daejeon 305-701, South Korea

2_{Department of Mathematics and Program of Integrated Biotechnology, Sogang University,}

Seoul 121-741, South Korea

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

Received 6 August 2008; Revised 15 December 2008; Accepted 14 January 2009

Recommended by Rigoberto Medina

We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.

Copyrightq2009 Y.-S. Lee and S.-Y. 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

One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam1. The case of approximately additive mappings was solved by Hyers2. In 1978, Rassias3generalized Hyers’ result to the unbounded Cauchy difference. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authorssee4–9. The terminology Hyers-Ulam-Rassias stability originates from these historical backgrounds and this terminology is also applied to the cases of other functional equations. For instance, Rassias 10 investigated stability properties of the following functional equation

fx2y fx−2y 6fx 4fxy 4fx−y 24fy. 1.1

It is easy to see thatfx x4_{is a solution of1.1by virtue of the identity}

For this reason, 1.1 is called a quartic functional equation. Also Chung and Sahoo 11

determined the general solution of1.1without assuming any regularity conditions on the unknown function. In fact, they proved that the functionf : R → Ris a solution of1.1if and only iffx Ax, x, x, x, where the functionA:R4 _{→ Ris symmetric and additive in}

each variable. Since the solution of1.1is even, we can rewrite1.1as

f2xy f2x−y 4fxy 4fx−y 24fx−6fy. 1.3

Lee et al. 12 obtained the general solution of 1.3 and proved the Hyers-Ulam-Rassias stability of this equation. Also Park 13 investigated the stability problem of 1.3 in the orthogonality normed space.

In this paper we consider the following quartic functional equation, which is a generalization of1.3,

faxy fax−y a2fxy a2fx−y 2a2a2−1fx−2a2−1fy, 1.4

for fixed integera with a /0,±1. In the cases of a 0,±1 in 1.4, homogeneity property of quartic functional equations does not hold. We dispense with this cases henceforth, and assume that a /0,±1. In Section 2, we show that for each fixed integer a with a /0,±1,

1.4is equivalent to1.3. Moreover, using the idea of G˘avrut¸a14, we prove the Hyers-Ulam-Rassias stability of1.4inSection 3. Finally, making use of the pullbacks and the heat kernels, we reformulate and prove the Hyers-Ulam-Rassias stability of1.4in the spaces of some generalized functions such asSRn_{of tempered distributions andFR}n_{of Fourier}

hyperfunctions inSection 4.

2. General Solution of

1.4

Throughout this section, we denoteE1andE2by real vector spaces. It is well known15that

a functionf :E1 → E2satisfies the quadratic functional equation

fxy fx−y 2fx 2fy 2.1

if and only if there exists a unique symmetric biadditive functionBsuch thatfx Bx, x

for allx∈E1. The biadditive functionBis given by

Bx, y 1

4

fxy−fx−y. 2.2

Stability problems of quadratic functional equations can be found in 16–19. Similarly, a functionf :E1 → E2satisfies the quartic functional equation1.3if and only if there exists

a symmetric biquadratic functionF : E1×E1 → E2 such thatfx Fx, xfor allx∈ E1

see12. We now present the general solution of1.4in the class of functions between real vector spaces.

Proof. Suppose thatfsatisfies1.3. Puttingxy0 in1.3we havef0 0. Also letting

x0 in1.3we getf−y fy. Using an induction argument we may assume that1.4

is true for allawith 1< a≤k. Replacingybyxyandabykin1.4we have

fk1xyfk−1x−y

k2f2xy k2fy 2k2k2−1fx−2k2−1fxy. 2.3

Substitutingyby−yin2.3and using the evenness offwe get

fk1x−yfk−1xy

k2f2x−y k2fy 2k2k2−1fx−2k2−1fx−y. 2.4

Adding2.3to2.4yields

fk1xyfk1x−y

−fk−1xyfk−1x−yk2f2xy f2x−y

4k2k2−1fx 2k2fy−2k2−1fxy fx−y.

2.5

According to the inductive assumption forak−1,2.5can be rewritten as

fk1xyfk1x−y

−k−12fxy k−12fx−y

2k−12k−12−1fx−2k−12−1fy

k24fxy 4fx−y 24fx−6fy

4k2_k2_{−1fx 2k}2_fy−2k2_{−1fxy fx−y}

k12fxy k12fx−y

2k12k12−1fx−2k12−1fy

2.6

which proves the validity of1.4forak1. For a negative integera <−1, replacingaby

−a > 1 one can easily prove the validity of1.4. Therefore1.3implies1.4for any fixed integerawitha /0,±1.

We now prove the converse. For each fixed integerawith a /0,±1, we assume that

f :E1 → E2satisfies1.4. Puttingx y 0 in1.4we havef0 0. Also lettingx 0 in1.4we getf−y fyfor ally ∈X. Settingy0 in1.4we obtain the homogeneity propertyfax a4_{fxfor allx∈X. Replacingybyxayin1.4we have}

faxy xfax−y−x

Interchangingyinto−yin2.7yields

fax−y xfaxy−x

a2f2x−ay a6fy 2a2a2−1fx−2a2−1fx−ay. 2.8

Replacingxandybyxyandxin1.4we get

faxy xfaxy−x

a2f2xy a2fy 2a2a2−1fxy−2a2−1fx. 2.9

Substitutingyby−yin2.9gives

fax−y xfax−y−x

a2f2x−y a2fy 2a2a2−1fx−y−2a2−1fx. 2.10

Plugging2.7into2.8, and using2.9and2.10we have

a2f2xay f2x−ay2a6fy 4a2a2−1fx −2a2−1fxay fx−ay

a2_{f2xy f2x−y2a}2_fy−4a2_−1fx

2a2a2−1fxy fx−y.

2.11

Replacingxandyby 2xandayin1.4, respectively, we get

a4f2xy a4f2x−y

a2f2xay a2f2x−ay 2a2a2−1f2x−2a4a2−1fy. 2.12

Settingybyayin1.4and dividing bya2_{we obtain}

a2_{fxy a}2_fx−y

fxay fx−ay 2a2−1fx−2a2a2−1fy. 2.13

It follows from2.12and2.13that2.11can be rewritten in the form

Using an induction argument in2.14, it is easy to see thatfsatisfies the following functional

Now we are going to prove the Hyers-Ulam-Rassias stability for quartic functional equations. LetXbe a real vector space and letYbe a Banach space.

Theorem 3.1. Letφ:X2 _{→ R: 0,∞be a mapping such that}

for allx, y ∈ X. Then there exists a unique quartic mappingT : X → Y which satisfies quartic functional equation1.4and the inequality

for allx∈X. To prove the uniqueness, let us assume that there exists another quartic mapping

S : X → Y which satisfies 1.4 and the inequality 3.12. Obviously, we haveTak_x

a4k_TxandSak_{x a}4k_{Sxfor allk∈N, x∈X. Thus, we have}

Tx−Sxa−4kTakx−Sakx

≤a−4kTakx−gakxgakx−Sakx

≤a−4

∞

jk

φajx,0

a4j

3.13

for all x ∈ X. Lettingk → ∞, we must have Sx Tx for allx. This completes the proof.

Corollary 3.2. Letabe fixed integer witha /0,±1and let, p, qbe real numbers such that≥0and either0≤p, q <4orp, q >4. Suppose that a mappingf:X → Ysatisfies the inequality

faxy fax−y−a2fxy−a2fx−y

−2a2a2−1fx 2a2−1fy≤ x p y q 3.14

for allx, y∈X. Then there exists a unique quartic mappingT :X → Y which satisfies1.4and the inequality

fx− f0

a2₁−Tx

≤

2a4_{− |a|}p x

p _3.15

for allx∈Xand for allx∈X\ {0}ifp <0. The mappingT is given by

Tx lim

k→ ∞

fak_x

a4k if p, q <4,

Tx lim

k→ ∞a

4k_f

x

ak if p, q >4

3.16

for allx∈X.

Corollary 3.3. Leta be fixed integer witha /0,±1 and ≥ 0 be a real number. Suppose that a mappingf:X → Ysatisfies the inequality

_{faxy fax−y−a}2_fxy−a2_fx−y

−2a2a2−1fx 2a2−1fy≤ 3.17

for allx, y∈X. Then there exists a unique quartic mappingT :X → Y defined by

Tx lim

k→ ∞

fak_x

which satisfies1.4and the inequality

fx− f0

a2₁−Tx

≤

2a4₋₁ 3.19

for allx∈X.

4. Stability of

1.4

in Generalized Functions

In this section, we reformulate and prove the stability theorem of the quartic functional equation 1.4 in the spaces of some generalized functions such as SRn _{of tempered}

distributions andFRn_{of Fourier hyperfunctions. We first introduce briefly spaces of some}

generalized functions. Here we use the multi-index notations,|α|α1· · ·αn,α!α1!· · ·αn!,

xα_xα1

1 · · ·x

αn

n and∂α∂α₁1· · ·∂αnn, forx x1, . . . , xn∈Rn,α α1, . . . , αn∈Nn0, whereN0is

the set of non-negative integers and∂j∂/∂xj.

Definition 4.1 see 20, 21. We denote by SRn _{the Schwartz space of all infinitely}

differentiable functionsϕinRn_satisfying

ϕ α,βsup x∈Rn

xα∂βϕx<∞ 4.1

for allα,β∈Nn

0, equipped with the topology defined by the seminorms · α,β. A linear form

uonSRn_{is said to be tempered distribution if there is a constantC≥0 and a nonnegative}

integerNsuch that

_{u, ϕ≤C}

|α|,|β|≤N

sup

x∈Rn

_xα_∂β_{ϕ 4.2}

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

Imposing growth conditions on · α,βin4.1a new space of test functions has emerged

as follows.

Definition 4.2 see22. We denote by FRn _{the Sato space of all infinitely differentiable}

functionsϕinRnsuch that

ϕ A,Bsup x,α,β

_xα_∂β_ϕx

A|α|B|β|α!β! <∞ 4.3

for some positive constantsA, B depending only onϕ. We say thatϕj → 0 asj → ∞if

ϕj A,B → 0 asj → ∞for someA, B >0, and denote byFRnthe strong dual ofFRnand

It can be verified that the seminorms4.3are equivalent to

ϕ h,k sup x∈Rn_{, α∈N}n

0

∂α_ϕxexpk|x|

h|α|α! <∞ 4.4

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

FRn_{→ SR}n_{, SR}n_{→ FR}n_{. 4.5}

From the above inclusions it suffices to say that we consider1.4in the spaceFRn_{. Note}

that3.14itself makes no sense in the spaces of generalized functions. Following the notions as in23–25, we reformulate the inequality3.14as

u◦A1u◦A2−a2u◦B1−a2u◦B2

−2a2a2−1u◦P12a2−1u◦P2≤|x|p|y|q, 4.6

whereA1x, y axy, A2x, y ax−y, B1x, y xy, B2x, y x−y, P1x, y x, P2x, y y. Here◦denotes the pullbacks of generalized functions. Also |·|denotes the

Euclidean norm and the inequality v ≤ψx, yin4.6means that|v, ϕ| ≤ ψϕ L1for all test functionsϕx, ydefined onR2n_{. We refer tosee20, Chapter VIfor pullbacks and to}

21,23–26for more details ofSRnandFRn.

Ifp <0, q <0, the right side of4.6does not define a distribution. Thus, the inequality

4.6makes no sense in this case. Also, ifp4, q4, it is not known whether Hyers-Ulam-Rassias stability of 1.4holds even in the classical case. Thus, we consider only the case 0≤p, q <4 orp, q >4.

In order to prove the stability problems of quartic functional equations in the space of

F_Rn_{we employ then-dimensional heat kernel, that is, the fundamental solutionE} txof

the heat operator∂t−ΔxinRnx×Rt given by

Etx

4πt−n/2exp− |x|2/4t, t >0,

0, t≤0. 4.7

Since for eacht >0,Et·belongs toFRn, the convolution

ux, t u∗Et

x uy, Etx−y

, x∈Rn, t >0 4.8

is well defined for eachu ∈ FRn_{, which is called the Gauss transform ofu. In connection}

with the Gauss transform it is well known that the semigroup property of the heat kernel

Et∗Es

x Etsx 4.9

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

∂

∂t−Δ ux, t 0 4.10

satisfying

iThere exist positive constantsC, MandNsuch that

ux, t≤Ct−M1|x|N in Rn_{×0, δ. 4.11}

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

u, ϕ lim

t→0

ux, tϕxdx. 4.12

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

4.11can be uniquely expressed asUx, t ux, tfor someu ∈ SRn_{. Similarly, we can}

represent Fourier hyperfunctions as initial values of solutions of the heat equation as a special case of the resultssee28. In this case, the estimate4.11is replaced by the following.

For everyε >0 there exists a positive constantCεsuch that

ux, t≤Cεexp

ε

|x|1 t

inRn×0, δ. 4.13

We note that the Gauss transform

ψpx, t:

|ξ|pEtx−ξdξ 4.14

is well defined andψpx, t → |x|plocally uniformly ast → 0. Alsoψpx, tsatisfies

semi-homogeneity property

ψp

rx, r2t|r|pψpx, t 4.15

for allr∈R.

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

Theorem 4.3. Letabe fixed integer witha /0,±1and let, p, qbe real numbers such that≥0and either0≤p, q <4orp, q >4. Suppose thatuinSRn_orFRn_{satisfies the inequality4.6. Then}

there exists a unique quartic mappingTxwhich satisfies1.4and the inequality

u− c

a2₁−Tx

≤

2a4_{− |a|}p|x|

p_{, 4.16}

Proof. Definev:u◦A1u◦A2−a2_u◦B1−a2_u◦B2−2a2_a2_−1u◦P12a2_−1u◦P2.

On the other hand, we figure out

where u is the Gauss transform of u. Thus, inequality4.6 is converted into the classical functional inequality

uaxy, a2tsuax−y, a2ts−a2uxy, ts

−a2ux−y, ts−2a2a2−1ux, t 2a2−1uy, s ≤ψpx, t ψqy, s

for allx, y∈Rn_{, t, s >0. In view of4.20, it can be verified that}

and dividing the result by 2a4_{we get}

which is written in the form

Using induction arguments and triangle inequalities we have

for allx ∈ Rn_{, t > 0. On the other hand, replacingx, y, t, sbya}k_{x, a}k_{y, a}2k_{t, a}2k_sin4.20,

respectively, and then dividing the result bya4k_{we get}

a−4kuakaxy, a2ka2tsuakax−y, a2ka2ts −a2uakxy, a2kts−a2uakxy, a2kts −2a2_a2_−1uak_{x, a}2k_t2a2_−1uak_{y, a}2k_s

≤a−4kψp

akx, a2ktψq

aky, a2ks

|a|p−4kψpx, t |a|q−4kψqy, s

.

4.28

Now lettingk → ∞we see by definition ofgthatgsatisfies

gaxy, a2tsgax−y, a2ts

a2gxy, ts a2gx−y, ts

2a2a2−1gx, t−2a2−1gy, s

4.29

for allx, y∈Rn_{, t, s >0. Lettingk → ∞in4.25yields}

vx, t−gx, t≤

2a4_{− |a|}pψpx, t. 4.30

To prove the uniqueness ofgx, t, we assume thathx, tis another function satisfying4.29

and4.30. Settingy0 ands → 0in4.29we have

gakx, a2kta4kgx, t, hakx, a2kta4khx, t 4.31

for allk∈R, x∈Rn_{, t >0. Then it follows from4.30and4.31that}

gx, t−hx, ta−4kgakx, a2kt−hakx, a2kt ≤a−4kgakx, a2kt−vakx, a2kt

a−4kvakx, a2kt−hakx, a2kt

≤

|a|4−pk_a4_{− |a|}pψpx, t

4.32

for allk ∈N, x ∈Rn_{, t >0. Lettingk → ∞, we havegx, t hx, tfor allx∈ R}n_{, t >0.}

This proves the uniqueness.

It follows from the inequality4.30that we get

vx, t−gx, t, ϕ≤

2a4_{− |a|}p

ψpx, t, ϕ

for all test functions ϕ. Since gx, t is given by the uniform limit of the sequence Lettingt → 0we have the inequality

respectively, and lettings → 0and then multiplying the result bya4_{we have}

Using induction argument and triangle inequality we obtain

As an immediate consequence, we have the following corollary.

Corollary 4.4. Letabe fixed integer witha /0,±1and ≥ 0be a real number. Suppose thatuin S_Rn_orFRn_{satisfies the inequality}

u◦A1u◦A2−a2u◦B1−a2u◦B2−2a2

a2−1u◦P12

a2−1u◦P2≤. 4.41

Then there exists a unique quartic mappingTxwhich satisfies1.4and the inequality

u− c

a2₁ −Tx

≤

2a4₋₁, 4.42

wherec:lim sup_t→0u0, t.

Acknowledgments

The first author was supported by the second stage of the Brain Korea 21 Project, The Development Project of Human Resources in Mathematics, KAIST, in 2009. The second author was supported by the Special Grant of Sogang University in 2005.

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