Approximate Quartic and Quadratic Mappings in Quasi-Banach Spaces

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Volume 2011, Article ID 734567,18pages doi:10.1155/2011/734567

Research Article

Approximate Quartic and Quadratic Mappings in Quasi-Banach Spaces

M. Eshaghi Gordji,

¹

H. Khodaei,

¹

and Hark-Mahn Kim

²

1Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran

2Department of Mathematics, Chungnam National University, 220 Yuseong-Gu, Daejeon 305-764, Republic of Korea

Correspondence should be addressed to Hark-Mahn Kim,[email protected] Received 17 March 2011; Accepted 13 May 2011

Academic Editor: Petru Jebelean

Copyrightq2011 M. Eshaghi Gordji et al. 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.

we establish the general solution for a mixed type functional equation of aquartic and a quadratic mapping in linear spaces. In addition, we investigate the generalized Hyers-Ulam stability inp- Banach spaces.

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam1in 1940, concerning the stability of group homomorphisms. LetG1,·be a group, and letG2,∗be a metric group with the metricd·,·. Given > 0, does there exist aδ > 0 such that if a mappingh: G₁ → G₂ satisfies the inequalitydhx·y, hx∗hy < δfor allx, y ∈ G₁, then there exists a homomorphismH :G1 → G2withdhx, Hx < for allx∈ G1? In other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers 2 gave a first aﬃrmative answer to the question of Ulam for Banach spaces. Let f:X → Xbe a mapping between Banach spaces such that

f xy

−fx−f

y≤δ, 1.1

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for allx, y∈X, and for someδ >0. Then, there exists a unique additive mappingT :X → X such that

fx−Tx≤δ, 1.2 for allx∈X.

The result of Hyers was generalized by Aoki3for approximate additive function and by Rassias4for approximate linear function by allowing the diﬀerence Cauchy equation fxy−fx−fy to be controlled by εx^p y^p. Taking into consideration a lot of influence of Ulam, Hyers and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias may be called the Hyers-Ulam-Rassias stabilitysee5,6. In 1994, a generalization of Rassias theorem was obtained by G˘avruta7, who replacedεx^py^pby a general control functionϕx, y.

The functional equation f

xy f

x−y

2fx 2f y

1.3 is related to a symmetric biadditive function8–10. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation1.3is said to be a quadratic function. It is well known that a functionf between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive functionB1 such that fx B1x, xfor allxin the vector space. The biadditive functionB1is given by

B1

x, y 1 4

f xy

−f x−y

. 1.4

A Hyers-Ulam stability problem for the quadratic functional equation1.3was proved by Skof for functionsf:X → Y, whereXis normed space andY is Banach spacesee11. In the paper12, Czerwik proved the Hyers-Ulam-Rassias stability of1.3.

Lee et al.13considered the following functional equation:

f 2xy

f 2x−y

4f xy

4f x−y

24fx−6f y

. 1.5

In fact, they proved that a functionfbetween two real vector spacesXandYis a solution of 1.5if and only if there exists a unique symmetric biquadratic functionB2:X×X → Ysuch thatfx B2x, xfor allx∈X. The biquadratic functionB2is given by

B2

x, y 1 12

f xy

f x−y

−2fx−2f y

. 1.6

It is easy to show that the functionfx ax⁴satisfies the functional equation1.5, which is called the quartic functional equationsee also14.

Jun and Kim15have obtained the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation. In addition, the generalized Hyers-Ulam stability for a mixed type of cubic, quadratic, and additive functional equation has been investigated by Gordji and Khodaei16 see also17,18. The stability problems for several mixed types

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of functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem19–27.

In this paper, we deal with the following functional equation derived from quartic and quadratic functions:

f kxy

f kx−y k²f

xy k²f

x−y k²

k²−1 6

f2x−4fx

−2 k²−1

f

y 1.7

for fixed integersk /0,±1. It is easy to see that the functionfx ax⁴bx²is a solution of the functional equation1.7. In the sequel, we investigate the general solution of functional equation1.7whenfis a function between vector spaces, and then we prove the generalized Hyers-Ulam stability of 1.7 in the spirit of Hyers, Ulam, and Rassias using the direct method.

We recall some basic facts concerning quasi-Banach spaces and some preliminary results.

Definition 1.1see28,29. LetXbe a real linear space. A quasinorm is a real-valued function onXsatisfying the following:

1x ≥0 for allx∈Xandx 0 if and only ifx 0, 2λ·x |λ| · xfor allλ∈^Êand allx∈X,

3there is a constantM≥1 such thatxy ≤Mxyfor allx, y∈X.

The pairX, · is called a quasinormed space if · is a quasinorm onX.

The smallest possibleMis called the modulus of concavity of · . A quasi-Banach space is a complete quasinormed space. A quasinorm · is called ap-norm0< p≤1if

xy^p≤ x^py^p, 1.8 for allx, y∈X. In this case, a quasi-Banach space is called ap-Banach space.

Given ap-norm, the formuladx, y: x−y^pgives us a translation invariant metric onX. By the Aoki-Rolewicz Theorem29, each quasinorm is equivalent to somep-normsee also28. Since it is much easier to work withp-norms, henceforth we restrict our attention mainly top-norms.

Lemma 1.2see17. Letx1, x2, . . . , xnbe nonnegative real numbers. Then, one has _n

i 1

x_i

p

≤ⁿ

i 1

x_i^p, 1.9

for a positive real numberpwithp≤1.

2. General Solution

We here present the general solution of1.7.

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Theorem 2.1. Let bothXandY be real vector spaces. A functionf :X → Y satisfies1.7for all x, y∈Xif and only if there exists a unique symmetric biquadratic functionB2 :X×X → Y and a unique symmetric biadditive functionB₁:X×X → Y such that

fx B₂x, x B₁x, x, 2.1

for allx∈X.

Proof. Letfsatisfy1.7and letg, h:X → Ybe functions defined by

gx: f2x−16fx, hx: f2x−4fx, 2.2

for allx∈X. We claim that the functionsgandhare quadratic and quartic, respectively.

Lettingx y 0 in1.7, we havef0 0. By puttingx 0 in1.7, one leads to the evennessf−y fyoff. Replacingybyxyin1.7, we have

f

k1xy f

k−1x−y

k²f 2xy

k²f

−y k²

k²−1 6

f2x−4fx 2

1−k² f

xy ,

2.3

for allx, y∈X. Replacingyby−yin2.3, we obtain

f

k1x−y f

k−1xy

k²f 2x−y

k²f y

k² k²−1

6

f2x−4fx 2

1−k² f

x−y ,

2.4

for allx, y∈X. Adding2.3to2.4, we get by evenness off,

f

k1xy f

k1x−y f

k−1xy f

k−1x−y

k² f

2xy f

2x−y 2k²

k²−1 6

f2x−4fx

2

1−k² f

xy f

x−y

2k²f y

,

2.5

for allx, y∈X. From the substitutiony kxyin1.7, we have by evenness off,

f

2kxy f

y k²f

k1xy k²f

k−1xy k²

k²−1 6

f2x−4fx 2

1−k² f

kxy ,

2.6

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for allx, y∈X. Replacingyby−yin2.6, we get

f

2kx−y f

−y k²f

k1x−y k²f

k−1x−y

k² k²−1

6

f2x−4fx 2

1−k² f

kx−y ,

2.7

for allx, y∈X. Adding2.6to2.7, we get by evenness off,

f

2kxy f

2kx−y k²

f

k1xy f

k1x−y f

k−1xy f

k−1x−y

2k² k²−1

6

f2x−4fx 2

1−k² f

kxy f

kx−y

−2f y

, 2.8

for allx, y∈X. By using1.7and2.5, it follows from2.8that

f

2kxy f

2kx−y k²

k²

f 2xy

f

2x−y 2k²

k²−1 6

f2x−4fx

2

1−k² f

xy f

x−y

2k²f y

2

1−k² k²f

xy k²f

x−y k²

k²−1 6

f2x−4fx 2

1−k² f

y 2k²

k²−1 6

f2x−4fx

−2f y

,

2.9

for allx, y∈X. If we replacexby 2xin1.7, then we get that

f

2kxy f

2kx−y k²f

2xy k²f

2x−y k²

k²−1 6

f4x−4f2x 2

1−k² f

y ,

2.10

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for allx, y∈X. It follows from2.9and2.10that

k²

k² f

2xy f

2x−y 2k²

k²−1 6

f2x−4fx

2

1−k² f

xy f

x−y

2k²f y

2

1−k² k²f

xy k²f

x−y k²

k²−1 6

f2x−4fx 2

1−k² f

y 2k²

k²−1 6

f2x−4fx

−2f y k²f

2xy k²f

2x−y k²

k²−1 6

f4x−4f2x 2

1−k² f

y ,

2.11

for allx, y∈X. On the other hand, puttingy 0 in1.7, we get

fkx k²fx k² k²−1

12

f2x−4fx

, 2.12

for allx∈X. Puttingy xin1.7, we get

fk1x fk−1x k²f2x k² k²−1

6

f2x−4fx 2

1−k² fx,

2.13

for allx∈X. Puttingy kxin1.7and using the evenness off, we obtain

f2kx k²

fk1x fk−1x k²

k²−1 6

f2x−4fx 2

1−k² fkx,

2.14

for allx∈X. Lettingy 0 in2.10, we have

f2kx k²f2x k² k²−1

12

f4x−4f2x

, 2.15

for allx∈X. It follows from2.14and2.15that k²

k²−1 12

f4x−4f2x k²

fk1x fk−1x k²

k²−1 6

f2x−4fx

2 1−k²

fkx−k²f2x,

2.16

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for allx∈X. Now, by using2.12,2.13and2.16, we lead to k²

k²−1 12

f4x−4f2x k²

k²f2x k² k²−1

6

f2x−4fx 2

1−k² fx

2

1−k²

k²fx k² k²−1

12

f2x−4fx

k² k²−1

6

f2x−4fx

−k²f2x,

2.17

for allx∈X. Finally, comparing2.11with2.17, then we conclude that f

2xy f

2x−y 4f

xy 4f

x−y 2

f2x−4fx

−6f y

, 2.18

for allx, y∈X. Replacingyby 2yin2.18, we get f

2x2y f

2x−2y 4f

x2y 4f

x−2y 2

f2x−4fx

−6f 2y

,

2.19

for allx, y∈X. Interchangingxwithyin2.18, one gets f

x2y f

x−2y 4f

xy 4f

x−y 2

f 2y

−4f y

−6fx, 2.20

for allx, y∈X. It follows from2.19and2.20that f

2 xy

−16f xy

f 2

x−y

−16f x−y 2

f2x−16fx 2

f 2y

−16f y

, 2.21

for allx, y∈X. This means that g

xy g

x−y

2gx 2g y

, 2.22

for allx, y∈X. So the functiong:X → Ydefined bygx: f2x−16fxis quadratic.

To prove thath:X → Y defined byhx: f2x−4fxis quartic, we need to show that

h 2xy

h 2x−y

4h xy

4h x−y

24hx−6h y

, 2.23

for allx, y∈X. Replacingxandyby 2xand 2yin2.18, respectively, we obtain f

2

2xy f

2

2x−y 4f

2 xy

4f 2

x−y 2

f4x−4f2x

−6f 2y

, 2.24

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for allx, y ∈ X. But, since g2x 4gxfor allx ∈ X, where g : X → Y is a quadratic function defined above, we see that

f4x 20f2x−64fx, 2.25

for allx∈X. Hence, according to2.24and2.25, we get f

2

2xy f

2

2x−y 4f

2 xy

4f 2

x−y 32

f2x−4fx

−6f 2y

, 2.26

for allx, y∈X. By multiplying 4 on both sides of2.18, we get that 4f

2xy 4f

2x−y 16f

xy 16f

x−y 8

f2x−4fx

−24f y

,

2.27

for allx, y∈X. If we subtract the last equation from2.26, then we arrive at f

2

2xy

−4f 2xy

f 2

2x−y

−4f 2x−y 4

f 2

xy

−4f xy

4 f

2 x−y

−4f x−y 24

f2x−4fx

−6 f

2y

−4f y

,

2.28

for allx, y ∈ X. This means that hsatisfies2.23and, therefore, the functionh : X → Y is quartic. Thus, there exists a unique symmetric biquadratic functionB₂ :X×X → Y and a unique symmetric biadditive functionB₁ : X ×X → Y such thathx 12B₂x, xand gx −12B1x, xfor allx∈Xsee8,13. Therefore, we obtain from2.2that

fx 1

12hx− 1

12gx B2x, x B1x, x, 2.29 for allx∈X.

The proof of the converse is trivial.

3. Generalized Hyers-Ulam Stability

From this point on, assume thatX is a quasinormed space with quasinorm · Xand thatY is ap-Banach space withp-norm · Y. LetMbe the modulus of concavity of · Y.

Before taking up the main subject, given a mapping f : X → Y, we define the diﬀerence operatorDf :X×X → Y by

Df

x, y : f

kxy f

kx−y

−k²f xy

−k²f x−y

−k² k²−1

6

f2x−4fx 2

k²−1 f

y ,

3.1

for allx, y∈X. Letϕ^px, y: ϕx, y^pfor notational convenience.

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Theorem 3.1. Letj ∈ {−1,1}be fixed and letϕq:X×X → 0,∞be a function such that

nlim→ ∞4^njϕ_q x

2^nj, y 2^nj

0, 3.2

for allx, y∈Xand

∞ i 1j/2

4^ipjϕ^p_q u

2^ij, y 2^ij

<∞, 3.3

for allu, y∈ {x,0,2x,0,x, x,x, kx:x∈X}. Suppose that an even functionf :X → Y withf0 0 satisfies the inequality

Df

x, y

Y ≤ϕq

x, y

, 3.4

for allx, y∈X. Then, there exists a unique quadratic functionQ:X → Ysuch that f2x−16fx−Qx

Y ≤ M² 4

ψqx_1/p

, 3.5

for allx∈X, where

ψ_qx: ^∞

i 1j/2

4^ipj k^2pk²−1^p

12k²_p ϕ^p_q

x 2^ij, x

2^ij

12

k²−1_p ϕ^p_q

x 2^ij,0

6^pϕ^p_q 2x

2^ij,0

12^pϕ^p_q x

2^ij,kx 2^ij

.

3.6

Proof. Letj 1. Settingy 0 in3.4, we have

2fkx−2k²fx−k² k²−1

6

f2x−4fx

Y

≤ϕqx,0, 3.7

for allx∈X. Puttingy xin3.4, we obtain

fk1x fk−1x−k²f2x−k² k²−1

6

f2x−4fx 2

k²−1 fx

Y

≤ϕqx, x,

3.8 for allx∈X. Replacingxby 2xin3.7, we see that

2f2kx−2k²f2x−k² k²−1

6

f4x−4f2x

Y

≤ϕ_q2x,0, 3.9

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for allx∈X. Settingybykxin3.4and using the evenness off, we get

f2kx−k²fk1x−k²fk−1x 2 k²−1

fkx−k² k²−1

6

f2x−4fx

Y

≤ϕ_qx, kx,

3.10

for allx∈X. It follows from3.9and3.10that

k²f2x k² k²−1

12

f4x−4f2x

−k²fk1x−k²fk−1x

2 k²−1

fkx− k² k²−1

6

f2x−4fx

Y

≤M 1

2ϕ_q2x,0 ϕ_qx, kx

,

3.11

for allx∈X. Also, it follows from3.7and3.8that

k²fk1x k²fk−1x−2 k²−1

fkx−k⁴f2x

−k² k²−1

6 f2x−4fx 4k²

k²−1 fx

Y

≤M

k²ϕ_qx, x k²−1

ϕ_qx,0 , 3.12

for allx∈X. Finally, using3.11and3.12, we obtain that f4x−20f2x 64fx

Y

≤ M² k²k²−1

12k²ϕqx, x 12 k²−1

ϕqx,0 6ϕq2x,0 12ϕqx, kx M²ψqx,

3.13

where

ψqx: 1 k²k²−1

12k²ϕqx, x 12 k²−1

ϕqx,0 6ϕq2x,0 12ϕqx, kx ,

3.14 for allx∈X. Letg :X → Y be a function defined bygx: f2x−16fxfor allx∈X.

From3.13, we conclude that

g2x−4gxY ≤M²ψ_qx, 3.15

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for allx∈X. If we replacexin3.15byx/2ⁿ¹and multiply both sides of3.15by 4ⁿ, then we get

4ⁿ¹g x

2ⁿ¹

−4ⁿgx 2ⁿ

Y

≤M²4ⁿψ_q x

2ⁿ¹

, 3.16

for allx∈Xand all non-negative integersn. SinceYis ap-Banach space, the inequality3.16 gives

4ⁿ¹g x

2ⁿ¹

−4^mgx 2^m

^p

Y

≤ⁿ

i m

4ⁱ¹g x

2ⁱ¹

−4ⁱg x

2ⁱ ^p

Y

≤M^2p n i m

4^ipψ_q^p x

2ⁱ¹

, 3.17 for all nonnegative integersnandmwithn≥mand allx∈X. Since 0< p≤1, by Lemma1.2 and3.14, we conclude that

ψ^p_qx≤ 1 k^2pk²−1^p

12k²_p

ϕ^p_qx, x 12

k²−1_p

ϕ^p_qx,0 6^pϕ^p_q2x,0 12^pϕ^p_qx, kx , 3.18

for allx∈X. Therefore, it follows from3.3and3.18that ∞

i 1

4^ipψ_q^p x

2ⁱ

<∞, 3.19

for allx∈X. It follows from3.17and3.19that the sequence{4ⁿgx/2ⁿ}is a Cauchy for allx∈X. SinceY is complete, the sequence{4ⁿgx/2ⁿ}converges for allx∈X. So one can define a functionQ:X → Y by

Qx lim

n→ ∞4ⁿgx 2ⁿ

, 3.20

for allx∈X. Lettingm 0 and passing the limitn → ∞in3.17, we get gx−Qx^p_Y ≤M^2p

∞ i 0

4^ipψ_q^p x

2ⁱ¹

M^2p 4^p

∞ i 1

4^ipψ^p_q x

2ⁱ

, 3.21

for allx∈X. Thus3.5follows from3.18and3.21. Now we show thatQis quadratic. It follows from3.16,3.19and3.20that

Q2x−4Qx_Y lim

n→ ∞

4ⁿg x

2ⁿ⁻¹

−4ⁿ¹gx 2ⁿ

Y

4 lim

n→ ∞

4ⁿ⁻¹g x

2ⁿ⁻¹

−4ⁿgx 2ⁿ

Y

≤M²lim

n→ ∞4ⁿψq

x 2ⁿ

0,

3.22

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for allx∈X. So,

Q2x 4Qx, 3.23

for allx∈X. On the other hand, it follows from3.2,3.4and3.20that DQ

x, y

Y lim

n→ ∞4ⁿDg

x 2ⁿ, y

2ⁿ

Y lim

n→ ∞4ⁿ Df

x 2ⁿ⁻¹, y

2ⁿ⁻¹

−16Df

x 2ⁿ, y

2ⁿ

Y

≤Mlim

n→ ∞4ⁿ Df

x 2ⁿ⁻¹, y

2ⁿ⁻¹

Y

16Df

x 2ⁿ, y

2ⁿ

Y

≤Mlim

n→ ∞4ⁿ

ϕq

x 2ⁿ⁻¹, y

2ⁿ⁻¹

16ϕq

x 2ⁿ, y

2ⁿ

0,

3.24 for all x, y ∈ X. Hence the function Q satisfies1.7. Thus, by Theorem 2.1, the function x Q2x−16Qxis quadratic. Therefore,3.23implies that the functionQis quadratic.

Now, to prove the uniqueness property ofQ, let Q : X → Y be another quadratic function satisfying3.5. It follows from3.3that

nlim→ ∞4^np ∞

i 1

4^ipϕ^p_q u

2ⁿⁱ, y 2ⁿⁱ

nlim→ ∞

∞ i n1

4^ipϕ^p_q u

2ⁱ,y 2ⁱ

0, 3.25

for allu, y∈ {x,0,2x,0,x, x,x, kx:x∈X}. Hence,

nlim→ ∞4^npψq

x 2ⁿ

0, 3.26

for allx∈X. It follows from3.5,3.20and3.26that Qx−Qx^p

Y lim

n→ ∞4^npgx 2ⁿ

−Qx 2ⁿ

^p

Y ≤ M^2p 4^p lim

n→ ∞4^npψq

x 2ⁿ

0, 3.27

for allx∈X. SoQ Q.

Forj −1, we can prove the theorem by a similar argument.

Corollary 3.2. Letθ, r, sbe nonnegative real numbers such thatr,s >2 orr,s <2. Suppose that an even functionf:X → Y withf0 0 satisfies the inequality

Df

x, y

Y ≤θ

x^r_Xy^s

X

, 3.28

for allx, y∈X. Then there exists a unique quadratic functionQ:X → Ysatisfying f2x−16fx−Qx

Y ≤ M²θ

k²k²−1γqx, 3.29

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γ_qx

⎛

⎜⎝12^p

k^2p k²−1^p

2^r−1p1

|4^p−2^rp| x^rp_X 12^p

k^2pk^sp

|4^p−2^sp| x^sp_X

⎞

⎟⎠

1/p

. 3.30

Proof. In Theorem3.1, puttingϕ_qx, y: θx^r_Xy^s_Xfor allx, y∈X, we get the desired result.

Corollary 3.3. Letθ≥0 andr, s >0 be real numbers such thatλ: rs /2. Suppose that an even functionf:X → Y withf0 0 satisfies the inequality

D_f x, y

Y ≤θx^r_Xy^s

X, 3.31

for allx, y∈X. Then there exists a unique quadratic functionQ:X → Ysatisfying f2x−16fx−Qx

Y ≤ M²θ

k²k²−1 12^p

k^2pk^sp 4^p−2^λp

1/p

x^λ_X, 3.32

for allx∈X.

Proof. In Theorem3.1, takingϕ_qx, y: θx^r_Xy^s_X, for allx, y∈X, we arrive at the desired result.

Theorem 3.4. Letj ∈ {−1,1}be fixed and letϕv:X×X → 0,∞be a function such that

nlim→ ∞16^njϕv

x 2^nj, y

2^nj

0, 3.33

for allx, y∈Xand

∞ i 1j/2

16^ipjϕ^p_v u

2^ij, y 2^ij

<∞, 3.34

Dfx,y

Y ≤ϕv

x, y

, 3.35

for allx, y∈X. Then there exists a unique quartic functionV :X → Y such that f2x−4fx−Vx

Y ≤ M² 16

ψ_vx_1/p

, 3.36

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ψ_vx: ^∞

i 1j/2

16^ipj k^2pk²−1^p

12k²_p ϕ^p_v

x 2^ij, x

2^ij

12

k²−1_p ϕ^p_v

x 2^ij,0

6^pϕ^pv

2x 2^ij,0

12^pϕ^pv

x 2^ij,kx

2^ij

.

3.37

Proof. Being similar to the proof of Theorem3.1, we omit its proof.

Corollary 3.5. Letθ, r, sbe nonnegative real numbers such thatr, s >4 orr, s <4. Suppose that an even functionf :X → Y withf0 0 satisfies the inequality3.28for allx, y ∈X. Then there exists a unique quartic functionV :X → Y satisfying

f2x−4fx−Vx

Y ≤ M²θ

k²k²−1γvx, 3.38 for allx∈X, where

γvx

⎛

⎜⎝12^p k^2p

k²−1_p

2^r−1_p 1

|16^p−2^rp| x^rp_X 12^p

k^2pk^sp

|16^p−2^sp| x^sp_X

⎞

⎟⎠

1/p

, 3.39

for allx∈X.

Corollary 3.6. Letθ≥0 andr, s >0 be real numbers such thatλ: rs /4. Suppose that an even functionf :X → Y withf0 0 satisfies the inequality3.31for allx, y ∈X. Then, there exists a unique quartic functionV :X → Ysatisfying

f2x−4fx−Vx

Y ≤ M²θ

k²k²−1 12^p

1/p

x^λ_X, 3.40

for allx∈X.

Now, we are ready to prove the main theorem concerning the stability problem for 1.7.

Theorem 3.7. Letj ∈ {−1,1}be fixed and letϕ:X×X → 0,∞be a function such that

nlim→ ∞

1−j 2

4^njϕ

x 2^nj, y

2^nj

1j

2

16^njϕ x

2^nj, y 2^nj

0, 3.41

for allx, y∈Xand ∞ i 1j/2

1−j 2

4^ipjϕ^p

u 2^ij, y

2^ij

1j

2

16^ipjϕ^p u

2^ij, y 2^ij

<∞, 3.42

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D_fx, y

Y ≤ϕ x, y

, 3.43

for allx, y ∈X. Then, there exists a unique quadratic functionQ : X → Y and a unique quartic functionV :X → Y such that

fx−Qx−Vx

Y ≤ M³ 192

4

ψ_qx_1/p

ψ_vx_1/p

, 3.44

ψqx: ^∞

i 1j/2

4^ipj k^2pk²−1^p

12k²p

ϕ^p x

2^ij, x 2^ij

12

k²−1p

ϕ^p x

2^ij,0

6^pϕ^p 2x

2^ij,0

12^pϕ^p x

2^ij,kx 2^ij

,

ψ_vx: ^∞

i 1j/2

16^ipj k^2pk²−1^p

12k²_p ϕ^p

x 2^ij, x

2^ij

12

k²−1_p ϕ^p

x 2^ij,0

6^pϕ^p 2x

2^ij,0

12^pϕ^p x

2^ij,kx 2^ij

.

3.45

Proof. By Theorems3.1and3.4, there exists a quadratic functionQ0 :X → Y and a quartic functionV₀:X → Y such that

f2x−16fx−Q0x

Y≤ M² 4

ψqx_1/p

, f2x−4fx−V0x

Y ≤ M² 16

ψvx_1/p , 3.46

for allx∈X. Therefore, it follows from3.46that fx 1

12Q0x− 1

12V0x Y

≤ M³ 192

4

ψqx_1/p

ψvx_1/p

, 3.47

for allx∈X. Thus we obtain3.44by lettingQx −1/12Q0xandVx 1/12V0x for allx∈X.

To prove the uniqueness property ofQandV, letQ, V:X → Ybe another quadratic and quartic functions satisfying3.44. LetQ Q−QandV V −V. Hence,

Qx Vx

Y ≤M fx−Qx−Vx

Y fx−Qx−Vx

Y

!

≤ M⁴ 96

4

ψ_qx_1/p

ψ_vx_1/p ,

3.48

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for allx∈X. Since limn→ ∞4^npjψqx/2ⁿ limn→ ∞16^npjψvx/2ⁿ 0, for allx∈X, we figure out that

nlim→ ∞16ⁿQx 2ⁿ

Vx 2ⁿ

Y 0, 3.49

for allx∈X. Therefore, we getV 0 and thenQ 0.

Corollary 3.8. Letθ, r, sbe nonnegative real numbers such thatr, s >4 or 2< r,s <4 orr, s <2.

Suppose that an even functionf : X → Y with f0 0 satisfies the inequality3.28, for all x, y∈X. Then, there exists a unique quadratic functionQ:X → Y and a unique quartic function V :X → Y such that

fx−Qx−Vx

Y ≤ M³θ

12k²k²−1

γqx γvx

, 3.50

for allx∈X, whereγqxandγvxare defined as in Corollaries3.2and3.5.

Corollary 3.9. Letθ≥0 andr, s >0 be non-negative real numbers such thatλ : rs∈0,2∪ 2,4∪4,∞. Suppose that an even function f : X → Y withf0 0 satisfies the inequality 3.31for allx, y∈X. Then there exist a unique quadratic functionQ:X → Yand a unique quartic functionV :X → Y such that

fx−Qx−Vx

Y

≤ M³θ 12k²k²−1

⎧⎨

⎩ 12^p

1/p

12^p

1/p⎫

⎬

⎭x^λ_X, 3.51

for allx∈X.

Corollary 3.10. Suppose that an even functionf:X → Ywithf0 0 satisfies the inequality Df

x, y

X ≤ε, 3.52

for allx, y∈Xwhereε >0. Then there exist a unique quadratic functionQ:X → Y and a unique quartic functionV :X → Ysuch that

fx−Qx−Vx

Y

≤ M³ε k²k²−1

⎧⎨

⎩

k^2p

k²−1_p

2^−p1 4^p−1

1/p

k^2p k²−1^p2^−p1 16^p−1

1/p⎫

⎬

⎭, 3.53 for allx∈X.

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Acknowledgment

Hark-Mahn Kim was supported by Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technologyno. 2011- 0002614.

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