On the Stability of a Generalized

Volume 2009, Article ID 153084,26pages doi:10.1155/2009/153084

Research Article

On the Stability of a Generalized

Quadratic and Quartic Type Functional Equation in Quasi-Banach Spaces

M. Eshaghi Gordji,

S. Abbaszadeh,

and Choonkil Park

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

2Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea

Correspondence should be addressed to Choonkil Park,baak@hanyang.ac.kr Received 31 May 2009; Accepted 9 September 2009

Recommended by Nikolaos Papageorgiou

We establish the general solution of the functional equationfnxy fnx−y n²fxy n²fx−y 2fnx−n²fx−2n²−1fyfor fixed integersnwithn /0,±1 and investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.

Copyrightq2009 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.

1. Introduction

The stability problem of functional equations originated from a question of Ulam1in 1940, concerning the stability of group homomorphisms. LetG1,·be a group and let G2,∗be a metric group with the metricd·,·.Givenε > 0, does there exist a δ > 0, such that if a mappingh: G₁ → G₂ satisfies the inequalitydhx·y, hx∗hy < δfor allx, y ∈ G₁, then there exists a homomorphismH :G₁ → G₂ withdhx, Hx< εfor allx∈ G₁? In other words, under what condition does there exist 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 2gave a first affirmative answer to the question of Ulam for Banach spaces. Let f:E → Ebe a mapping between Banach spaces such that

f xy

−fx−f

y≤δ 1.1

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

fx−Tx≤δ 1.2

for allx∈E.Moreover, ifftxis continuous int∈Rfor each fixedx∈E,thenT isR-linear.

In 1978, Th. M. Rassias3 provided a generalization of Hyers’ theorem which allows the Cauchy difference to be unbounded. The functional equation

f xy

f x−y

2fx 2f y

1.3

is related to a symmetric biadditive mapping4–7. It is natural that this functional equation is called a quadratic functional equation. In particular, every solution of the quadratic equation 1.3is said to be a quadratic mapping. It is well known that a mappingfbetween real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mappingBsuch thatfx Bx, xfor allxsee4,7. The biadditive mappingBis given by

B x, y

1 4

f xy

−f x−y

. 1.4

The generalized Hyers-Ulam stability problem for the quadratic functional equation1.3was proved by Skof for mappingsf:A → B, whereAis a normed space andBis a Banach space see8. Cholewa9noticed that the theorem of Skof is still true if relevant domainAis replaced by an abelian group. In10, Czerwik proved the generalized Hyers-Ulam stability of the functional equation1.3. Grabiec11has generalized these results mentioned above.

In12, Park and Bae considered the following quartic functional equation:

f x2y

f x−2y

4 f

xy f

x−y 6f

y

−6fx. 1.5

In fact, they proved that a mappingf between two real vector spacesX andY is a solution of1.5if and only if there exists a unique symmetric multiadditive mappingD:X×X×X× X → Y such thatfx Dx, x, x, xfor allx. It is easy to show thatfx x⁴satisfies the functional equation1.5, which is called a quartic functional equationsee also13.

In addition, Kim14has obtained the generalized Hyers-Ulam stability for a mixed type of quartic and quadratic functional equation between two real linear Banach spaces.

Najati and Zamani Eskandani15have established the general solution and the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation, wheneverf is a mapping between two quasi-Banach spacessee also16,17.

Now we introduce the following functional equation for fixed integersnwithn /0,±1:

f nxy

f nx−y

n²f xy

n²f x−y

2fnx−2n²fx−2 n²−1

f y

1.6

in quasi-Banach spaces. It is easy to see that the function fx ax⁴ bx² is a solution of the functional equation1.6. In the present paper we investigate the general solution of the functional equation1.6whenf is a mapping between vector spaces, and we establish the generalized Hyers-Ulam stability of this functional equation wheneverf is a mapping between two quasi-Banach spaces.

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

Definition 1.1See18,19. LetXbe a real linear space. A quasinorm is a real-valued function onXsatisfying the following.

1x ≥0 for allx∈Xandx0 if and only ifx0.

2λ·x|λ| · xfor allλ∈Rand allx∈X.

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

It follows from the condition3that

2m

i1

xi

≤M^m

2m i1

xi,

2m1

i1

xi

≤M^m1

2m1

i1

xi 1.7

for allm≥1 and allx1, x2, . . . , x_2m1∈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 quasi-normed space.

A quasi-norm · 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 on X. By the Aoki-Rolewicz theorem 19 see also 18, each quasi-norm is equivalent to some p-norm. Since it is much easier to work withp-norms, henceforth we restrict our attention mainly top-norms. In20, Tabor has investigated a version of Hyers-Rassias-Gajda theoremsee3,21in quasi-Banach spaces.

2. General Solution

Throughout this section, X and Y will be real vector spaces. We here present the general solution of1.6.

Lemma 2.1. If a mappingf :X → Y satisfies the functional equation1.6, thenfis a quadratic and quartic mapping.

Proof. Lettingxy0 in1.6, we getf0 0. Settingx0 in1.6, we getfy f−y for ally∈X. So the mappingfis even. Replacingxbyxyin1.6and thenxbyx−yin 1.6, we get

f

nx n1y f

nx n−1y n²f

x2y

n²fx 2f

nxny

−2n²f xy

−2 n²−1

f y

, 2.1

f

nx−n−1y f

nx−n1y n²fx n²f

x−2y 2f

nx−ny

−2n²f x−y

−2 n²−1

f

y 2.2

for allx, y∈X. Interchangingxandyin1.6and using the evenness off, we get the relation

f xny

f x−ny n²f

xy n²f

x−y 2f

ny

−2n²f y

−2 n²−1

fx 2.3

for allx, y∈X. Replacingybynyin1.6and then using2.3, we have

f

nxny f

nx−ny n⁴f

xy n⁴f

x−y 2f

ny

2fnx−2n⁴fx−2n⁴f

y 2.4

for allx, y∈X. If we add2.1to2.2and use2.4, then we have

f

nx n1y f

nx−n1y f

nx n−1y f

nx−n−1y n²f

x2y n²f

x−2y 2n²

n²−1 f

xy 2n²

n²−1 f

x−y 4f

ny

4fnx

−4n⁴2n²

fx

−4n⁴−4n²4 f

y

2.5

for allx, y ∈ X. Replacing ybyxy in1.6and then ybyx−y in1.6and using the evenness off, we obtain that

f

n1xy f

n−1x−y n²f

2xy n²f

y

2fnx−2n²fx−2 n²−1

f xy

, 2.6

f

n1x−y f

n−1xy n²f

2x−y n²f

y

2fnx−2n²fx−2 n²−1

f

x−y 2.7

for allx, y∈X. Interchangingxwithyin2.6and2.7and using the evenness off, we get the relations

f

x n1y f

x−n−1y n²f

x2y

n²fx 2f ny

−2n²f y

−2 n²−1

f xy

, 2.8

f

x−n1y f

x n−1y n²f

x−2y

n²fx 2f ny

−2n²f y

−2 n²−1

f

x−y 2.9

for allx, y∈X. Replacingybyn1yin1.6and thenybyn−1yin1.6, we have f

nx n1y f

nx−n1y n²f

x n1y n²f

x−n1y

2fnx−2n²fx−2 n²−1

f

n1y , 2.10 f

nx n−1y f

nx−n−1y n²f

x n−1y n²f

x−n−1y

2fnx−2n²fx−2 n²−1

f

n−1y 2.11

for allx, y∈X. Replacingxbyyin1.6, we obtain

f

n1y f

n−1y n²f

2y

−2

2n²−1 f

y 2f

ny

2.12

for ally∈X. Adding2.10to2.11and using2.8,2.9, and2.12, we get f

nx n1y f

nx−n1y f

nx n−1y f

nx−n−1y n⁴f

x2y n⁴f

x−2y

−2n² n²−1

f xy

−2n² n²−1

f x−y 4f

ny

4fnx−2n² n²−1

f 2y

2n⁴−4n²

fx

4n⁴−12n²4 f

y 2.13

for allx, y∈X. By2.5and2.13, we obtain f

x2y f

x−2y 4f

xy 4f

x−y 2f

2y

−8f y

−6fx 2.14

for allx, y ∈ X. Interchanging xand y in2.14and using the evenness of f, we get the relation

f 2xy

f 2x−y

4f xy

4f x−y

2f2x−8fx−6f y

2.15

for allx, y∈X.

Now we show that2.15is a quadratic and quartic functional equation. To get this, we show that the mappingsg :X → Y, defined bygx f2x−16fx, andh:X → Y, defined byhx f2x−4fx, are quadratic and quartic, respectively.

Replacingyby 2yin2.15and using the evenness off, we have f

2x2y f

2x−2y 4f

2yx 4f

2y−x

2f2x−8fx−6f 2y

2.16

for allx, y ∈ X. Interchangingxwith y in2.16 and then using2.15, we obtain by the evenness off

f

2x2y f

2x−2y 4f

2xy 4f

2x−y 2f

2y

−8f y

−6f2x 16f

xy 16f

x−y

2f2x 2f 2y

−32fx−32f

y 2.17

for allx, y∈X. By2.17, we have f

2x2y

−16f

xy

f

2x−2y

−16f

x−y 2

f2x−16fx 2 f

2y

−16f y 2.18 for allx, y∈X. This means that

g xy

g x−y

2gx 2g y

2.19

for allx, y∈X. Thus the mappingg:X → Y is quadratic.

To prove thath:X → Y is quartic, we have to show that h

2xy h

2x−y 4h

xy 4h

x−y

24hx−6h y

2.20

for allx, y∈X. Replacingxandyby 2xand 2yin2.15, respectively, we get f

4x2y f

4x−2y 4f

2x2y 4f

2x−2y

2f4x−8f2x−6f 2y

2.21

for allx, y∈X. Sinceg2x 4gxfor allx∈Xandg:X → Y is a quadratic mapping, we have

f4x 20f2x−64fx 2.22

for allx∈X. So it follows from2.15,2.21, and2.22that h

2xy h

2x−y

f

4x2y

−4f

2xy

f

4x−2y

−4f 2x−y 4

f

2x2y

−4f

xy 4 f

2x−2y

−4f x−y 24

f2x−4fx −6 f

2y

−4f y 4h

xy 4h

x−y

24hx−6h y

2.23

for allx, y∈X. Thush:X → Yis a quartic mapping.

Theorem 2.2. A mappingf : X → Y satisfies1.6if and only if there exist a unique symmetric multiadditive mapping D : X ×X ×X×X → Y and a unique symmetric bi-additive mapping B:X×X → Y such that

fx Dx, x, x, x Bx, x 2.24

for allx∈X.

Proof. We first assume that the mappingf : X → Y satisfies1.6. Let g, h : X → Y be mappings defined by

gx:f2x−16fx, hx:f2x−4fx 2.25

for allx∈X.ByLemma 2.1, we achieve that the mappingsgandhare quadratic and quartic, respectively, and

fx: 1

12hx− 1

12gx 2.26

for allx∈X.Thus there exist a unique symmetric multiadditive mappingD:X×X×X×X → Y and a unique symmetric bi-additive mappingB : X×X → Y such thatDx, x, x, x 1/12hxandBx, x −1/12gxfor allx∈Xsee citead, ki. So

fx Dx, x, x, x Bx, x 2.27

for allx∈X.

Conversely assume that

fx Dx, x, x, x Bx, x 2.28

for allx∈X,where the mappingD :X×X×X×X → Y is symmetric multi-additive and B:X×X → Y is bi-additive. By a simple computation, one can show that the mappingsD andBsatisfy the functional equation1.6, so the mapping f satisfies1.6.

3. Generalized Hyers-Ulam Stability of 1.6

From now on, letX andY be a quasi-Banach space with quasi-norm · _X and ap-Banach space withp-norm · _Y, respectively. LetM be the modulus of concavity of · _Y. In this section, using an idea of G˘avruta22, we prove the stability of1.6in the spirit of Hyers, Ulam, and Rassias. For convenience we use the following abbreviation for a given mapping

f:X → Y:

Δf x, y

f nxy

f nx−y

−n²f xy

−n²f x−y

−2fnx 2n²fx 2 n²−1

f

y 3.1

for allx, y∈X. Letϕ^p_qx, y: ϕqx, y^p. We will use the following lemma in this section.

Lemma 3.1see15. Let 0< p≤1 and letx₁, x₂, . . . , x_nbe nonnegative real numbers. Then _n

i1

xi

_p

≤ⁿ

i1

xip. 3.2

Theorem 3.2. Letϕ_q :X×X → 0,∞be a function such that

mlim→ ∞4^mϕq

x 2^m, y

2^m

0 3.3

for allx, y∈Xand

∞ i1

4^piϕqp

x 2ⁱ,y

2ⁱ

<∞ 3.4

for allx∈X and ally∈ {x,2x,3x, nx,n1x,n−1x,n2x,n−2x,n−3x}.Suppose that a mappingf:X → Ywithf0 0 satisfies the inequality

Δf x, y

Y ≤ϕ_q x, y

3.5

for allx, y∈X.Then the limit

Qx: lim

m→ ∞4^m

f x

2^m−1

−16f x 2^m

3.6

exists for allx∈XandQ:X → Y is a unique quadratic mapping satisfying

f2x−16fx−Qx

Y ≤ M¹¹ 4

ψ_qx ^1/p 3.7

for allx∈X,where

ψ_qx:^∞

i1

4^pi

1 n^2pn²−1^p

ϕ^p_q

x

2ⁱ,n2x 2ⁱ

ϕ^p_q

x

2ⁱ,n−2x 2ⁱ

4^pϕ^p_q

x

2ⁱ,n1x 2ⁱ

4^pϕ^p_q x

2ⁱ,n−1x 2ⁱ

4^pϕ^p_q

x 2ⁱ,nx

2ⁱ

ϕ^p_q 2x

2ⁱ ,2x 2ⁱ

4^pϕ^p_q

2x 2ⁱ,x

2ⁱ

n^2pϕ^p_q x

2ⁱ,3x 2ⁱ

2^p

3n²−1_p ϕ^p_q

x 2ⁱ,2x

2ⁱ

17n²−8_p ϕ^p_q

x 2ⁱ,x

2ⁱ

n^2p n²−1^p

ϕ^p_q

0,xn1x 2ⁱ

ϕ^p_q

0,n−3x 2ⁱ

10^pϕ^p_q

0,n−1x 2ⁱ

4^pϕ^p_q

0,nx 2ⁱ

4^pϕ^p_q

0,n−2x 2ⁱ

n⁴1p

n²−1^pϕ^p_q

0,2x 2ⁱ

2

3n⁴−n²2_p n²−1^p ϕ^p_q

0,x

2ⁱ

.

3.8 Proof. Settingx0 in3.5and then interchangingxandy, we get

n²−1

fx− n²−1

f−x≤ϕ_q0, x 3.9

for allx∈X. Replacingybyx, 2x,nx,n1xandn−1xin3.5, respectively, we get fn1x fn−1x−n²f2x−2fnx

4n²−2

fx≤ϕqx, x, 3.10 fn2x fn−2x−n²f3x−n²f−x−2fnx 2n²fx

2 n²−1

f2x≤ϕqx,2x,

3.11 f2nx−n²fn1x−n²f1−nx 2

n²−2

fnx 2n²fx≤ϕqx, nx, 3.12 f2n1x f−x−n²fn2x−n²f−nx−2fnx 2n²fx

2 n²−1

fn1x≤ϕ_qx,n1x,

3.13 f2n−1x fx−n²f2−nx−

n²2

fnx 2n²fx 2

n²−1

fn−1x≤ϕ_qx,n−1x, 3.14

f2n1x f−2x−n²fn3x−n²f−n1x−2fnx 2n²fx 2

n²−1

fn2x≤ϕ_qx,n2x, 3.15

f2n−1x f2x−n²fn−1x−n²f−n−3x−2fnx 2n²fx 2

n²−1

fn−2x≤ϕ_qx,n−2x, 3.16

fn3x fn−3x−n²f4x−n²f−2x−2fnx 2n²fx 2

n²−1

f3x≤ϕqx,3x 3.17

for all x ∈ X. Combining 3.9 and 3.11–3.17, respectively, yields the following ine- qualities:

fn2x fn−2x−n²f3x−n²fx−2fnx 2n²fx 2 n²−1

f2x

≤ϕqx,2x n²

n²−1ϕq0, x,

3.18 f2nx−n²fn1x−n²fn−1x 2

n²−2

fnx 2n²fx

≤ϕqx, nx n²

n²−1ϕq0,n−1x,

3.19 f2n1x fx−n²fn2x−n²fnx−2fnx 2n²fx 2

n²−1

fn1x

≤ϕ_qx,n1x n²

n²−1ϕ_q0, nx 1

n²−1ϕ_q0, x,

3.20 f2n−1x fx−n²fn−2x−

n²2

fnx 2n²fx 2 n²−1

fn−1x

≤ϕ_qx,n−1x n²

n²−1ϕ_q0,n−2x,

3.21 f2n1xf2x−n²fn3x−n²fn1x−2fnx2n²fx2

n²−1

fn2x

≤ϕ_qx,n2x n²

n²−1ϕ_q0,n1x ϕ_q0,2x, 3.22

f2n−1xf2x−n²fn−1x−n²fn−3x−2fnx2n²fx2 n²−1

fn−2x

≤ϕ_qx,n−2x n²

n²−1ϕ_q0,n−3x, 3.23

fn3x fn−3x−n²f4x−n²f2x−2fnx 2n²fx 2 n²−1

f3x

≤ϕ_qx,3x n²

n²−1ϕ_q0,2x

3.24 for allx∈X.

Replacingxandyby 2xandxin3.5, respectively, we obtain f2n1x f2n−1x−n²f3x−2f2nx 2n²f2x

n²−2

fx≤ϕ_q2x, x 3.25

for allx∈X. Putting 2xand 2yinstead ofxandyin3.5, respectively, we have f2n1x f2n−1x−n²f4x−2f2nx 2

2n²−1

f2x≤ϕq2x,2x 3.26

for allx∈X. It follows from3.10,3.18,3.19,3.20,3.21, and3.25that f3x−6f2x 15fx

≤ M⁵ n²n²−1

ϕ_qx,n1x ϕ_qx,n−1x ϕ_q2x, x 2ϕ_qx, nx n²ϕ_qx,2x

4n²−2

ϕ_qx, x n² n²−1

2ϕ_q0,n−1x ϕ_q0, nx ϕ_q0,n−2x

n⁴1

n²−1ϕ_q0, x

3.27

for allx∈X. Also, from3.10,3.18,3.19,3.22,3.23,3.24, and3.26, we conclude f4x−4f3x 4f2x 4fx

≤ M⁶ n²n²−1

ϕqx,n2x ϕqx,n−2x ϕq2x,2x 2ϕqx, nx n²

ϕ_qx,3x ϕ_qx, x 2

n²−1

ϕ_qx,2x n²

n²−1

2ϕ_q0,n−1x ϕ_q0,n−3x ϕ_q0,n1x

n⁴1

n²−1ϕq0,2x 2n²ϕq0, x

3.28

for allx∈X. Finally, combining3.27and3.28yields f4x−24f2x 64fx

≤ M⁸ n²n²−1

ϕqx,n2x ϕqx,n−2x 4ϕqx,n1x 4ϕ_qx,n−1x 10ϕ_qx, nx ϕ_q2x,2x 4ϕ_q2x, x n²ϕqx,3x 2

3n²−1

ϕqx,2x

17n²−8

ϕqx, x n²

n²−1

ϕq0,n1x ϕq0,n−3x 10ϕq0,n−1x

4ϕq0, nx4ϕq0,n−2x

n⁴1

n²−1ϕ_q0,2x 2

3n⁴−n²2

n²−1 ϕq0, x

3.29

for allx∈X. Let

ψqx: 1 n²n²−1

ϕqx,n2x ϕqx,n−2x 4ϕqx,n1x

4ϕ_qx,n−1x 10ϕ_qx, nx ϕ_q2x,2x 4ϕ_q2x, x n²ϕqx,3x 2

3n²−1

ϕqx,2x

17n²−8

ϕqx, x n²

n²−1

ϕ_q0,n1xϕq0,n−3x10ϕq0,n−1x4ϕq0, nx

4ϕq0,n−2x

n⁴1

n²−1ϕq0,2x2

3n⁴−n²2

n²−1 ϕq0, x

. 3.30

Then the inequality3.29implies that

f4x−20f2x 64fx≤M⁸ψ_qx 3.31

for allx∈X.

Letg :X → Y be a mapping defined bygx : f2x−16fxfor allx ∈X.From 3.31, we conclude that

g2x−4gx≤M⁸ψqx 3.32

for allx∈X.If we replacexin3.32byx/2^m1and multiply both sides of3.32by 4^m,then we get

4^m1g x

2^m1

−4^mg x 2^m

Y

≤M⁸4^mψ_q x

2^m1

3.33

for allx∈Xand all nonnegative integersm. SinceYis a p-Banach space, the inequality3.33 gives

4^m1g x

2^m1

−4^kg x

2^{k p}

Y

≤^m

ik

4ⁱ¹g x

2ⁱ¹

−4ⁱg x

2^{i p}

Y

≤M^8p m ik

4^ipψ_q^p x

2ⁱ¹

3.34

for all nonnegative integersmandkwithm≥kand allx∈X.Since 0< p≤1, byLemma 3.1 and3.30, we conclude that

ψ_q^px≤ 1 n^2pn²−1^p

ϕ^p_qx,n2x ϕ^p_qx,n−2x 4^pϕ^p_qx,n1x 4^pϕ^p_qx,n−1x10^pϕ^p_qx, nxϕ^pq2x,2x 4^pϕ^p_q2x, x n^2pϕ^pqx,3x2^p

3n²−1p

ϕ^pqx,2x

17n²−8p

ϕ^pqx, x n^2p n²−1^p

×

ϕ^p_q0,n1xϕ^pq0,n−3x10^pϕ^p_q0,n−1x4^pϕ^p_q0, nx 4^pϕ^p_q0,n−2x

n⁴1p

n²−1^pϕ^p_q0,2x 2

3n⁴−n²2p

n²−1^p ϕ^p_q0, x

3.35

for allx∈X.Therefore, it follows from3.4and3.35that ∞

i1

4^ipψqp

x 2ⁱ

<∞ 3.36

for allx∈X.It follows from3.34and3.36that the sequence{4^mgx/2^m}is Cauchy for allx∈X.SinceY is complete, the sequence{4^mgx/2^m}converges for allx∈X.So one can define the mappingQ:X → Yby

Qx lim

m→ ∞4^mg x 2^m

3.37

for allx∈X.Lettingk0 and passing the limitm → ∞in3.34, we get gx−Qx^p

Y ≤M^8p ∞

i0

4^ipψ_q^p x

2ⁱ¹

M^8p 4^p

∞ i1

4^ipψ_q^p x

2ⁱ

3.38

for allx∈X.Thus3.7follows from3.4and3.38.

Now we show thatQis quadratic. It follows from3.3,3.33and3.37that

Q2x−4Qx_Y lim

m→ ∞

4^mg x

2^m−1

−4^m1gx 2^m

Y

4 lim

m→ ∞

4^m−1g x

4^m−1

−4^mg x 2^m

Y

≤M¹¹ lim

m→ ∞4^mψ_q x 2^m

0

3.39

for allx∈X.So

Q2x 4Qx 3.40

for allx∈X.On the other hand, it follows from3.3,3.5,3.6and3.37that ΔQx, y

Y lim

m→ ∞4^mΔgx 2^m, y

2^m

Y

lim

m→ ∞4^m Δf

x 2^m−1, y

2^m−1

−16Δfx 2^m, y

2^m

Y

≤Mlim

m→ ∞4^m Δf

x 2^m−1, y

2^m−1

Y

16Δf x 2^m, y

2^m

Y

≤Mlim

m→ ∞4^m

ϕq

x 2^m−1, y

2^m−1

16ϕq

x 2^m, y

2^m 0

3.41

for allx, y ∈ X.Hence the mappingQ satisfies1.6. ByLemma 2.1, the mappingQ2x− 4Qxis quadratic. Hence3.40implies that the mappingQis quadratic.

It remains to show that Q is unique. Suppose that there exists another quadratic mappingQ : X → Y which satisfies1.6and 3.7. Since Qx/2^m 1/4^mQxand Qx/2^m 1/4^mQxfor allx∈X, we conclude from3.7that

Qx−Qx^p

Y lim

m→ ∞4^mpgx 2^m

−Q x 2^m

Y

p≤ M^8p 4^p lim

m→ ∞4^mpψq

x 2^m

3.42

for allx∈X.On the other hand, since

mlim→ ∞4^mp ∞ i1

4^ipϕ_q^p x

2^mi, y 2^mi

lim

m→ ∞

∞ im1

4^ipϕ_q^p x

2ⁱ,y 2ⁱ

0 3.43

for allx∈Xand ally∈ {x,2x,3x, nx,n1x,n−1x,n2x,n−2x,n−3x},then

mlim→ ∞4^mpψ_qx 2^m

0 3.44

for allx∈X. Using3.44and3.42, we getQQ,as desired.

Theorem 3.3. Letϕ_q :X×X → 0,∞be a function such that

mlim→ ∞

1 4^mϕq

2^mx,2^my

0 3.45

for allx, y∈Xand

∞ i0

1 4^piϕ_q^p

2ⁱx,2ⁱy

<∞ 3.46

for allx∈X and ally∈ {x,2x,3x, nx,n1x,n−1x,n2x,n−2x,n−3x}.Suppose that a mappingf:X → Ywithf0 0 satisfies the inequality

Δf x, y

Y ≤ϕq

x, y

3.47

for allx, y∈X.Then the limit

Qx: lim

m→ ∞

1 4^m

f

2^m1x

−16f2^mx

3.48

exists for allx∈XandQ:X → Y is a unique quadratic mapping satisfying

f2x−16fx−Qx

Y ≤ M⁸ 4

ψ_qx ^1/p 3.49

for allx∈X, where

ψ_qx:^∞

i0

1 4^pi

1 n^2pn²−1^p

ϕ^p_q

2ⁱx,2ⁱn2x ϕ^p_q

2ⁱx,2ⁱn−2x 4^pϕ^p_q

2ⁱx,2ⁱn1x

4^pϕ^pq

2ⁱx,2ⁱn−1x 10^pϕ^pq

2ⁱx,2ⁱnx ϕ^pq

2ⁱ2x,2ⁱ2x 4^pϕ^pq

2ⁱ2x,2ⁱx

n^2pϕ^p_q

2ⁱx,2ⁱ3x 2^p

3n²−1_p ϕ^p_q

2ⁱx,2ⁱ2x

17n²−8_p ϕ^p_q

2ⁱx,2ⁱx

n^2p n²−1^p

ϕ^p_q

0,2ⁱn1x ϕ^p_q

0,2ⁱn−3x 10^pϕ^p_q

0,2ⁱn−1x

4^pϕ^p_q

0,2ⁱnx

4^pϕ^p_q

0,2ⁱn−2x

n⁴1p

n²−1^pϕ^p_q 0,2ⁱ2x

2

3n⁴−n²2p

n²−1^p ϕ^p_q

0,2ⁱx .

3.50

Proof. The proof is similar to the proof ofTheorem 3.2.

Corollary 3.4. Letθ, r, sbe nonnegative real numbers such thatr, s > 2 ors < 2. Suppose that a mappingf:X → Ywithf0 0 satisfies the inequality

Δfx, y

Y ≤

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

θ, r s0,

θx^r_X, r >0, s0, θy^s

X, r 0, s >0, θ

x^r_Xy^s

X

, r, s >0

3.51

for allx, y∈X.Then there exists a unique quadratic mappingQ:X → Y satisfying

f2x−16fx−Qx

Y ≤ M⁸θ n²n²−1

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

δ_q, rs0,

α_qx, r >0, s0, βqx, r0, s >0,

α^p_qx β^p_qx_1/p

, r, s >0

3.52

for allx∈X,where

δq

1 4^p−1n²−1^p

6n²−2p

n²−1p

17n²−8p

n²−1^p

6n⁴−2n²4p

n^2p210^p2∗4^p

n⁴1_p n^2p

n²−1_p

3∗4^p

n²−1_p

10^p

n²−1_p 3

n²−1_p1/p ,

α_qx

4^p22^rp 10^p

6n²−2_p

17n²−8_p

2^rpn^2p

|4^p−2^rp|

_1/p x^r_X,

β_qx

1

n²−1^p|4^p−2^sp|

2^sp

6n²−2_p

n²−1_p

17n²−8^pn²−1^p

6n⁴−2n²4_p n^2p

n1^sp n−3^sp10^pn−1^sp4^pn^sp4^pn−2^sp 2^sp

n⁴1p

3^spn^2pn²−1^p4^pn²−1^pn2^sp

n²−1p

n−2^sp

n²−1p

4^pn1^spn²−1^p4^pn−1^sp

n²−1p

10^pn^sp

n²−1_p1/p x^s_X.

3.53 Proof. InTheorem 3.2, puttingϕqx, y:θx^r_Xy^s_Xfor allx, y ∈X, we get the desired result.

Corollary 3.5. Letθ≥0 andr, s >0 be nonnegative real numbers such thatλ:rs /2. Suppose that a mapingf:X → Y withf0 0 satisfies the inequality

Δf x, y

Y ≤θx^r_Xy^s

X 3.54

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

Y

≤ M⁸θ n²n²−1

1 4^p−2^λp

n2^sp n−2^sp4^pn1^sp

4^pn−1^sp10^pn^sp2^rsp4^p2^rpn^2p3^sp 2^sp

6n²−2p

17n²−8p_1/p x^λ_X

3.55

for allx∈X.

Proof. InTheorem 3.2, puttingϕqx, y : θx^r_Xy^s_X for allx, y ∈ X, we get the desired result.

Theorem 3.6. Letϕ_t:X×X → 0,∞be a function such that

m→ ∞lim16^mϕt

x 2^m, y

2^m

0 3.56

for allx, y∈Xand

∞ i1

16^piϕtp

x 2ⁱ,y

2ⁱ

<∞ 3.57

for allx∈X and ally∈ {x,2x,3x, nx,n1x,n−1x,n2x,n−2x,n−3x}.Suppose that a mappingf:X → Ywithf0 0 satisfies the inequality

Δf x, y

Y ≤ϕt

x, y

3.58 for allx, y∈X.Then the limit

Tx: lim

m→ ∞16^m

f x

2^m−1

−4f x 2^m

3.59

exists for allx∈XandT :X → Y is a unique quartic mapping satisfying f2x−4fx−Tx

Y ≤ M⁸ 16

ψ_tx ^1/p 3.60

for allx∈X,where

ψtx:^∞

i1

16^pi

1 n^2pn²−1^p

ϕ^p_t

x

2ⁱ,n2x 2ⁱ

ϕ^p_t

x

2ⁱ,n−2x 2ⁱ

4^pϕ^p_t

x

2ⁱ,n1x 2ⁱ

4^pϕ^p_t x

2ⁱ,n−1x 2ⁱ

10^pϕ^p_t

x 2ⁱ,nx

2ⁱ

ϕ^p_t 2x

2ⁱ,2x 2ⁱ

4^pϕ^p_t

2x 2ⁱ, x

2ⁱ

n^2pϕ^p_t x

2ⁱ,3x 2ⁱ

2^p

3n²−1p

ϕ^p_t x

2ⁱ,2x 2ⁱ

17n²−8p

ϕ^p_t x

2ⁱ,x 2ⁱ

n^2p n²−1^p

ϕ^p_t

0,xn1x 2ⁱ

ϕ^p_t

0,n−3x 2ⁱ

10^pϕ^p_t

0,n−1x 2ⁱ

4^pϕ^p_t

0,nx 2ⁱ

4^pϕ^p_t

0,n−2x 2ⁱ

n⁴1_p n²−1^pϕ^p_t

0,2x

2ⁱ

2

3n⁴−n²2p

n²−1^p ϕ^p_t

0,x 2ⁱ

.

3.61

Proof. Similar to the proofTheorem 3.2, we have

f4x−20f2x 64fx≤M⁸ψ_tx 3.62 for allx∈X,where

ψ_tx 1 n²n²−1

ϕ_tx,n2x ϕ_tx,n−2x 4ϕ_tx,n1x

4ϕ_tx,n−1x 10ϕ_tx, nx ϕ_t2x,2x 4ϕ_t2x, x n²ϕtx,3x 2

3n²−1

ϕtx,2x

17n²−8 ϕtx, x n²

n²−1

ϕ_t0,n1x ϕ_t0,n−3x 10ϕ_t0,n−1x 4ϕ_t0, nx

4ϕt0,n−2x

n⁴1

n²−1ϕt0,2x 2

3n⁴−n²2 n²−1 ϕt0, x

. 3.63

Leth:X → Y be a mapping defined byhx:f2x−4fx. Then we conclude that

h2x−16hx ≤M⁸ψ_tx 3.64

for allx ∈ X.If we replacexin3.65byx/2^m1 and multiply both sides of3.65by 16^m, then we get

16^m1h x

2^m1

−16^mh x 2^m

Y

≤M⁸16^mψt

x 2^m1

3.65

for allx∈Xand all nonnegative integersm. SinceYis ap-Banach space, the inequality3.66 gives

16^m1h x

2^m1

−16^kh x

2^{k p}

Y

≤^m

ik

16ⁱ¹h x

2ⁱ¹

−16ⁱh x

2^{i p}

Y

≤M^8p m ik

16^piψ_t^p x

2ⁱ¹

3.66