On the Stability of a Generalized Quadratic and Quartic Type Functional Equation in Quasi Banach Spaces

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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,

1

_{S. Abbaszadeh,}

1

_{and Choonkil Park}

2

1_{Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran} 2_{Department of Mathematics, Research Institute for Natural Sciences, Hanyang University,}

Seoul 133-791, South Korea

Correspondence should be addressed to Choonkil Park,[email protected]

Received 31 May 2009; Accepted 9 September 2009

Recommended by Nikolaos Papageorgiou

We establish the general solution of the functional equationfnxy fnx−y n2_f_x_y

n2_f_x₋_y ₂_f_nx₋_n2_f_x₋₂_n2₋₁_f_y_{for fixed integers}_n_with_{n /}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

fxy−fx−fy≤δ 1.1

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

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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

fxyfx−y2fx 2fy 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

Bx, y 1 4

fxy−fx−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:

fx2yfx−2y4fxyfx−y6fy−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 x4_{satisfies 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:

fnxyfnx−yn2fxyn2fx−y2fnx−2n2fx−2n2−1fy

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.

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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

≤Mm

2m

i1 xi,

2m1

i1

xi

≤Mm1

2m1

i1

xi 1.7

for allm≥1 and allx1, x2, . . . , x2m1∈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

xyp≤ xpyp 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−ypgives 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

fnx n1yfnx n−1y

n2fx2yn2fx 2fnxny−2n2fxy−2n2−1fy, 2.1

fnx−n−1yfnx−n1y

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

fxnyfx−ny

n2fxyn2fx−y2fny−2n2fy−2n2−1fx 2.3

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

fnxnyfnx−ny

n4fxyn4fx−y2fny2fnx−2n4fx−2n4fy

2.4

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

fnx n1yfnx−n1yfnx n−1yfnx−n−1y

n2fx2yn2fx−2y2n2n2−1fxy2n2n2−1fx−y

4fny4fnx −4n42n2fx −4n4−4n24fy

2.5

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

fn1xyfn−1x−y

n2f2xyn2fy2fnx−2n2fx−2n2−1fxy, 2.6

fn1x−yfn−1xy

n2f2x−yn2fy2fnx−2n2fx−2n2−1fx−y 2.7

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

fx n1yfx−n−1y

n2fx2yn2fx 2fny−2n2fy−2n2−1fxy, 2.8

fx−n1yfx n−1y

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

fnx n1yfnx−n1y

n2fx n1yn2fx−n1y2fnx−2n2fx−2n2−1fn1y,

2.10

fnx n−1yfnx−n−1y

n2fx n−1yn2fx−n−1y2fnx−2n2fx−2n2−1fn−1y

2.11

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

fn1yfn−1yn2f2y−22n2−1fy2fny 2.12

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

fnx n1yfnx−n1yfnx n−1yfnx−n−1y

n4fx2yn4fx−2y−2n2n2−1fxy−2n2n2−1fx−y

4fny4fnx−2n2n2−1f2y2n4−4n2fx 4n4−12n24fy

2.13

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

fx2yfx−2y4fxy4fx−y2f2y−8fy−6fx 2.14

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

f2xyf2x−y4fxy4fx−y2f2x−8fx−6fy 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

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

f2x2yf2x−2y

4f2xy4f2x−y2f2y−8fy−6f2x

16fxy16fx−y2f2x 2f2y−32fx−32fy

2.17

for allx, y∈X. By2.17, we have

f2x2y−16fxy f2x−2y−16fx−y 2f2x−16fx 2f2y−16fy

2.18

for allx, y∈X. This means that

gxygx−y2gx 2gy 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

h2xyh2x−y4hxy4hx−y24hx−6hy 2.20

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

f4x2yf4x−2y4f2x2y4f2x−2y2f4x−8f2x−6f2y 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

h2xyh2x−y

f4x2y−4f2xy f4x−2y−4f2x−y

4f2x2y−4fxy 4f2x−2y−4fx−y

24f2x−4fx −6f2y−4fy

4hxy4hx−y24hx−6hy

2.23

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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

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f:X → Y:

Δfx, yfnxyfnx−y−n2fxy−n2fx−y

−2fnx 2n2fx 2n2−1fy 3.1

for allx, y∈X. Letϕpqx, y: ϕqx, yp. We will use the following lemma in this section.

Lemma 3.1see15. Let0< p≤1and letx1, x2, . . . , xnbe nonnegative real numbers. Then

_n

i1

xi

p

≤n

i1

xip. 3.2

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

lim

m→ ∞4

m_ϕ q

_x

2m,

y 2m

0 3.3

for allx, y∈Xand

∞

i1

4piϕqp

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 0satisfies the inequality

_Δfx, y_Y ≤ϕq

x, y 3.5

for allx, y∈X.Then the limit

Qx: lim

m→ ∞4

m

f

x 2m−1

−16f x 2m

3.6

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

f2x−16fx−Qx_Y ≤ M

11

4

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for allx∈X,where

ψqx:

∞

i1

4pi

1 n2p_n2₋₁p

ϕpq

x 2i,

n2x 2i

ϕpq

x 2i,

n−2x 2i

4p_ϕp q

x 2i,

n1x 2i

4pϕpq

x 2i,

n−1x 2i

4pϕpq

x 2i,

nx 2i

ϕpq

2x 2i ,

2x 2i

4pϕpq

2x 2i,

x 2i

n2pϕpq

x 2i,

3x 2i

2p3n2−1pϕpq

x 2i,

2x 2i

17n2−8pϕpq

x 2i,

x 2i

n2p n2₋₁p

ϕpq

0,xn1x 2i

ϕpq

0,n−3x 2i

10pϕpq

0,n−1x 2i

4pϕpq

0,nx 2i

4pϕpq

0,n−2x 2i

n41p

n2−₁pϕ

p q 0,2x 2i

23n4₋_n2₂p n2₋₁p ϕ

p q 0,x 2i . 3.8

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

n2₋₁_f_x₋_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 4n2−2fx≤ϕqx, x, 3.10

fn2x fn−2x−n2f3x−n2f−x−2fnx 2n2fx

2n2−1f2x≤ϕqx,2x,

3.11

f2nx−n2fn1x−n2f1−nx 2n2−2fnx 2n2fx≤ϕqx, nx, 3.12

f2n1x f−x−n2fn2x−n2f−nx−2fnx 2n2fx

2n2−1fn1x≤ϕqx,n1x,

3.13

f2n−1x fx−n2f2−nx−n22fnx 2n2fx

2n2−1fn−1x≤ϕqx,n−1x,

3.14

f2n1x f−2x−n2fn3x−n2f−n1x−2fnx

2n2_f_x ₂_n2₋₁_f_n₂_x_≤_ϕ

qx,n2x,

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f2n−1x f2x−n2_f_n₋₁_x₋_n2_f₋_n₋₃_x₋_2f_nx _2n2_f_x

2n2₋₁_f_n₋₂_x_≤_ϕ

qx,n−2x,

3.16

fn3x fn−3x−n2_f_4x₋_n2_f₋_2x₋_2f_nx _2n2_f_x

2n2−1f3x≤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 2n2−1f2x

≤ϕqx,2x n

2

n2₋₁ϕq0, x,

3.18

f2nx−n2fn1x−n2fn−1x 2n2−2fnx 2n2fx

≤ϕqx, nx

n2

n2₋₁ϕq0,n−1x,

3.19

f2n1x fx−n2fn2x−n2fnx−2fnx 2n2fx 2n2−1fn1x

≤ϕqx,n1x

n2

n2₋₁ϕq0, nx

1

n2₋₁ϕq0, x,

3.20

f2n−1x fx−n2fn−2x−n22fnx 2n2fx 2n2−1fn−1x

≤ϕqx,n−1x

n2

n2₋₁ϕq0,n−2x,

3.21

f2n1xf2x−n2_f_n₃_x₋_n2_f_n₁_x₋_2f_nx_2n2_f_x₂_n2₋₁_f_n₂_x

≤ϕqx,n2x

n2

n2₋₁ϕq0,n1x ϕq0,2x,

3.22

f2n−1xf2x−n2_f_n₋₁_x₋_n2_f_n₋₃_x₋_2f_nx_2n2_f_x₂_n2₋₁_f_n₋₂_x

≤ϕqx,n−2x

n2

n2₋₁ϕq0,n−3x,

3.23

fn3x fn−3x−n2f4x−n2f2x−2fnx 2n2fx 2n2−1f3x

≤ϕqx,3x

n2

n2₋₁ϕq0,2x

3.24

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Replacingxandyby 2xandxin3.5, respectively, we obtain

f2n1x f2n−1x−n2_f_3x₋_2f_2nx _2n2_f_2x _n2₋₂_f_x_≤_ϕ

q2x, x

3.25

for allx∈X. Putting 2xand 2yinstead ofxandyin3.5, respectively, we have

f2n1x f2n−1x−n2f4x−2f2nx 22n2−1f2x≤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

n2_n2₋₁

ϕqx,n1x ϕqx,n−1x ϕq2x, x 2ϕqx, nx n2ϕqx,2x

4n2−2ϕqx, x

n2

n2−₁

2ϕq0,n−1x ϕq0, nx ϕq0,n−2x

n41

n2₋₁ϕq0, x

3.27

for allx∈X. Also, from3.10,3.18,3.19,3.22,3.23,3.24, and3.26, we conclude

_f_4x₋_4f_3x _4f_2x _4f_x

≤ M6

n2_n2₋₁

ϕqx,n2x ϕqx,n−2x ϕq2x,2x 2ϕqx, nx

n2ϕqx,3x ϕqx, x

2n2−1ϕqx,2x

n2

n2₋₁

2ϕq0,n−1x ϕq0,n−3x ϕq0,n1x

n41

n2−₁ϕq0,2x 2n 2_ϕ

q0, x

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for allx∈X. Finally, combining3.27and3.28yields

_f_4x₋_24f_2x _64f_x

≤ M8

n2_n2−₁

ϕqx,n2x ϕqx,n−2x 4ϕqx,n1x

4ϕqx,n−1x 10ϕqx, nx ϕq2x,2x 4ϕq2x, x

n2ϕqx,3x 2

3n2−1ϕqx,2x

17n2−8ϕqx, x

n2

n2₋₁

ϕq0,n1x ϕq0,n−3x 10ϕq0,n−1x

4ϕq0, nx4ϕq0,n−2x

n41

n2₋₁ϕq0,2x

2

3n4₋_n2₂

n2₋₁ ϕq0, x

3.29

for allx∈X. Let

ψqx: 1

n2_n2₋₁

ϕqx,n2x ϕqx,n−2x 4ϕqx,n1x

4ϕqx,n−1x 10ϕqx, nx ϕq2x,2x 4ϕq2x, x

n2ϕqx,3x 2

3n2−1ϕqx,2x

17n2−8ϕqx, x

n2

n2₋₁

ϕq0,n1xϕq0,n−3x10ϕq0,n−1x4ϕq0, nx

4ϕq0,n−2x

n41

n2₋₁ϕq0,2x

23n4₋_n2₂

n2₋₁ ϕq0, x

.

3.30

Then the inequality3.29implies that

f4x−20f2x 64fx≤M8_ψ

qx 3.31

for allx∈X.

Letg :X → Y be a mapping defined bygx : f2x−16fxfor allx ∈X.From

3.31, we conclude that

_g_2x₋_4g_x_≤_M8_ψ

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for allx∈X.If we replacexin3.32byx/2m1_{and multiply both sides of}_3.32_{by 4}m_,_then

we get

4m1g

x 2m1

−4mg x 2m

Y

≤M84mψq

x 2m1

3.33

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

4m1g

x 2m1

−4kg

x 2k

p

Y

≤m

ik

4i1g

x 2i1

−4ig

x 2i

p

Y

≤M8pm ik

4ip_ψ qp

x 2i1

3.34

for all nonnegative integersmandkwithm≥kand allx∈X.Since 0< p≤1, byLemma 3.1

and3.30, we conclude that

ψqpx≤

1 n2p_n2₋₁p

ϕpqx,n2x ϕpqx,n−2x 4pϕpqx,n1x

4pϕpqx,n−1x10pϕpqx, nxϕpq2x,2x 4pϕpq2x, x

n2pϕpqx,3x2p

3n2−1pϕpqx,2x

17n2−8pϕpqx, x

n2p

n2₋₁p

×ϕpq0,n1xϕpq0,n−3x10pϕpq0,n−1x4pϕpq0, nx

4pϕpq0,n−2x

n41p

n2−₁pϕ

p

q0,2x

23n4−n22p

n2−₁p ϕ

p q0, x

3.35

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

∞

i1

4ipψqp

x 2i

<∞ 3.36

for allx∈X.It follows from3.34and3.36that the sequence{4m_g_x/2m_}_{is Cauchy for}

allx∈X.SinceY is complete, the sequence{4m_g_x/2m_}_{converges for all}_x_∈_X._{So one can}

define the mappingQ:X → Yby

Qx lim

m→ ∞4

m_g x

2m

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for allx∈X.Lettingk0 and passing the limitm → ∞in3.34, we get

gx−Qxp_Y ≤M8p∞ i0

4ip_ψ qp

x 2i1

M8p

4p

∞

i1

4ip_ψ qp 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→ ∞

4m_g

x 2m−1

−4m1_gx

2m

Y

4 lim

m→ ∞

4m−1g

x 4m−1

−4mg x 2m

Y

≤M11 lim

m→ ∞4

m_ψ q _x 2m 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

_Δ_Q_{x, y}

Y _mlim_{→ ∞}4mΔg

_x

2m,

y 2m

Y

lim

m→ ∞4

m_Δ_f

x 2m−1,

y 2m−1

−16Δfx 2m,

y 2m

Y

≤Mlim

m→ ∞4

m_Δ_f

x 2m−1,

y 2m−1

Y

16Δf x 2m,

y 2m

Y

≤Mlim

m→ ∞4

m

ϕq

x 2m−1,

y 2m−1

16ϕq

_x

2m,

y 2m

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/2m_1/4m_Q_x_and

Qx/2m_1/4m_Q_x_{for all}_x_∈_{X, we conclude from}_3.7_that

_Q_x₋_Q_xp

Y _mlim_{→ ∞}4mpg

_x

2m

−Q x 2m

Y p

≤ M8p

4p _mlim_{→ ∞}4mpψq

_x

2m

(15)

for allx∈X.On the other hand, since

lim

m→ ∞4

mp∞

i1

4ipϕqp

x 2mi,

y 2mi

lim

m→ ∞ ∞

im1

4ipϕqp

x 2i,

y 2i

0 3.43

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

lim

m→ ∞4

mp_ψ q

_x

2m

0 3.44

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

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

lim

m→ ∞

1 4mϕq

2mx,2my0 3.45

∞

i0

1 4piϕq

p₂i_x,₂i_y_<_∞ _3.46

_Δ_f

x, y_Y ≤ϕq

x, y 3.47

Qx: lim

m→ ∞

1 4m

f2m1x−16f2mx 3.48

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

8

4

(16)

for allx∈X, where

ψqx:

∞

i0

1 4pi

1 n2p_n2₋₁p

ϕpq

2i_x,₂i_n₂_x_ϕp q

2i_x,₂i_n₋₂_x₄p_ϕp q

2i_x,₂i_n₁_x

4pϕpq

2ix,2in−1x10pϕpq

2ix,2inxϕpq

2i2x,2i2x4pϕpq

2i2x,2ix

n2pϕpq

2ix,2i3x2p3n2−1pϕpq

2ix,2i2x17n2−8pϕpq

2ix,2ix

n2p n2₋₁p

ϕpq

0,2in1xϕpq

0,2in−3x

10pϕpq

0,2in−1x4pϕpq

0,2inx4pϕpq

0,2in−2x

n4₁p n2−₁pϕ

p q

0,2i2x

23n4₋_n2₂p n2−₁p 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 > 2or s < 2. Suppose that a mappingf:X → Ywithf0 0satisfies the inequality

_Δfx, y_Y ≤

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

θ, r s0,

θxr_X, r >0, s0,

θys_X, r 0, s >0,

θxr_Xys_X, r, s >0

3.51

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

8_θ

n2_n2₋₁

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

δq, rs0,

αqx, r >0, s0,

βqx, r0, s >0,

αpqx βpqx

1/p

, r, s >0

(17)

δq

1 4p₋₁_n2₋₁p

6n2−2pn2−1p17n2−8pn2−1p6n4−2n24p

n2p210p2∗4pn41pn2pn2−1p3∗4pn2−1p

10pn2−1p3n2−1p

1/p

,

αqx

4p₂₂rp ₁₀p_6n2₋₂p_17n2₋₈p₂rp_n2p

|4p₋₂rp_|

1/p

xr_X,

βqx

1

n2₋₁p_|₄p₋₂sp_|

2sp6n2−2pn2−1p 17n2−8pn2−1p6n4−2n24p

n2pn1sp n−3sp10pn−1sp4pnsp4pn−2sp

2sp_n4₁p₃sp_n2p_n2₋₁p₄p_n2₋₁p_n₂sp_n2₋₁p

n−2spn2₋₁p₄p_n₁sp_n2₋₁p₄p_n₋₁sp_n2₋₁p

10pnspn2−1p

1/p

xs_X.

3.53

Proof. InTheorem 3.2, puttingϕqx, y:θxrXysXfor allx, y ∈X, we get the desired

result.

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

_Δ_f

x, y_Y ≤θx_Xrys_X 3.54

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

≤ M8θ

n2_n2₋₁

1

₄p₋₂λp

n2sp n−2sp4pn1sp

4pn−1sp10pnsp2rsp4p2rpn2p3sp

2sp6n2−2p 17n2−8p

1/p

xλ_X

3.55

(18)

Proof. InTheorem 3.2, puttingϕqx, y : θxrXysX for allx, y ∈ X, we get the desired

result.

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

lim

m→ ∞16

m_ϕ t

_x

2m,

y 2m

0 3.56

∞

i1

16piϕtp

x 2i,

y 2i

<∞ 3.57

_Δ_f

x, y_Y ≤ϕt

x, y 3.58

Tx: lim

m→ ∞16

m

f

x 2m−1

−4f x 2m

3.59

exists for allx∈XandT :X → Y is a unique quartic mapping satisfying

f2x−4fx−Tx_Y ≤ M

8

16

ψtx 1/p 3.60

ψtx:

∞

i1

16pi

1 n2p_n2₋₁p

ϕp_t

x 2i,

n2x 2i

ϕp_t

x 2i,

n−2x 2i

4pϕp_t

x 2i,

n1x 2i

4pϕp_t

x 2i,

n−1x 2i

10pϕp_t

x 2i,

nx 2i

ϕp_t

2x 2i,

2x 2i

4pϕp_t

2x 2i,

x 2i

n2pϕp_t

x 2i,

3x 2i

2p3n2−1pϕp_t

x 2i,

2x 2i

17n2−8pϕp_t

x 2i,

x 2i

n2p n2−₁p

ϕp_t

0,xn1x 2i

ϕp_t

0,n−3x 2i

10pϕp_t

0,n−1x 2i

4p_ϕp t

0,nx 2i

4p_ϕp t

0,n−2x 2i

n4₁p n2₋₁pϕ

p t 0,2x 2i

23n4₋_n2₂p n2₋₁p ϕ

(19)

Proof. Similar to the proofTheorem 3.2, we have

f4x−20f2x 64fx≤M8ψtx 3.62

for allx∈X,where

ψtx

1 n2_n2₋₁

ϕtx,n2x ϕtx,n−2x 4ϕtx,n1x

4ϕtx,n−1x 10ϕtx, nx ϕt2x,2x 4ϕt2x, x

n2ϕtx,3x 2

3n2−1ϕtx,2x

17n2−8ϕtx, x

n2

n2₋₁

ϕt0,n1x ϕt0,n−3x 10ϕt0,n−1x 4ϕt0, nx

4ϕt0,n−2x

n41

n2₋₁ϕt0,2x

23n4₋_n2₂

n2₋₁ ϕt0, x

.

3.63

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

h2x−16hx ≤M8ψtx 3.64

for allx ∈ X.If we replacexin3.65byx/2m1 _{and multiply both sides of}_3.65_{by 16}m_,

then we get

16m1h

x 2m1

−16mh x 2m

Y

≤M816mψt

x 2m1

3.65

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

16m1h

x 2m1

−16kh

x 2k

p

Y

≤m

ik

16i1h

x 2i1

−16ih

x 2i

p

Y

≤M8pm ik

16pi_ψ tp

x 2i1

(20)

for all nonnegative integersmandkwithm≥kand allx∈X.Since 0< p≤1, byLemma 3.1, we conclude from3.64that

ψp_tx≤ 1 n2p_n2₋₁p

ϕp_tx,n2x ϕp_tx,n−2x 4pϕp_tx,n1x

4pϕp_tx,n−1x 10pϕp_tx, nx ϕp_t2x,2x 4pϕp_t2x, x

n2pϕp_tx,3x2p3n2−1pϕp_tx,2x17n2−8pϕp_tx, x

n2p n2₋₁p

ϕp_t0,n1x ϕp_t0,n−3x 10pϕp_t0,n−1x

4pϕp_t0, nx 4pϕ_tp0,n−2x

n4₁p n2₋₁pϕ

p t0,2x

23n4₋_n2₂p n2₋₁p ϕ

p t0, x

3.67

for allx∈X.It follows from3.57and3.67that

∞

i1

16piψtp

x 2i

<∞ 3.68

for allx ∈ X.Thus we conclude from3.67and3.69that the sequence{16m_h_x/2m_}_is

Cauchy for all x ∈ X. Since Y is complete, the sequence{16m_h_x/2m_} _{converges for all}

x∈X.So one can define the mappingT :X → Y by

Tx lim

m→ ∞16

m_h x

2m

3.69

for allx∈X.Lettingk0 and passing the limitm → ∞in3.67, we get

hx−Txp_Y ≤M8p

∞

i0

16piψtp

x 2i1

M11p

16p

∞

i1

16piψtp

x 2i

3.70

for allx∈X.Thus3.60follows from3.58and3.70.

Now we show thatTis quartic. From3.57,3.66, and3.70, it follows that

T2x−16Tx_Y lim

m→ ∞

16mh

x 2m−1

−16m1h x 2m

Y

16 lim

m→ ∞

16m−1h

x 16m−1

−16mh x 2m

Y

≤M8 lim

m→ ∞16

m_ψ t

_x

2m

0

(21)

for allx∈X.So

T2x 16Tx 3.72

for allx∈X.On the other hand, by3.59,3.69, and3.70, we have

_ΔTx, y_Y lim

m→ ∞16

m_Δ_hx

2m,

y 2m

Y

lim

m→ ∞16 m_Δ_f

x 2m−1,

y 2m−1

−4Δfx 2m,

y 2m

Y

≤Mlim

m→ ∞16

m_Δ_f

x 2m−1,

y 2m−1

Y

4Δf x 2m,

y 2m

Y

≤Mlim

m→ ∞16

m

ϕt

x 2m−1,

y 2m−1

4ϕt

_x

2m,

y 2m

0

3.73

for all x, y ∈ X.Hence the mappingT satisfies1.6. By Lemma 2.1, the mappingT2x− 16Txis quartic. Therefore,3.75implies that the mappingT is quartic.

To prove the uniqueness property ofT,let T : X → Y be another quartic mapping satisfying3.61. Since

lim

m→ ∞16 mp∞

i1

16piϕtp

x 2mi,

x 2mi

lim

m→ ∞

∞

im1

16piϕtp

x 2i,

x 2i

0 3.74

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

lim

m→ ∞16

mp_ψ t

_x

2m

0 3.75

for allx∈X. It follows from3.61and3.86that

_T_x₋_T_x

Y _mlim_{→ ∞}16mph

_x

2m

−Tx 2m

Y p

≤ M8p

16p _mlim_{→ ∞}16 mp_ψ

t

_x

2m

0 3.76

for allx∈X.SoT T,as desired.

Theorem 3.7. Letϕt:X×X → 0,∞be a function such that

lim

m→ ∞

1 16mϕt

2mx,2my0 3.77

∞

i0

1 16piϕt

(22)

_Δ_f

x, y_Y ≤ϕt

x, y 3.79

Tx: lim

m→ ∞

1 16m

f2m1x−4f2mx 3.80

exists for allx∈XandT :X → Y is a unique quartic mapping satisfying

8

16

ψtx 1/p 3.81

ψtx:

∞

i0

1 16pi

1 n2p_n2−₁p

ϕp_t2ix,2in2xϕp_t2ix,2in−2x4pϕp_t2ix,2in1x

4pϕp_t2ix,2in−1x10pϕp_t2ix,2inxϕp_t2i2x,2i2x4pϕp_t2i2x,2ix

n2pϕp_t2ix,2i3x2p3n2−1pϕ_tp2ix,2i2x17n2−8pϕp_t2ix,2ix

n2p n2₋₁p

ϕp_t0,2in1xϕp_t0,2in−3x10pϕp_t0,2in−1x

4pϕp_t0,2inx4pϕp_t0,2in−2x

n4₁p n2₋₁pϕ

p t

0,2i2x

23n4₋_n2₂p n2₋₁p ϕ

p t

0,2ix.

3.82

Proof. The proof is similar to the proof ofTheorem 3.6.

Corollary 3.8. Letθ, r, sbe nonnegative real numbers such thatr, s >4or0≤r, s <4. Suppose that a mappingf :X → Ywithf0 0satisfies the inequality3.51for allx, y∈X.Then there exists a unique quartic mappingT :X → Ysatisfying

8_θ

n2_n2₋₁

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

δt, r s0,

αtx, r >0, s0,

βtx, r 0, s >0,

αp_tx βp_tx1/p, r, s >0

(23)

δt

1

16p₋₁_n2₋₁p

6n2₋₂p_n2₋₁p_17n2₋₈p_n2₋₁p_6n4₋_2n2₄p

n2p210p2·4pn41pn2pn2−1p3·4pn2−1p

10pn2−1p3n2−1p

1/p

,

αtx

4p22rp 10p6n2−2p17n2−8p2rpn2p

|16p₋₂rp_|

1/p

xr_X,

βtx

1

n2₋₁p_|₁₆p₋₂sp_|

2sp6n2−2pn2−1p17n2−8pn2−1p6n4−2n24p

n2pn1spn−3sp10pn−1sp10pnsp4pn−2sp

2spn41p3spn2pn2−1p4pn2−1p

n2spn2−1pn−2spn2−1p4pn1spn2−1p

4pn−1spn2−1p4pnspn2−1p

1/p

xs_X.

3.84

Proof. InTheorem 3.6, puttingϕtx, y:θxrXysXfor allx, y∈X, we get the desired

result.

Corollary 3.9. Letθ≥0andr, s >0be nonnegative real numbers such thatλ:rs /4. Suppose that a mappingf :X → Y withf0 0satisfies the inequality3.56for allx, y∈X.Then there exists a unique quartic mappingT :X → Y satisfying

_f_2x₋_4f_x₋_T_x

Y

≤ M8θ

n2_n2₋₁

1

16p₋₂λp

n2sp n−2sp4pn1sp

2sp6n2−2p 17n2−8p1/pxλ_X

3.85

for allx∈X.

Proof. InTheorem 3.6, puttingϕtx, y : θxrXysX for all x, y ∈ X, we get the desired

(24)

Theorem 3.10. Letϕ:X×X → 0,∞be a function such that

lim

m→ ∞4

m_ϕ x

2m,

y 2m

0 lim

m→ ∞

1 16mϕ

2mx,2my 3.86

∞

i1

4piϕp

x 2i,

y 2i

<∞,

∞

i0

1 16piϕ

p₂i_x,₂i_y_<_∞

3.87

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

_Δfx, y_Y ≤ϕx, y 3.88

for allx, y ∈ X.Then there exist a unique quadratic mappingQ : X → Y and a unique quartic mappingT :X → Y such that

_f_x₋_Q_x₋_T_x

Y ≤

M9

192

4ψqx 1/p

ψtx 1/p

3.89

for allx∈X,whereψqxandψtxare defined in Theorems3.2and3.7, respectively.

Proof. By Theorems3.2and3.7, there exist a quadratic mappingQ0 : X → Y and a quartic

mappingT0:X → Y such that

_f_2x₋_16f_x₋_Q₀_x

Y ≤

M8

4

ψqx 1/p, f2x−4fx−T0x_Y ≤

M8

16

ψtx 1/p

3.90

for allx∈X.It follows from the last inequalities that

fx 1

12Q0x− 1 12T0x

Y

≤ M9

192

4ψqx 1/p

ψtx 1/p

3.91

for allx∈X.So we obtain3.92by lettingQx −1/12Q0xandTx 1/12T0xfor

allx∈X.

(25)

be another quadratic and quartic mappings satisfying3.92 and letQ2 1/12Q0,T2 1/12T0,Q3Q2−Q1andT3T2−T1.So

Q3x−T3xY ≤Mfx−Q2x−T2x_Y fx−Q1x−T1x_Y

≤ M10

96

4ψqx 1/p

ψtx 1/p

3.92

for allx∈x.Since

lim

m→ ∞4

mp_ψ q

_x

2m

lim

m→ ∞

1 16mpψt2

m_x ₀ _3.93

for allx∈X,3.62implies that limm→ ∞4mQ3x/2m 1/16mT32mxY 0 for allx∈X.

ThusT3Q3.ButT3is only a quartic function andQ3is only a quadratic function.

Therefore, we haveT3 Q3 0 and this completes the uniqueness property ofQand

T.We can prove the other results similarly.

Corollary 3.11. Letθ, r, sbe nonnegative real numbers such thatr, s >4or2< r, s <4or0≤r, s < 2. Suppose that a mappingf:X → Y satisfies the inequality3.51for allx, y∈X.Then there exist a unique quadratic mappingQ:X → Y and a unique quartic mappingT :X → Ysuch that

_f_x₋_Q_x₋_T_x

Y

≤ M9θ

12n2_n2₋₁

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

δqδt, rs0,

αqx αtx, r >0, s0,

βqx βtx, r0, s > 0,

αpqx βpqx

1/p

αp_tx β_tpx1/p, r, s >0

3.94

for allx∈X,whereδq, δt, αqx, αtx, βqx, andβtxare defined as in Corollaries3.4and3.8.

Corollary 3.12. Letθ≥0andr, s >0be nonnegative real numbers such thatλ :rs∈0,2∪

2,4∪4,∞. Suppose that a mappingf :X → Y satisfies the inequality3.56for allx, y ∈X.

Then there exist a unique quadratic mappingQ:X → Y and a unique quartic functionT :X → Y

such that

fx−Qx−Tx_Y

≤ M9θ

12n2_n2₋₁

1

₄p₋₂λp

n2sp n−2sp4pn1sp

2sp6n2−2p17n2−8p

xλ_X

3.95

(26)

Acknowledgment

The third and corresponding author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology NRF-2009-0070788. Also, the second author would like to thank the oﬃce of gifted students at Semnan University for its financial support.

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