Solution and Stability of a Mixed Type Additive, Quadratic, and Cubic Functional Equation

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doi:10.1155/2009/826130

Research Article

Solution and Stability of a Mixed Type Additive, Quadratic, and Cubic Functional Equation

M. Eshaghi Gordji,

¹

S. Kaboli Gharetapeh,

²

J. M. Rassias,

³

and S. Zolfaghari

¹

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

2Department of Mathematics, Payame Noor University of Mashhad, Mashhad, Iran

3Section of Mathematics and Informatics, Pedagogical Department, National and Capodistrian University of Athens, 4 Agamemnonos St., Aghia Paraskevi, Athens 15342, Greece

Correspondence should be addressed to M. Eshaghi Gordji,[email protected] Received 24 January 2009; Revised 13 April 2009; Accepted 26 June 2009

Recommended by Patricia J. Y. Wong

We obtain the general solution and the generalized Hyers-Ulam-Rassias stability of the mixed type additive, quadratic, and cubic functional equationfx2y−fx−2y 2fxy−fx−y 2f3y−6f2y 6fy.

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 letG2,∗be a metric group with the metricd·,·.Given > 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 → G₂withdhx, Hx< for allx∈ G₁? In other words, under what condition does there exist a homomorphism near an approximate homomorphism?

In 1941, Hyers2gave a first aﬃrmative answer to the question of Ulam for Banach spaces. Letf :E → Ebe a mapping between Banach spaces such that

f xy

−fx−f

y≤δ, 1.1

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

fx−Tx≤δ, 1.2

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for allx ∈E.Moreover ifftxis continuous intfor each fixedx ∈E,thenT is linearsee also3. In 1950, Aoki4generalized Hyers’ theorem for approximately additive mappings.

In 1978, Th. M. Rassias5 provided a generalization of Hyers’ theorem which allows the Cauchy diﬀerence to be unbounded. This new concept is known as Hyers-Ulam-Rassias stability of functional equationssee2–24.

The functional equation f

xy f

x−y

2fx 2f y

1.3 is related to symmetric biadditive function. In the real case it has fx x² among its solutions. Thus, it has been called quadratic functional equation, and each of its solutions is said to be a quadratic function. Hyers-Ulam-Rassias stability for the quadratic functional equation1.3was proved by Skof for functionsf:A → B, whereAis normed space andB Banach spacesee25–28.

The following cubic functional equation was introduced by the third author of this paper, J. M. Rassias29,30 in 2000-2001:

f x2y

3fx 3f xy

f x−y

6f y

. 1.4

Jun and Kim13introduced the following cubic functional equation:

f 2xy

f 2x−y

2f xy

2f x−y

12fx, 1.5

and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation1.5.

The functionfx x³satisfies the functional equation1.5, which explains why it is called cubic functional equation.

Jun and Kim proved that a functionfbetween real vector spacesXandY is a solution of1.5if and only if there exists a unique functionC : X×X ×X → Y such thatfx Cx, x, xfor allx ∈ X,and Cis symmetric for each fixed one variable and is additive for fixed two variablessee also31–33.

We deal with the following functional equation deriving from additive, cubic and quadratic functions:

f x2y

−f x−2y

2 f

xy

−f x−y

2f 3y

−6f 2y

6f y

. 1.6

It is easy to see that the functionfx ax³bx²cxis a solution of the functional equation1.6. In the present paper we investigate the general solution and the generalized Hyers-Ulam-Rassias stability of the functional equation1.6.

2. General Solution

In this section we establish the general solution of functional equation1.6.

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Theorem 2.1. LetX,Y be vector spaces, and letf :X → Y be a function. Thenfsatisfies1.6if and only if there exists a unique additive functionA :X → Y, a unique symmetric and biadditive functionQ:X×X → Y,and a unique symmetric and 3-additive functionC:X×X×X → Ysuch thatfx Ax Qx, x Cx, x, xfor allx∈X.

Proof. Suppose thatfx Ax Qx, x Cx, x, xfor allx ∈ X, whereA : X → Y is additive,Q:X×X → Y is symmetric and biadditive, andC:X×X×X → Y is symmetric and 3-additive. Then it is easy to see thatf satisfies1.6. For the converse letfsatisfy1.6.

We decomposefinto the even part and odd part by setting f_ex 1

2

fx f−x

, f_ox 1 2

fx−f−x

, 2.1

for allx∈X.By1.6, we have fe

x2y

−fe

x−2y

1 2

f x2y

f

−x−2y

−f x−2y

−f

−x2y 1

2 f

x2y

−f

x−2y 1

2 f

−x

−2y

−f

−x−

−2y 1

2 2f

xy

−2f x−y

2f 3y

−6f 2y

6f y 1

2 2f

−x−y

−2f

−xy 2f

−3y

−6f

−2y 6f

−y 2

1 2

f xy

f

−x−y

−2 1

2 f

x−y f

−xy 2

1 2

f 3y

f

−3y

−6 1

2 f

2y f

−2y 6

1 2

f y

f

−y 2

f_e xy

−f_e x−y

2f_e 3y

−6f_e 2y

6f_e y

,

2.2

for allx, y∈X.This means thatf_esatisfies1.6, that is, f_e

x2y

−f_e x−2y

2 f_e

xy

−f_e x−y

2f_e 3y

−6f_e 2y

6f_e y

. 2.3

Now puttingxy0 in2.3, we getfe0 0. Settingx0 in2.3, by evenness offewe obtain

3f_e 2y

f_e 3y

3f_e y

. 2.4

Replacingxbyyin2.3, we obtain 4fe

2y fe

3y 7fe

y

. 2.5

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Comparing2.4with2.5, we get f_e

3y 9f_e

y

. 2.6

By utilizing2.5with2.6, we obtain f_e

2y 4f_e

y

. 2.7

Hence, according to2.6and2.7,2.3can be written as fe

x2y

−fe

x−2y 2fe

xy

−2fe

x−y

. 2.8

With the substitutionx:xy, y:x−yin2.8, we have f_e

3x−y

−f_e x−3y

8f_ex−8f_e y

. 2.9

Replacingyby−yin above relation, we obtain f_e

3xy

−f_e x3y

8f_ex−8f_e y

. 2.10

Settingxyinstead ofxin2.8, we get fe

x3y

−fe

x−y 2fe

x2y

−2fex. 2.11

Interchangingxandyin2.11, we get f_e

3xy

−f_e x−y

2f_e 2xy

−2f_e y

. 2.12

If we subtract2.12from2.11and use2.10, we obtain fe

x2y

−fe

2xy 3fe

y

−3fex, 2.13

which, by puttingy:2yand using2.7, leads to fe

x4y

−4fe

xy 12fe

y

−3fex. 2.14

Let us interchangexandyin2.14. Then we see that f_e

4xy

−4f_e xy

12f_ex−3f_e y

, 2.15

and by adding2.14and2.15, we arrive at fe

x4y fe

4xy 8fe

xy

9fex 9fe

y

. 2.16

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Replacingybyxyin2.8, we obtain f_e

3x2y

−f_e x2y

2f_e 2xy

−2f_e y

. 2.17

Let us Interchangexandyin2.17. Then we see that fe

2x3y

−fe

2xy 2fe

x2y

−2fex. 2.18

Thus by adding2.17and2.18, we have f_e

2x3y f_e

3x2y 3f_e

x2y 3f_e

2xy

−2f_ex−2f_e y

. 2.19

Replacingxby 2xin2.11and using2.7we have f_e

2x3y

−f_e 2x−y

8f_e xy

−8f_ex, 2.20

and interchangingxandyin2.20yields fe

3x2y

−fe

x−2y 8fe

xy

−8fe

y

. 2.21

If we add2.20to2.21, we have f_e

2x3y f_e

3x2y f_e

2x−y f_e

x−2y

16f_e xy

−8f_ex−8f_e y

. 2.22

Interchangingxandyin2.8, we get fe

2xy

−fe

2x−y 2fe

xy

−2fe

x−y

, 2.23

and by adding the last equation and2.8with2.19, we get f_e

2x3y f_e

3x2y

−f_e 2x−y

−f_e x−2y 2fe

x2y 2fe

2xy 4fe

xy

−4fe

x−y

−2fex−2fe

y

. 2.24

Now according to2.22and2.24, it follows that fe

x2y fe

2xy 6fe

xy 2fe

x−y

−3fex−3fe

y

. 2.25

From the substitutiony−yin2.25it follows that fe

x−2y fe

2x−y 6fe

x−y 2fe

xy

−3fex−3fe

y

. 2.26

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Replacingyby 2yin2.25we have fe

x4y 4fe

xy 6fe

x2y 2fe

x−2y

−3fex−12fe

y

, 2.27

and interchangingxandyyields f_e

4xy 4f_e

xy 6f_e

2xy 2f_e

2x−y

−12f_ex−3f_e y

. 2.28

By adding2.27and2.28and then using2.25and2.26, we lead to f_e

x4y f_e

4xy

32f_e xy

24f_e x−y

−39f_ex−39f_e y

. 2.29

If we compare2.16and2.29, we conclude that fe

xy fe

x−y

2fex 2fe

y

. 2.30

This means thatf_eis quadratic. Thus there exists a unique quadratic functionQ:X×X → Y such thatfex Qx, x,for allx∈X.On the other hand we can show thatfosatisfies1.6, that is,

fo

x2y

−fo

x−2y 2

fo

xy

−fo

x−y 2fo

3y

−6fo

2y 6fo

y

. 2.31 Now we show that the mappingg :X → Y defined bygx :f_o2x−8f_oxis additive and the mappingh : X → Y defined byhx : fo2x−2foxis cubic. Puttingx 0 in 2.31, then by oddness off_o,we have

4fo

2y 5fo

y fo

3y

. 2.32

Hence2.31can be written as f_o

x2y

−f_o x−2y

2f_o xy

−2f_o x−y

2f_o 2y

−4f_o y

. 2.33

From the substitutiony:−yin2.33it follows that fo

x−2y

−fo

x2y 2fo

x−y

−2fo

xy

−2fo

2y 4fo

y

. 2.34

Interchangexandyin2.33, and it follows that f_o

2xy f_o

2x−y 2f_o

xy 2f_o

x−y

2f_o2x−4f_ox. 2.35

With the substitutionsx:x−yandy:xyin2.35, we have fo

3x−y fo

x−3y 2fo

2x−2y

−4fo

x−y

2fo2x−2fo

2y

. 2.36

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Replacexbyx−yin2.34. Then we have f_o

x−3y

−f_o xy

2f_o x−2y

−2f_ox−2f_o 2y

4f_o y

. 2.37

Replacingyby−yin2.37gives fo

x3y

−fo

x−y 2fo

x2y

−2fox 2fo

2y

−4fo

y

. 2.38

Interchangingxandyin2.38, we get f_o

3xy f_o

x−y 2f_o

2xy

−2f_o y

2f_o2x−4f_ox. 2.39

If we add2.38to2.39, we have fo

x3y fo

3xy 2f_o

x2y 2f_o

2xy

2f_o2x 2f_o 2y

−6f_ox−6f_o y

. 2.40

x3y fo

3xy 2fo

2x2y

−4fo

xy

2fo2x 2fo

2y

. 2.41

By comparing2.40with2.41, we arrive at f_o

x2y f_o

2xy f_o

2x2y

−2f_o xy

3f_ox 3f_o y

. 2.42

x−2y fo

2x−y fo

2x−2y

−2fo

x−y

3fox−3fo

y

. 2.43

With the substitutiony:xyin2.43, we have f_o

x−y

−f_o x2y

−fo

2y

−3f_o xy

3f_ox 2f_o y

, 2.44

and replacing−ybyygives fo

xy

−fo

x−2y fo

2y

−3fo

x−y

3fox−2fo

y

. 2.45

Let us interchangexandyin2.45. Then we see that fo

xy fo

2x−y

fo2x 3fo

x−y

−2fox 3fo

y

. 2.46

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If we add2.45to2.46, we have fo

2x−y

−fo

x−2y

fo2x−2fo

xy

fox fo

2y fo

y

. 2.47

Adding2.42to2.47and using2.33and2.35, we obtain f_o

2 xy

−8f_o xy

f_o2x−8f_ox

f_o 2y

−8f_o y

, 2.48

for allx, y∈X.The last equality means that g

xy

gx g y

, 2.49

for allx, y∈X.Therefore the mappingg:X → Y is additive. With the substitutionsx:2x andy:2yin2.35, we have

f_o

4x2y f_o

4x−2y 2f_o

2x2y 2f_o

2x−2y

2f_o4x−4f_o2x. 2.50

Letg : X → Y be the additive mapping defined above. It is easy to show thatfois cubic- additive function. Then there exists a unique function C : X ×X ×X → Y and a unique additive functionA : X → Y such thatf_ox Cx, x, x Ax,for allx ∈ X, andCis symmetric and 3-additive. Thus for allx∈X, we have

fx fex fox Qx, x Cx, x, x Ax. 2.51

This completes the proof of theorem.

The following corollary is an alternative result ofTheorem 2.1.

Corollary 2.2. LetX,Y be vector spaces, and letf :X → Ybe a function satisfying1.6. Then the following assertions hold.

aIffis even function, thenfis quadratic.

bIffis odd function, thenfis cubic-additive.

3. Stability

We now investigate the generalized Hyers-Ulam-Rassias stability problem for functional equation 1.6. From now on, let X be a real vector space, and let Y be a Banach space.

Now before taking up the main subject, givenf :X → Y, we define the diﬀerence operator Df :X×X → Yby

D_f x, y

f x2y

−f x−2y

−2 f

xy

−f x−y

−2f 3y

6f 2y

−6f y

, 3.1

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for allx, y∈X.We consider the following functional inequality:

D_f

x, y≤φ x, y

, 3.2

for an upper boundφ:X×X → 0,∞.

Theorem 3.1. Lets∈ {1,−1}be fixed. Suppose that an even mappingf :X → Ysatisfiesf0 0, and

Df

x, y≤φ x, y

, 3.3

for allx, y∈X.If the upper boundφ:X×X → 0,∞is a mapping such that

∞ i0

4^si

φ

2^−six,2^−six 1

2φ

0,2^−six

<∞ 3.4

and that

limn 4^snφ

2^−snx,2^−sny

0, 3.5

for allx, y∈X,then the limit

Qx:lim

n 4^snf 2^−snx

3.6

exists for allx∈X,andQ:X → Y is a unique quadratic function satisfying1.6, and fx−Qx≤ 1

8

∞ is1/2

4^si

φ

2^−six,2^−six 1

2φ

0,2^−six

, 3.7

for allx∈X.

Proof. Lets1.Puttingx0 in3.3, we get 2

f 3y

−3f 2y

3f

y≤φ 0, y

, 3.8

for ally∈X.On the other hand by replacingybyxin3.3, it follows that −f

3y 4f

2y

−7f

y≤φ y, y

, 3.9

for ally∈X.Combining3.8and3.9, we lead to 2f

2y

−8f

y≤2φ y, y

φ 0, y

, 3.10

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for all y ∈ X. With the substitution y : x/2 in 3.10 and then dividing both sides of inequality by 2, we get

fx−4fx 2

≤ 1 2

2φx 2,x

2 φ

0,x 2

. 3.11

Now, using methods similar as in 8, 34,35, we can easily show that the function Q:X → Y defined byQx lim_{n→ ∞}4ⁿfx/2ⁿfor allx∈Xis unique quadratic function satisfying1.6and3.7. Lets−1.Then by3.10we have

f2x

4 −fx ≤ 1

8

2φx, x φ0, x

, 3.12

for allx∈X.And analogously, as in the cases1, we can show that the functionQ:X → Y defined byQx : lim_{n→ ∞}4⁻ⁿf2ⁿx is unique quadratic function satisfying1.6and 3.7.

Theorem 3.2. Lets∈ {1,−1}be fixed. Letφ:X×X → 0,∞is a mapping such that

∞ i1

2^si

φ x

2^si, x 2^si1

φ

0, x

2^si1

<∞ 3.13

and that

nlim→ ∞2^snφ x 2^sn, y

2^sn

0, 3.14

for allx, y∈X.

Suppose that an odd mappingf:X → Ysatisfies Df

x, y≤φ x, y

, 3.15

for allx, y∈X.

Then the limit

Ax: lim

n→ ∞2^sn

f x

2^sn−1

−8f x 2^sn

3.16

exists, for allx∈X,andA:X → Y is a unique additive function satisfying1.6, and f2x−8fx−Ax≤ ^∞

i|s−1|/2

2^siφ x

2^si, x 2^si1

2

∞ i|s−1|/2

2^siφ

0, x 2^si1

, 3.17

for allx∈X.

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Proof. Lets1.setx0 in3.15. Then by oddness offwe have 2f

3y

−8f 2y

16f

y≤φ 0, y

, 3.18

for ally∈X.Replacingxby 2yin3.15we get f

4y

−4f 3y

6f 2y

−4f

y≤φ 2y, y

. 3.19

Combining3.18and3.19, we lead to f

4y

−10f 2y

16f

y≤φ 2y, y

2φ 0, y

, 3.20

for ally∈X.Puttingy:x/2 andgx:f2x−8fx,for allx∈X.Then we get gx−2gx

2

≤φ x,x

2

2φ 0,x

2

, 3.21

for allx ∈ X.Now, in a similar way as in8,34,35, we can show that the limitAx : lim_n_{→ ∞}2ⁿgx/2ⁿ exists, for all x ∈ X, and A is the unique function satisfying 1.6and 3.17. Ifs−1, then the proof is analogous.

Theorem 3.3. Lets∈ {1,−1}be fixed. Suppose that an odd mappingf :X → Y satisfies D_f

x, y≤φ x, y

, 3.22

∞ i1

8^siφ x

2^si, x 2^si1

^∞

i1

8^siφ

0, x 2^si1

<∞ 3.23

and that lim_n_{→ ∞}8^snφx/2^sn, y/2^sn 0,for allx, y∈X,then the limit Cx: lim

n→ ∞8^sn

f x

2^sn−1

−2f x 2^sn

3.24

exists, for allx∈X,andC:X → Yis a unique cubic function satisfying1.6and f2x−2fx−Cx≤ ^∞

i|s−1|/2

8^siφ x

2^si, x 2^si1

2

∞ i|s−1|/2

8^siφ

0, x 2^si1

, 3.25

for allx∈X.

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Proof. We prove the theorem fors1.Whens−1 we have a similar proof. It is easy to see thatfsatisfies3.20. Sethx:f2x−2fxthen by puttingy:x/2 in3.20, it follows that

hx−8hx 2

≤φ x,x

2

2φ 0,x

2

, 3.26

for all x ∈ X. By using 3.26, we may define a mapping C : X → Y as Cx : lim_n_{→ ∞}8ⁿhx/2ⁿ,for allx ∈ X.Similar toTheorem 3.1, we can show thatCis the unique cubic function satisfying1.6and3.25.

Theorem 3.4. Suppose that an odd mappingf:X → Y satisfies Df

x, y≤φ x, y

, 3.27

∞ i1

8ⁱφ x

2ⁱ, x 2ⁱ¹

^∞

i1

8ⁱφ

0, x 2ⁱ¹

<∞, 3.28

and that lim_n_{→ ∞}8ⁿφx/2ⁿ, y/2ⁿ 0,for allx, y ∈ X,then there exists a unique cubic function C:X → Y and a unique additive functionA:X → Y such that

fx−Cx−Ax≤ 1 6

∞ i0

2ⁱ8ⁱ

φ x

2ⁱ, x 2ⁱ¹

1

3

∞ i0

2ⁱ8ⁱ

φ

0, x 2ⁱ¹

, 3.29

for allx∈X.

Proof. By Theorems3.2and3.3, there exist an additive mappingA_o : X → Y and a cubic mappingCo:X → Y such that

f2x−8fx−A_ox≤ ^∞

i|s−1|/2

2^siφ x

2^si, x 2^si1

2

∞ i|s−1|/2

2^siφ

0, x 2^si1

,

f2x−2fx−Cox≤ ^∞

i|s−1|/2

8^siφ x

2^si, x 2^si1

2

∞ i|s−1|/2

8^siφ

0, x 2^si1

,

3.30

for allx∈X.Combine the two equations of3.30to obtain fx−1

6Cox 1

6Aox ≤ 1

6

∞ i0

2ⁱ8ⁱ

φ x

2ⁱ, x 2ⁱ¹

1

3

∞ i0

2ⁱ8ⁱ

φ

0, x 2ⁱ¹

, 3.31

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for allx∈X.So we get3.29by lettingAx −1/6Aox,andCx 1/6Cox,for all x∈X.To prove the uniqueness ofAandC,letA₁, C₁:X → Ybe another additive and cubic maps satisfying3.29. LetAA−A₁, and letCC−C₁.So

Ax−Cx≤fx−Ax−Cxfx−A₁x−C₁x

≤2 1

30

∞ i0

2ⁱ8ⁱ

φ x

2ⁱ, x 2ⁱ¹

1

15

∞ i0

2ⁱ8ⁱ

φ

0, x 2ⁱ¹

, 3.32

for allx∈X.Since

nlim→ ∞

_∞

i1

8ⁱⁿφ x

2ⁱⁿ, x 2ⁱⁿ¹

^∞

i1

8ⁱⁿφ

0, x 2ⁱⁿ¹

0, 3.33

then

nlim→ ∞

_∞

i1

2ⁱⁿφ x

2ⁱⁿ, x 2ⁱⁿ¹

^∞

i1

2ⁱⁿφ

0, x 2ⁱⁿ¹

0, 3.34

for allx∈X.Hence3.32implies that

nlim→ ∞8ⁿAx 2ⁿ

−Cx 2ⁿ

0, 3.35 for allx∈X.On the other handCandC₁are cubic, thenCx/2ⁿ 1/8ⁿCx.Therefore by3.35we obtain thatAx 0,for allx ∈X.Again by3.35we haveCx 0,for all x∈X.

Theorem 3.5. Suppose that an odd mappingf:X → Y satisfies D_f

x, y≤φ x, y

, 3.36

∞ i1

1 2ⁱφ

2ⁱx,2ⁱ⁻¹x ^∞

i1

2ⁱφ

0,2ⁱ⁻¹x

<∞ 3.37

and that lim_{n→ ∞}1/2ⁿφ2ⁿx,2ⁿy 0,for allx, y ∈X,then there exist a unique cubic function C:X → Y and a unique additive functionA:X → Y such that

fx−Cx−Ax

≤ 1 30

∞ i1

1 2ⁱ 1

8ⁱ φ

2ⁱx,2ⁱ⁻¹x 1

15

∞ i1

1 2ⁱ 1

8ⁱ φ

0,2ⁱ⁻¹x

, 3.38

for allx∈X.

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Proof. The proof is similar to the proof ofTheorem 3.4.

Now we establish the generalized Hyers-Ulam-Rassias stability of functional equation 1.6as follows.

Theorem 3.6. Suppose that a mappingf:X → Y satisfiesf0 0 andDfx, y ≤φx, y,for allx, y∈X.If the upper boundφ:X×X → 0,∞is a mapping such that

∞ i0

8ⁱ

φ

x 2ⁱ, x

2ⁱ¹

φ

0, x 2ⁱ¹

4ⁱφ

x 2ⁱ,x

2ⁱ

<∞ 3.39

and that lim_n_{→ ∞}8ⁿφx/2ⁿ, y/2ⁿ 0,for allx, y ∈X,then there exist a unique additive function A:X → Ya unique quadratic functionQ:X → Y and a unique cubic functionC:X → Y such that

fx−Ax−Qx−Cx

≤ 1 6

∞ i0

2ⁱ8ⁱ φ

x 2ⁱ, x

2ⁱ¹

2φ

0, x 2ⁱ¹

1

8

∞ i1

4ⁱ

φ x

2ⁱ,x 2ⁱ

1

2φ

0,x 2ⁱ

, 3.40

for allx∈X.

Proof. Letfex 1/2fx f−x,for all x ∈ X.Thenfe0 0, fe−x fex,and Dfex, y ≤1/2φx, y φ−x,−y,for allx, y∈X.Hence in view ofTheorem 3.1there exists a unique quadratic function Q : X → Y satisfying 3.7. Let f_ox 1/2fx− f−x,for all x ∈ X. Thenfo0 0, fo−x −fox,andDfox, y ≤ 1/2φx, y φ−x,−y, for all x, y ∈ X. From Theorem 3.4, it follows that there exist a unique cubic function C : X → Y and a unique additive functionA : X → Y satisfying3.29. Now it is obvious that3.40holds true for allx∈X,and the proof of theorem is complete.

Corollary 3.7. Letpq >3, θ≥0.Suppose that a mappingf:X → Y satisfiesf0 0,and D_f

x, y≤θ

x^py^q

, 3.41

for allx, y∈X.Then there exist a unique additive functionA:X → Y,a unique quadratic function Q:X → Y,and a unique cubic functionC:X → Y satisfying

fx−Ax−Qx−Cx≤θx^pq 1

6×2^q

2

2−2^pq 8 8−2^pq

1

8

2^pq 4−2^pq

, 3.42 for allx∈X.

Proof. It follows fromTheorem 3.6by takingφx, y θx^py^q,for allx, y∈X.

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Theorem 3.8. Suppose thatf : X → Y satisfies f0 0, and Dfx, y ≤ φx, y, for all x, y∈X.If the upper boundφ:X×X → 0,∞is a mapping such that

∞ i1

1 2ⁱ

φ

2ⁱx,2ⁱ⁻¹x φ

0,2ⁱ⁻¹x 1

4ⁱφ

2ⁱx,2ⁱx

<∞ 3.43

and that lim_n_{→ ∞}1/2ⁿφ2ⁿx,2ⁿy 0,for allx, y∈X,then there exists a unique additive function A:X → Y,a unique quadratic functionQ:X → Y,and a unique cubic functionC:X → Y such that

fx−Ax−Qx−Cx

≤ 1 6

_∞

i1

1 2ⁱ 1

8ⁱ

φ

2ⁱx,2ⁱ⁻¹x 2φ

0,2ⁱ⁻¹x 1

8

∞ i0

1 4ⁱ

φ

2ⁱx,2ⁱx 1

2φ 0,2ⁱx

, 3.44

for allx∈X.

By Theorem 3.8, we are going to investigate the following stability problem for functional equation1.6.

Corollary 3.9. Letpq <1, θ >0.Suppose thatf :X → Y satisfiesf0 0,and Df

x, y≤θ

x^py^q

, 3.45

for allx, y∈X,then there exist a unique additive functionA:X → Y,a unique quadratic function Q:X → Y,and a unique cubic functionC:X → Y satisfying

fx−Ax−Qx−Cx

≤θx^pq 1

6×2^q

2^pq

2−2^pq 2^pq 8−2^pq

1 8−2^pq3

,

3.46

for allx∈X.

ByCorollary 3.9, we solve the following Hyers-Ulam stability problem for functional equation1.6.

Corollary 3.10. Letbe a positive real number. Suppose that a mappingf :X → Ysatisfiesf0 0,andDfx, y ≤,for allx, y ∈ X,then there exist a unique additive functionA: X → Y,a unique quadratic functionQ:X → Y,and a unique cubic functionC:X → Y such that

fx−Ax−Qx−Cx≤ 5

14, 3.47

for allx∈X.

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Acknowledgments

The authors would like to express their sincere thanks to referees for their invaluable comments. The first author would like to thank the Semnan University for its financial support. Also, the fourth author would like to thank the oﬃce of gifted students at Semnan University for its financial support.

References

1 S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1940.

2 D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Diﬀerential Equations and Their Applications, 34, Birkh¨auser, Boston, Mass, USA, 1998.

3 D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951.

4 T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.

5 Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.

6 J. Acz´el and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.

7 P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984.

8 G.-L. Forti, “Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 127–133, 2004.

9 Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991.

10 P. G˘avrut¸a, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.

11 D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.

12 G. Isac and Th. M. Rassias, “On the Hyers-Ulam stability of ψ-additive mappings,” Journal of Approximation Theory, vol. 72, no. 2, pp. 131–137, 1993.

13 K.-W. Jun and H.-M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 267–278, 2002.

14 L. Maligranda, “A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions—a question of priority,” Aequationes Mathematicae, vol. 75, no. 3, pp. 289–296, 2008.

15 J. M. Rassias, “On a new approximation of approximately linear mappings by linear mappings,”

Discussiones Mathematicae, vol. 7, pp. 193–196, 1985.

16 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques. Deuxième Série, vol. 108, no. 4, pp. 445–446, 1984.

17 J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,” Discussiones Mathematicae, vol. 14, pp. 101–107, 1994.

18 J. M. Rassias, “On the stability of a multi-dimensional Cauchy type functional equation,” in Geometry, Analysis and Mechanics, pp. 365–376, World Scientific, River Edge, NJ, USA, 1994.

19 J. M. Rassias, “Solution of a stability problem of Ulam,” in Functional Analysis, Approximation Theory and Numerical Analysis, pp. 241–249, World Scientific, River Edge, NJ, USA, 1994.

20 J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95–103, 1992.

21 J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp.

268–273, 1989.

22 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.

23 Th. M. Rassias, Ed., Functional Equations and Inequalities, vol. 518 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.

(17)

24 Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264–284, 2000.

25 A. Grabiec, “The generalized Hyers-Ulam stability of a class of functional equations,” Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 217–235, 1996.

26 J. M. Rassias and M. J. Rassias, “Refined Ulam stability for Euler-Lagrange type mappings in Hilbert spaces,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 126–132, 2007.

27 J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185–190, 1992.

28 F. Skof, “Proprieta’ locali e approssimazione di operatori,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.

29 J. M. Rassias, “Solution of the Ulam stability problem for cubic mappings,” Glasnik Matematiˇcki. Serija III, vol. 3656, no. 1, pp. 63–72, 2001.

30 J. M. Rassias, “Solution of the Ulam problem for cubic mappings,” Analele Universit˘at¸ii din Timis¸oara.

Seria Matematic˘a-Informatic˘a, vol. 38, no. 1, pp. 121–132, 2000.

31 M. E. Gordji, “Stability of a functional equation deriving from quartic and additive functions,” to appear in Bulletin of the Korean Mathematical Society.

32 M. E. Gordji, A. Ebadian, and S. Zolfaghari, “Stability of a functional equation deriving from cubic and quartic functions,” Abstract and Applied Analysis, vol. 2008, Article ID 801904, 17 pages, 2008.

33 M. E. Gordji and H. Khodaei, “Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 71, pp. 5629–5643, 2009.

34 J. Brzde¸k and A. Pietrzyk, “A note on stability of the general linear equation,” Aequationes Mathematicae, vol. 75, no. 3, pp. 267–270, 2008.

35 A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,” Demonstratio Mathematica, vol. 39, no. 3, pp. 523–530, 2006.