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

doi:10.1155/2009/826130

Research Article

Solution and Stability of a Mixed Type Additive,

Quadratic, and Cubic Functional Equation

M. Eshaghi Gordji,

_{S. Kaboli Gharetapeh,}

_{J. M. Rassias,}

and S. Zolfaghari

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

2_{Department of Mathematics, Payame Noor University of Mashhad, Mashhad, Iran}

3_{Section 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

fxy−fx−fy≤δ, 1.1

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

such that

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

fxyfx−y2fx 2fy 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:

fx2y3fx 3fxyfx−y6fy. 1.4

Jun and Kim13introduced the following cubic functional equation:

f2xyf2x−y2fxy2fx−y12fx, 1.5

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

The functionfx x3_{satisfies 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:

fx2y−fx−2y2fxy−fx−y2f3y−6f2y6fy. 1.6

It is easy to see that the functionfx ax3_bx2_{cxis 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

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

Comparing2.4with2.5, we get

fe

3y9fe

y. 2.6

By utilizing2.5with2.6, we obtain

fe

2y4fe

y. 2.7

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

fe

x2y−fe

x−2y2fe

xy−2fe

x−y. 2.8

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

fe

3x−y−fe

x−3y8fex−8fe

y. 2.9

Replacingyby−yin above relation, we obtain

fe

3xy−fe

x3y8fex−8fe

y. 2.10

Settingxyinstead ofxin2.8, we get

fe

x3y−fe

x−y2fe

x2y−2fex. 2.11

Interchangingxandyin2.11, we get

fe

3xy−fe

x−y2fe

2xy−2fe

y. 2.12

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

fe

x2y−fe

2xy3fe

y−3fex, 2.13

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

fe

x4y−4fe

xy12fe

y−3fex. 2.14

Let us interchangexandyin2.14. Then we see that

fe

4xy−4fe

xy12fex−3fe

y, 2.15

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

fe

x4yfe

4xy8fe

xy9fex 9fe

Replacingybyxyin2.8, we obtain

Let us Interchangexandyin2.17. Then we see that

fe

Thus by adding2.17and2.18, we have

fe

and interchangingxandyin2.20yields

fe

Interchangingxandyin2.8, we get

fe

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

fe

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

fe

From the substitutiony−yin2.25it follows that

Replacingyby 2yin2.25we have

and interchangingxandyyields

fe

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

fe

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

fe

This means thatfeis 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,

Hence2.31can be written as

fo

From the substitutiony:−yin2.33it follows that

fo

Interchangexandyin2.33, and it follows that

Replacexbyx−yin2.34. Then we have

Interchangingxandyin2.38, we get

fo

By comparing2.40with2.41, we arrive at

fo

and replacing−ybyygives

fo

Let us interchangexandyin2.45. Then we see that

If we add2.45to2.46, we have

fo

2x−y−fo

x−2yfo2x−2fo

xyfox fo

2yfo

y. 2.47

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

fo

2xy−8fo

xyfo2x−8fox

fo

2y−8fo

y, 2.48

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

gxygx gy, 2.49

for allx, y∈X.Therefore the mappingg:X → Y is additive. With the substitutionsx:2x

andy:2yin2.35, we have

fo

4x2yfo

4x−2y2fo

2x2y2fo

2x−2y2fo4x−4fo2x. 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 thatfox 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

Df

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

Df

x, y≤φx, 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≤φx, y, 3.3

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

∞

i0 4si

φ2−six,2−six1

2φ

0,2−six<∞ 3.4

and that

lim

n 4

sn_φ2−sn_x,2−sn_{y0, 3.5}

for allx, y∈X,then the limit

Qx:lim

n 4

sn_f2−sn_x

3.6

exists for allx∈X,andQ:X → Y is a unique quadratic function satisfying1.6, and

_fx−Qx≤ 1 8

∞

is1/2 4si

φ2−six,2−six1

2φ

0,2−six, 3.7

for allx∈X.

Proof. Lets1.Puttingx0 in3.3, we get

2f3y−3f2y3fy≤φ0, y, 3.8

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

_−f

3y4f2y−7fy≤φy, y, 3.9

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

_2f

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

φ0,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→ ∞}4n_fx/2n_{for allx∈Xis unique quadratic function}

satisfying1.6and3.7. Lets−1.Then by3.10we have

f₄2x−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 : limn→ ∞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

2si

φ

x

2si,

x

2si1

φ

0, x

2si1

<∞ 3.13

and that

lim

n→ ∞2 sn_φ x

2sn,

y

2sn

0, 3.14

for allx, y∈X.

Suppose that an odd mappingf:X → Ysatisfies _Df

x, y≤φx, y, 3.15

for allx, y∈X.

Then the limit

Ax: lim

n→ ∞2 sn

f

x

2sn−1

−8f x

2sn

3.16

exists, for allx∈X,andA:X → Y is a unique additive function satisfying1.6, and

f2x−8fx−Ax≤

∞

i|s−1|/2 2si_φ

x

2si,

x

2si1

2

∞

i|s−1|/2 2si_φ

0, x

2si1

, _3.17

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

_2f

3y−8f2y16fy≤φ0, y, 3.18

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

f4y−4f3y6f2y−4fy≤φ2y, y. 3.19

Combining3.18and3.19, we lead to

f4y−10f2y16fy≤φ2y, y2φ0, 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

2φ0,x

2

, 3.21

for allx ∈ X.Now, in a similar way as in8,34,35, we can show that the limitAx : limn→ ∞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

Df

x, y≤φx, y, 3.22

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

∞

i1

8siφ

x

2si,

x

2si1

∞

i1

8siφ

0, x

2si1

<∞ 3.23

and thatlimn→ ∞8snφx/2sn, y/2sn 0,for allx, y∈X,then the limit

Cx: lim

n→ ∞8 sn

f

x

2sn−1

−2f x

2sn

3.24

exists, for allx∈X,andC:X → Yis a unique cubic function satisfying1.6and

f2x−2fx−Cx≤

∞

i|s−1|/2 8siφ

x

2si,

x

2si1

2

∞

i|s−1|/2 8siφ

0, x

2si1

, 3.25

Proof. We prove the theorem fors1.Whens−1 we have a similar proof. It is easy to see

cubic function satisfying1.6and3.25.

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

_Df

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

for allx∈X.So we get3.29by lettingAx −1/6Aox,andCx 1/6Cox,for all

for allx∈X.Hence3.32implies that

lim

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

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. 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

Df

x, y≤θxpyq, 3.41

for allx, y∈X.Then there exist a unique additive functionA:X → Y,a unique quadratic function

Theorem 3.8. Suppose that f : 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

φ2ix,2i−1xφ0,2i−1x 1

4iφ

2ix,2ix<∞ 3.43

and thatlimn→ ∞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

φ2ix,2i−1x2φ0,2i−1x 1

8

∞

i0 1 4i

φ2ix,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≤θxpyq, 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}

≤θxpq

1 6×2q

2pq 2−2pq

2pq 8−2pq

1

8−2pq3

,

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

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.

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