A Fixed Point Approach to the Stability of an Additive-Quadratic-Cubic-Quartic Functional Equation

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Volume 2010, Article ID 185780,16pages doi:10.1155/2010/185780

Research Article

A Fixed Point Approach to the Stability of

an Additive-Quadratic-Cubic-Quartic

Functional Equation

Jung Rye Lee,

1

_{Ji-hye Kim,}

2

_{and Choonkil Park}

2

1_{Department of Mathematics, Daejin University, Kyeonggi 487-711, South Korea}

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 24 August 2009; Revised 16 November 2009; Accepted 10 January 2010

Academic Editor: Fabio Zanolin

Copyrightq2010 Jung Rye Lee 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.

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equationfx2y fx−2y 4fxy 4fx−y− 6fx f2y f−2y−4fy−4f−yin Banach spaces.

1. Introduction and Preliminaries

The stability problem of functional equations is originated from a question of Ulam 1

concerning the stability of group homomorphisms. Hyers2gave a first aﬃrmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki

3for additive mappings and by Th. M. Rassias4for linear mappings by considering an unbounded Cauchy diﬀerence. The paper of Th. M. Rassias4has provided a lot of influence in the development of what we callgeneralized Hyers-Ulam stabilityor asHyers-Ulam-Rassias stabilityof functional equations. A generalization of the Th. M. Rassias theorem was obtained by G˘avrut¸a5by replacing the unbounded Cauchy diﬀerence by a general control function in the spirit of Th. M. Rassias’ approach.

The functional equation

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is called aquadratic functional equation. In particular, every solution of the quadratic functional equation is said to be aquadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof6for mappingsf :X → Y, whereXis a normed space andY is a Banach space. Cholewa7noticed that the theorem of Skof is still true if the relevant domainX is replaced by an Abelian group. Czerwik8proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problemsee9–19.

In20, Jun and Kim considered the following cubic functional equation

f2xyf2x−y2fxy2fx−y12fx, 1.2

which is called acubic functional equation,and every solution of the cubic functional equation is said to be acubic mapping.

In21, Lee et al. considered the following quartic functional equation

f2xyf2x−y4fxy4fx−y24fx−6fy, 1.3

which is called a quartic functional equation and every solution of the quartic functional equation is said to be aquartic mapping. Quartic functional equations have been investigated in22,23.

LetXbe a set. A functiond :X×X → 0,∞is called ageneralized metriconX ifd

satisfies

1dx, y 0 if and only ifxy;

2dx, y dy, xfor allx, y∈X;

3dx, z≤dx, y dy, zfor allx, y, z∈X.

We recall a fundamental result in fixed point theory.

Theorem 1.1see24,25. LetX, dbe a complete generalized metric space and letJ :X → X

be a strictly contractive mapping with Lipschitz constantL <1. Then for each given elementx∈X, either

dJnx, Jn1x∞ 1.4

for all nonnegative integersnor there exists a positive integern0such that

1dJn_{x, J}n1_x_<_∞_{, for all}_n_≥_n 0;

2the sequence{Jn_x_}_{converges to a fixed point}_y∗_of_J_;

3y∗is the unique fixed point ofJin the setY {y∈X|dJn0_{x, y}_<∞}_;

4dy, y∗≤1/1−Ldy, Jyfor ally∈Y.

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This paper is organized as follows. InSection 2, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation

fx2yfx−2y4fxy4fx−y−6fx f2yf−2y−4fy−4f−y

1.5

in Banach spaces for an odd case. InSection 3, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation1.5in Banach spaces for an even case.

Throughout this paper, assume thatXis a vector space and thatY is a Banach space.

2. Generalized Hyers-Ulam Stability of the Functional Equation

1.5 :

An Odd Case

For a given mappingf:X → Y, we define

Dfx, y:fx2yfx−2y−4fxy−4fx−y6fx

−f2y−f−2y4fy4f−y 2.1

for allx, y∈X.

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equationDfx, y 0 in Banach spaces: an odd case.

Note that the fundamental ideas in the proofs of the main results in Sections2and3

are contained in24,27,28.

Theorem 2.1. Letϕ:X2 _→ ₀_,_∞_{be a function such that there exists an}_{L <}₁_with

ϕx, y≤ L

8ϕ

2x,2y 2.2

for allx, y∈X. Letf:X → Ybe an odd mapping satisfying

Dfx, y≤ϕx, y 2.3

for allx, y∈X. Then there is a unique cubic mappingC:X → Y such that

f2x−2fx−Cx≤ L

8−8L

4ϕx, x ϕ2x, x 2.4

for allx∈X.

Proof. Lettingxyin2.3, we get _f

3y−4f2y5fy≤ϕy, y 2.5

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Replacingxby 2yin2.3, we get

f4y−4f3y6f2y−4fy≤ϕ2y, y 2.6

for ally∈X.

By2.5and2.6, _f

4y−10f2y16fy

≤4f3y−4f2y5fy

f4y−4f3y6f2y−4fy

≤4ϕy, yϕ2y, y

2.7

for ally∈X. Lettingy:x/2 andgx:f2x−2fxfor allx∈X, we get

gx−8gx

2

_≤₄_ϕ_x 2,

x

2

ϕx,x

2

2.8

for allx∈X.

Consider the set

S:g:X−→Y, 2.9

and introduce the generalized metric onS:

dg, hinfμ∈R:gx−hx≤μ4ϕx, x ϕ2x, x, ∀x∈X, 2.10

where, as usual, infφ ∞. It is easy to show thatS, dis completesee the proof of Lemma 2.1 of33.

Now we consider the linear mappingJ:S → Ssuch that

Jgx:8gx

2

2.11

for allx∈X.

Letg, h∈Sbe given such thatdg, h ε. Then

_g_x₋_h_x_≤₄_ϕ_{x, x} _ϕ₂_{x, x} _2.12

for allx∈X. Hence

Jgx−Jhx8gx

2

−8hx

2

_≤_L

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for allx∈X. Sodg, h εimplies thatdJg, Jh≤Lε. This means that

dJg, Jh≤Ldg, h 2.14

for allg, h∈S.

It follows from2.8that

gx−8gx

2 _≤ L

8

4ϕx, x ϕ2x, x 2.15

for allx∈X. Sodg, Jg≤L/8.

ByTheorem 1.1, there exists a mappingC:X → Y satisfying the following.

1Cis a fixed point ofJ, that is,

Cx

2

1

8Cx 2.16

for allx∈X. Sinceg :X → Y is odd,C :X → Y is an odd mapping. The mappingCis a unique fixed point ofJin the set

Mg∈S:df, g<∞. 2.17

This implies thatCis a unique mapping satisfying2.16such that there exists aμ∈0,∞

satisfying

_g_x₋_C_x_≤_μ

4ϕx, x ϕ2x, x 2.18

for allx∈X.

2dJn_{g, C} _→ _{0 as}_n _{→ ∞. This implies the equality}

lim n→ ∞8

n_gx 2n

Cx 2.19

for allx∈X.

3dg, C≤1/1−Ldg, Jg, which implies the inequality

dg, C≤ L

8−8L. 2.20

This implies that the inequality2.4holds. By2.3,

8nDgx

2n,

y

2n

_≤₈n _ϕ 2x 2n,

2y

2n

2ϕx

2n,

y

2n

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for allx, y∈Xand alln∈N. So

8n_Dgx 2n,

y

2n

_≤_Ln_ϕ₂_x,₂_y₂_ϕ_{x, y} _2.22

for allx, y∈Xand alln∈N. So

_DC

x, y0 2.23

for allx, y∈X. Thus the mappingC:X → Y is cubic, as desired.

Corollary 2.2. Letθ≥0and letpbe a real number withp >3. LetXbe a normed vector space with norm · . Letf :X → Ybe an odd mapping satisfying

_Df

x, y≤θ x pyp 2.24

for allx, y∈X. Then there is a unique cubic mappingC:X → Y such that

f2x−2fx−Cx≤ 2

p₉ 2p₋₈θ x

p _2.25

for allx∈X.

Proof. The proof follows fromTheorem 2.1by taking

ϕx, y:θ x pyp 2.26

for allx, y∈X. Then we can chooseL23−p_{and we get the desired result.} Theorem 2.3. Letϕ:X2 _→ ₀_,_∞_{be a function such that there exists an}_{L <}₁_with

ϕx, y≤8Lϕx

2,

y

2

2.27

for allx, y ∈ X. Letf :X → Y be an odd mapping satisfying2.3. Then there is a unique cubic mappingC:X → Y such that

f2x−2fx−Cx≤ 1

8−8L

4ϕx, x ϕ2x, x 2.28

for allx∈X.

Proof. LetS, dbe the generalized metric space defined in the proof ofTheorem 2.1. Consider the linear mappingJ:S → Ssuch that

Jgx: 1

8g2x 2.29

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gx−1

8g2x ≤ 1

8

4ϕx, x ϕ2x, x 2.30

for allx∈X. Sodg, Jg≤1/8.

The rest of the proof is similar to the proof ofTheorem 2.1.

Corollary 2.4. Letθ≥0and letpbe a real number with0< p <3. LetXbe a normed vector space with norm · . Letf :X → Y be an odd mapping satisfying2.24. Then there is a unique cubic mappingC:X → Y such that

_f₂_x₋₂_f_x₋_C_x_≤ 92p 8−2pθ x

p _2.31

for allx∈X.

Proof. The proof follows fromTheorem 2.3by taking

ϕx, y:θ x pyp 2.32

for allx, y∈X. Then we can chooseL2p−3_{and we get the desired result.}

ϕx, y≤ L

2ϕ

2x,2y 2.33

for allx, y∈X. Letf :X → Y be an odd mapping satisfying2.3. Then there is a unique additive mappingA:X → Y such that

f2x−8fx−Ax≤ L

2−2L

4ϕx, x ϕ2x, x 2.34

for allx∈X.

Proof. LetS, dbe the generalized metric space defined in the proof ofTheorem 2.1. Lettingy:x/2 andhx:f2x−8fxfor allx∈Xin2.7, we get

hx−2hx

2

_≤₄_ϕ_x 2,

x

2

ϕx,x

2

2.35

for allx∈X.

Jhx:2hx

2

2.36

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hx−2hx

2

_≤₂_Lϕ_{x, x} L

2ϕ2x, x 2.37

for allx∈X. Sodh, Jh≤L/2.

Corollary 2.6. Letθ ≥ 0and let pbe a real number withp > 1. LetX be a normed vector space with norm · . Letf :X → Y be an odd mapping satisfying2.24. Then there is a unique additive mappingA:X → Y such that

_f₂_x₋₈_f_x₋_A_x_≤ 2p₉ 2p₋₂θ x

p _2.38

for allx∈X.

ϕx, y≤2Lϕx

2,

y

2

2.39

for allx, y∈X. Letf :X → Y be an odd mapping satisfying2.3. Then there is a unique additive mappingA:X → Y such that

_f₂_x₋₈_f_x₋_A_x_≤ 1 2−2L

4ϕx, x ϕ2x, x 2.40

for allx∈X.

Jhx: 1

2h2x 2.41

for allx∈X.

hx−1

2h2x

≤2ϕx, x 1

2ϕ2x, x 2.42

for allx∈X. Sodh, Jh≤1/2.

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Corollary 2.8. Letθ≥0and letpbe a real number with0< p <1. LetXbe a normed vector space with norm · . Letf :X → Y be an odd mapping satisfying2.24. Then there is a unique additive mappingA:X → Y such that

f2x−8fx−Ax≤ 92

p

2−2pθ x

p _2.43

for allx∈X.

3. Generalized Hyers-Ulam Stability of the Functional Equation

1.5 :

An Even Case

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equationDfx, y 0 in Banach spaces: an even case.

Theorem 3.1. Letϕ:X2 → 0,∞be a function such that there exists anL <1with

ϕx, y≤ L

16ϕ

2x,2y 3.1

for allx, y ∈X. Letf :X → Y be an even mapping satisfyingf0 0and2.3. Then there is a unique quartic mappingQ:X → Ysuch that

_f₂_x₋₄_f_x₋_Q_x_≤ L

16−16L

4ϕx, x ϕ2x, x 3.2

for allx∈X.

Proof. Lettingxyin2.3, we get _f

3y−6f2y15fy≤ϕy, y 3.3

for ally∈X.

Replacingxby 2yin2.3, we get _f

4y−4f3y4f2y4fy≤ϕ2y, y 3.4

for ally∈X.

By3.4and3.5,

_f₄_x₋₂₀_f₂_x ₆₄_f_x

≤4f3x−6f2x 15fxf4x−4f3x 4f2x 4fx

≤4ϕx, x ϕ2x, x

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for allx∈X. Lettinggx:f2x−4fxfor allx∈X, we get

gx−16gx

2

_≤₄_ϕ_x 2,

x

2

ϕx,x

2

3.6

for allx∈X.

LetS, dbe the generalized metric space defined in the proof ofTheorem 2.1. It follows from3.16that

gx−16gx

2

_≤ L

16

4ϕx, x ϕ2x, x 3.7

for allx∈X. Sodg, Jg≤L/16.

Corollary 3.2. Letθ≥0and letpbe a real number withp >4. LetXbe a normed vector space with norm · . Letf :X → Y be an even mapping satisfyingf0 0and2.24. Then there is unique quartic mappingQ:X → Y such that

f2x−4fx−Qx≤ 2

p₉ 2p₋₁₆θ x

p _3.8

for allx∈X.

Theorem 3.3. Letϕ:X2 → 0,∞be a function such that there exists anL <1with

ϕx, y≤16Lϕx

2,

y

2

3.9

for allx, y ∈X. Letf :X → Y be an even mapping satisfyingf0 0and2.3. Then there is a unique quartic mappingQ:X → Ysuch that

f2x−4fx−Qx≤ 1

16−16L

4ϕx, x ϕ2x, x 3.10

for allx∈X.

Jgx: 1

16g2x 3.11

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gx− 1

16g2x ≤ 1

16

4ϕx, x ϕ2x, x 3.12

for allx∈X. Sodg, Jg≤1/16.

Corollary 3.4. Letθ≥0and letpbe a real number with0< p <4. LetXbe a normed vector space with norm · . Letf :X → Y be an even mapping satisfyingf0 0and2.24. Then there is a unique quartic mappingQ:X → Ysuch that

f2x−4fx−Qx≤ 92

p

16−2pθ x

p _3.13

for allx∈X.

ϕx, y≤ L

4ϕ

2x,2y 3.14

for allx, y ∈X. Letf :X → Y be an even mapping satisfyingf0 0and2.3. Then there is a unique quadratic mappingT :X → Y such that

f2x−16fx−Tx≤ L

4−4L

4ϕx, x ϕ2x, x 3.15

for allx∈X.

Proof. LetS, dbe the generalized metric space defined in the proof ofTheorem 2.1. Lettinghx:f2x−16fxfor allx∈Xin3.6, we get

hx−4hx

2

_≤₄_ϕ_x 2,

x

2

ϕx,x

2

3.16

for allx∈X.

Jhx:4hx

2

3.17

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hx−4hx

2

_≤_Lϕ_{x, x} L

4ϕ2x, x 3.18

for allx∈X. Sodh, Jh≤L/4.

Corollary 3.6. Letθ≥0and letpbe a real number withp >2. LetXbe a normed vector space with norm · . Letf:X → Y be an even mapping satisfyingf0 0and2.24. Then there is a unique quadratic mappingT:X → Ysuch that

_f₂_x₋₁₆_f_x₋_T_x_≤ 2p₉ 2p₋₄θ x

p _3.19

for allx∈X.

ϕx, y≤4Lϕx

2,

y

2

3.20

for allx, y ∈X. Letf :X → Y be an even mapping satisfyingf0 0and2.3. Then there is a unique quadratic mappingT :X → Y such that

_f₂_x₋₁₆_f_x₋_T_x_≤ 1 4−4L

4ϕx, x ϕ2x, x 3.21

for allx∈X.

Jhx: 1

4h2x 3.22

for allx∈X.

hx−1

4h2x

≤ϕx, x 1

4ϕ2x, x 3.23

for allx∈X. Sodh, Jh≤1/4.

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Corollary 3.8. Letθ≥0and letpbe a real number with0< p <2. LetXbe a normed vector space with norm · . Letf :X → Y be an even mapping satisfyingf0 0and2.24. Then there is a unique quadratic mappingT :X → Y such that

f2x−16fx−Tx≤ 92

p

4−2pθ x

p _3.24

for allx∈X.

4. Generalized Hyers-Ulam Stability of the Functional Equation

1.5

One can easily show that an odd mappingf : X → Y satisfies1.5if and only if the odd mappingf:X → Y is an additive-cubic mapping, that is,

fx2yfx−2y4fxy4fx−y−6fx. 4.1

It was shown in of34, Lemma 2.2thatgx:f2x−2fxandhx:f2x−8fxare cubic and additive, respectively, and thatfx 1/6gx−1/6hx.

One can easily show that an even mappingf :X → Y satisfies1.5if and only if the even mappingf:X → Yis a quadratic-quartic mapping, that is,

fx2yfx−2y4fxy4fx−y−6fx 2f2y−8fy. 4.2

It was shown in of35, Lemma 2.1thatgx : f2x−4fxandhx : f2x−16fx

are quartic and quadratic, respectively, and thatfx 1/12gx−1/12hx. Functional equations of mixed type have been investigated in36,37.

Let fox : fx−f−x/2 and fex : fx f−x/2. Thenfo is odd and

fe is even.foand fe satisfy the functional equation 1.5. Letgox : fo2x−2foxand

hox:fo2x−8fox. Thenfox 1/6gox−1/6hox. Letgex:fe2x−4fex andhex:fe2x−16fex. Thenfex 1/12gex−1/12hex. Thus

fx 1

6gox− 1 6hox

1

12gex− 1

12hex. 4.3

So we obtain the following results.

ϕx, y≤ L

16ϕ

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for allx, y ∈ X. Letf :X → Y be a mapping satisfyingf0 0and2.3. Then there exist an additive mappingA:X → Y, a quadratic mappingT :X → Y, a cubic mappingC:X → Y and a quartic mappingQ:X → Y such that

fx−1

6Ax− 1 12Tx−

1 6Cx−

1 12Qx

≤ L 12−12L

L

48−48L L

192−192L

4ϕx, x ϕ2x, x

4.5

for allx∈X.

Proof. Sinceϕx, y ≤ L/16ϕ2x,2y,ϕx, y ≤ L/8ϕ2x,2y,ϕx, y ≤ L/4ϕ2x,2y

andϕx, y≤L/2ϕ2x,2y. The result follows from Theorems2.1,2.5,3.1, and3.5.

Corollary 4.2. Letθ ≥ 0 and letp be a real number withp > 4. Letf : X → Y be a mapping satisfyingf0 0 and 2.24. Then there exist an additive mapping A : X → Y, a quadratic mappingT :X → Y, a cubic mappingC:X → Y and a quartic mappingQ:X → Ysuch that

fx−1

6Ax− 1 12Tx−

1 6Cx−

1 12Qx

≤ 2p9 62p₋₂

2p₉ 122p₋₄

2p₉ 62p₋₈

2p₉ 122p₋₁₆

θ x p

4.6

for allx∈X.

ϕx, y≤2Lϕx

2,

y

2

4.7

for allx, y ∈ X. Letf :X → Y be a mapping satisfyingf0 0and2.3. Then there exist an additive mappingA:X → Y, a quadratic mappingT :X → Y, a cubic mappingC:X → Y, and a quartic mappingQ:X → Ysuch that

fx−1

6Ax− 1 12Tx−

1 6Cx−

1 12Qx

≤ 1 12−12L

1 48−48L

1 192−192L

4ϕx, x ϕ2x, x

4.8

for allx∈X.

Proof. Sinceϕx, y≤2Lϕx/2, y/2,ϕx, y≤4Lϕx/2, y/2,ϕx, y≤8Lϕx/2, y/2and

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Corollary 4.4. Letθ ≥0and letpbe a real number with0 < p <1. Letf :X → Y be a mapping satisfyingf0 0 and 2.24. Then there exist an additive mapping A : X → Y, a quadratic mappingT :X → Y, a cubic mappingC:X → Y,and a quartic mappingQ:X → Y such that

fx−1

6Ax− 1 12Tx−

1 6Cx−

1 12Qx

≤ 2p9 62−2p

2p₉ 124−2p

2p₉ 68−2p

2p₉ 1216−2p

θ x p

4.9

for allx∈X.

Acknowledgments

The first and third authors were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and TechnologyNRF-2009-0071229andNRF-2009-0070788, respectively.

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