Stability of a Bi-Jensen Functional Equation II

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Volume 2009, Article ID 976284,10pages doi:10.1155/2009/976284

Research Article

Stability of a Bi-Jensen Functional Equation II

Kil-Woung Jun,

¹

Il-Sook Jung,

¹

and Yang-Hi Lee

²

1Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea

2Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, South Korea

Correspondence should be addressed to Kil-Woung Jun,[email protected] Yang-Hi Lee,[email protected]

Received 7 October 2008; Accepted 24 January 2009 Recommended by Alexander Domoshnitsky

We investigate the stability of the bi-Jensen functional equation IIfxy/2, z−fx, z−fy, z 0,2fx,yz/2−fx, y−fx, z 0 in the spirit of G˘avruta.

Copyrightq2009 Kil-Woung Jun 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

In 1940, Ulam1raised a question concerning the stability of homomorphisms. LetG₁ be a group and letG2be a metric group with the metricd·,·. Givenε >0, does there exist aδ >0 such that if a mappingh:G₁ → G₂satisfies the inequality

d

hxy, hxhy

< δ 1.1

for allx, y∈G₁, then there is a homomorphismH:G₁ → G₂with d

hx, Hx

< ε 1.2

for allx∈G₁. The case of approximately additive mappings was solved by Hyers2under the assumption thatG1andG2are Banach spaces. In 1978, Rassias3gave a generalization of Hyers’ theorem by allowing the Cauchy diﬀerence to be controlled by a sum of powers like

hxy−hx−hy≤ε

x^py^p

. 1.3

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G˘avruta4provided a further generalization of Rassias’ theorem in which he replaced the boundεx^py^pby a general function.

Throughout this paper, let X and Y be a normed space and a Banach space, respectively. A mappingg :X → Y is called a Cauchy mappingresp., a Jensen mappingif gsatisfies the functional equationgxy gx gy resp., 2gxy/2 gx gy.

For a given mappingf:X×X → Y, we define

J1fx, y, z:2f xy

2 , z

−fx, z−fy, z,

J2fx, y, z:2f

x,yz 2

−fx, y−fx, z

1.4

for allx, y, z∈X. A mappingf :X×X → Y is called a bi-Jensen mapping iffsatisfies the functional equationsJ1f0 andJ2f0.

Bae and Park 5 obtained the generalized Hyers-Ulam stability of a bi-Jensen mapping. Jun et al.6improved the results of Bae and Park in the spirit of Rassias.

In this paper, we investigate the stability of a bi-Jensen functional equationJ1f 0, J₂f0 in the spirit of G˘avruta.

2. Stability of a Bi-Jensen Functional Equation

Jun et al.7established the basic properties of a bi-Jensen mapping in the following lemma.

Lemma 2.1. Letf :X×X → Ybe a bi-Jensen mapping. Then, the following equalities hold:

fx, y 1 4ⁿf

2ⁿx,2ⁿy

1 2ⁿ − 1

4ⁿ f

2ⁿx,0 f

0,2ⁿy

1− 1

2ⁿ ₂

f0,0

f

2ⁿx,2ⁿy

2ⁿ 2ⁿ−1 2²ⁿ¹

f

2ⁿx,−2ⁿy f

−2ⁿx,2ⁿy

1− 1

2ⁿ 2

f0,0,

fx, y 1 4ⁿf

2ⁿx,2ⁿy

2ⁿ−1 f

x 2ⁿ,0

f

0, y

2ⁿ

−

2ⁿ¹−3 1 4ⁿ

f0,0,

fx, y 4ⁿf x

2ⁿ, y 2ⁿ

2ⁿ−4ⁿ f

x 2ⁿ,0

f

0, y

2ⁿ

2ⁿ−12

f0,0,

fx, y 1 2ⁿf

2ⁿx, y 1

2ⁿ

1− 1 2ⁿ

f

0,2ⁿy

1− 1

2ⁿ ₂

f0,0

2.1 for allx, y∈Xandn∈N.

Now we have the stability of a bi-Jensen mapping.

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Theorem 2.2. Letϕ, ψ:X×X×X → 0,∞be two functions satisfying

∞ j1

ϕ

2^jx,2^jy,2^jz

2^j <∞, 2.2

∞ j1

ψ

2^jx,2^jy,2^jz

2^j <∞ 2.3

for allx, y, z∈X. Letf :X×X → Ybe a mapping such that J₁fx, y, z≤ϕx, y, z,

J₂fx, y, z≤ψx, y, z 2.4

for allx, y, z∈X. Then, there exists a unique bi-Jensen mappingF :X×X → Ysuch that

fx, y−Fx, y≤^∞

j1

Φ

2^jx,2^jy

4^j ^∞

j1

ψ

0,0,2^jy ϕ

2^jx,0,0

2^j 2.5

for allx, y∈XwithF0,0 f0,0, where

Φx, y ϕx,0, y ψx,0, y

2 ψ

x 2,0, y

ϕ

x,0,y

2

3ψ0,0, y 3ϕx,0,0

2 . 2.6

The mappingF:X×X → Y is given by

Fx, y: lim

j→ ∞

f

2^jx,2^jy

4^j f

2^jx,0 f

0,2^jy

2^j f0,0 2.7

for allx, y∈X.

Proof. Letf, f, fbe the maps defined by

fx, y fx, y−f0, y, fx, y fx, y−fx,0,

fx, y fx, y−fx,0−f0, y f0,0

2.8

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for allx, y∈X. By2.4, we get

f

2^jx,0 2^j −f

2^j1x,0 2^j1

J1f

2^j1x,0,0 2^j1

≤ ϕ

2^j1x,0,0 2^j1 ,

f

0,2^jy 2^j −f

0,2^j1y 2^j1

J2f

0,0,2^j1y 2^j1

≤ ψ

0,0,2^j1y 2^j1 ,

f

2^jx,2^jy 4^j − f

2^j1x,2^j1y 4^j1

J₁f

2^j1x,0,2^j1y J₂f

2^j1x,0,2^j1y

2J₂f

2^jx,0,2^j1y 2·4^j1

2J₁f

2^j1x,0,2^jy

−3J₂f

0,0,2^j1y

−3J₁f

2^j1x,0,0 2·4^j1

≤ Φ

2^j1x,2^j1y 4^j1

2.9

for allx, y∈Xandj∈N. For given integersl, m0≤l < m,

f

2^lx,0 2^l − f

2^mx,0 2^m

≤^m−1

jl

ϕ

2^j1x,0,0 2^j1 ,

f

0,2^ly 2^l −f

0,2^my 2^m

≤^m−1

jl

ψ

0,0,2^j1y 2^j1 ,

f

2^lx,2^ly 4^l −f

2^mx,2^my 4^m

≤^m−1

jl

Φ

2^j1x,2^j1y 4^j1

2.10

for all x, y ∈ X. By 2.2 and 2.3, the sequences {1/2^jf2^jx,0 − f0,0}, {1/2^jf0,2^jy−f0,0}, and{1/4^jf2^jx,2^jy}are Cauchy sequences for allx, y ∈X.

SinceY is complete, the sequences{1/2^jf2^jx,0−f0,0},{1/2^jf0,2^jy−f0,0}, and{1/4^jf2^jx,2^jy}converge for allx, y∈X. DefineF1, F2, F3:X×X → Y by

F₁x, y: lim

j→ ∞

f 2^jx,0

2^j , F2x, y: lim

j→ ∞

f 0,2^jy

2^j , F₃x, y: lim

j→ ∞

f

2^jx,2^jy 4^j

2.11

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for allx, y∈X. Puttingl0 and takingm → ∞in2.10, one can obtain the inequalities fx,0−f0,0−F1x, y≤^∞

j1

ϕ

2^jx,0,0 2^j , f0, y−f0,0−F2x, y≤^∞

j1

ψ

0,0,2^jy 2^j , fx, y−fx,0−f0, y f0,0−F3x, y≤^∞

j1

Φ

2^jx,2^jy 4^j

2.12

for allx, y∈X. By2.4and the definitions ofF₁andF₂, we get

J1F1x, y, z lim

j→ ∞

1 2^jJ1f

2^jx,2^jy,0 0, J₂F₁x, y, z 0,

J₁F₂x, y, z 0, J₂F₂x, y, z lim

j→ ∞

1 2^jJ₂f

0,2^jy,2^jz 0,

J1F3x, y, z lim

j→ ∞

J₁f

2^jx,2^jy,2^jz

4^j 0,

J₂F₃x, y, z lim

j→ ∞

J2f

2^jx,2^jy,2^jz

4^j 0

2.13

for allx, y, z∈X. SoFis a bi-Jensen mapping satisfying2.5, whereFis given by

Fx, y F₁x, y F₂x, y F₃x, y f0,0. 2.14

Now, letF:X×X → Y be another bi-Jensen mapping satisfying2.5withF0,0 f0,0.

ByLemma 2.1, we have Fx, y−Fx, y

F−F

2ⁿx,2ⁿy

4ⁿ

1 2ⁿ − 1

4ⁿ

F−F 2ⁿx,0

F−F

0,2ⁿy

≤

F−f

2ⁿx,2ⁿy 4ⁿ

F−f 0,2ⁿy 2ⁿ

F−f 2ⁿx,0 2ⁿ

f−F

2ⁿx,2ⁿy 4ⁿ

f−F 0,2ⁿy 2ⁿ

f−F 2ⁿx,0 2ⁿ

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≤ ^∞

jn1

Φ

2^jx,2^jy 2^j−1 ^∞

jn1

ψ

0,0,2^jy ϕ

2^jx,0,0 2^j−2

^∞

jn1

Φ 2^jx,0

Φ 0,2^jy

2^j−1 ϕ0,0,0

2ⁿ⁻¹ ψ0,0,0 2ⁿ⁻¹

2.15

for allx, y∈Xandn∈N. Asn → ∞, we may conclude thatFx, y Fx, yfor allx, y∈X.

Thus such a bi-Jensen mappingF:X×X → Y is unique.

Remark 2.3. Letϕ, ψ:X×X×X → 0,∞be the functions defined by

ϕx, y, z ψx, y, z: ε

3 2.16

for allx, y, z∈X. Letf, F, F:X×X → Ybe the bi-Jensen mappings defined by

fx, y:0, Fx, y:ε, Fx, y:−ε 2.17

for all x, y ∈ X. Then, ϕ, ψ, andf satisfy2.2,2.3,2.4for all x, y, z ∈ X. In addition, f, F satisfy2.5 for allx, y ∈ X and f, F also satisfy2.5 for allx, y ∈ X. But we get F/F. Hence, the condition F0,0 f0,0 is necessary to show that the mapping F is unique.

We have another stability result applying for several cases.

∞ j0

4^j

ϕ x

2^j, y 2^j, z

2^j

ψ x

2^j, y 2^j, z

2^j

<∞ 2.18

for allx, y, z∈X. Letf :X×X → Y be a mapping satisfying2.4for allx, y, z∈X. Then, there exists a unique bi-Jensen mappingF :X×X → Ysatisfying

fx, y−Fx, y≤^∞

j0

4^jΦ

x 2^j, y

2^j

2^jψ

0,0, y 2^j

2^jϕ

x 2^j,0,0

2.19

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for allx, y∈X. The mappingF:X×X → Y is given by

Fx, y: lim

j→ ∞

4^jf

x 2^j, y

2^j

−

4^j−2^j f

x 2^j,0

f

0,y

2^j

2^j−12

f0,0 2.20

for allx, y∈X.

Proof. By2.4and the similar method inTheorem 2.2, we define the mapsF1, F2, F3:X×X → Y by

F1x, y: lim

j→ ∞2^j

f x

2^j,0

−f0,0

,

F₂x, y: lim

j→ ∞2^j

f

0, y 2^j

−f0,0

,

F3x, y: lim

j→ ∞4^j

f x

2^j, y 2^j

−f x

2^j,0

−f

0,y 2^j

−f0,0

2.21

for allx, y∈X. By2.4and the definitions ofF1,F2, andF3, we get

J1F1x, y, z lim

j→ ∞2^jJ1f x

2^j, y 2^j,0

0, J2F1x, y, z 0,

J₁F₂x, y, z 0, J₂F₂x, y, z lim

j→ ∞2^jJ₂f

0, y 2^j, z

2^j

0,

J1F3x, y, z lim

j→ ∞4^j

J1f x

2^j, y 2^j, z

2^j

−J1f x

2^j, y

2^j,0 0, J₂F₃x, y, z lim

j→ ∞4^j

J₂f x

2^j, y 2^j, z

2^j

−J₂f

0, y 2^j, z

2^j 0

2.22

for allx, y, z∈X. By the similar method inTheorem 2.2,Fis a bi-Jensen mapping satisfying 2.19, whereFis given by

Fx, y F1x, y F2x, y F3x, y f0,0. 2.23

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Now, letF :X×X → Y be another bi-Jensen mapping satisfying2.19. UsingLemma 2.1, ϕ0,0,0 ψ0,0,0 0, andF0,0 f0,0 F0,0, we have

Fx, y−Fx, y

4ⁿ

F−F x 2ⁿ, y

2ⁿ

2ⁿ−4ⁿ

F−F x 2ⁿ,0

F−F 0, y

2ⁿ

≤4ⁿ F−f

x 2ⁿ, y

2ⁿ

f−F x 2ⁿ, y

2ⁿ

F−f x

2ⁿ,0

f−F x 2ⁿ,0

F−f

0, y

2ⁿ

f−F 0, y

2ⁿ

≤2 ∞ jn

4^j

Φ x

2^j,y 2^j

2ψ

0,0,y

2^j

2ϕ x

2^j,0,0

Φ x

2^j,0

Φ

0,y 2^j

2.24 for allx, y∈Xandn∈N. Asn → ∞, we may conclude thatFx, y Fx, yfor allx, y∈X.

Thus such a bi-Jensen mappingF:X×X → Y is unique.

∞ j0

2^jϕ x

2^j, y 2^j, z

2^j

^∞

j0

ϕ

2^jx,2^jy,2^jz

4^j <∞, 2.25

∞ j0

2^jψ x

2^j, y 2^j, z

2^j

^∞

j0

ψ

2^jx,2^jy,2^jz

4^j <∞ 2.26

for allx, y, z∈X. Letf :X×X → Y be a mapping satisfying2.4for allx, y, z∈X. Then, there exists a bi-Jensen mappingF:X×X → Y satisfying

fx, y−Fx, y≤^∞

j1

Φ

2^jx,2^jy

4^j ^∞

j0

2^j

ψ

0,0, y 2^j

ϕ

x 2^j,0,0

2.27

for allx, y∈X, where the mappingF:X×X → Y is given by

Fx, y: lim

j→ ∞

1 4^j

f

2^jx,2^jy

−f 2^jx,0

−f 0,2^jy

lim

j→ ∞

2^j

f

x 2^j,0

f

0,y

2^j

−

2^j1−1 f0,0

2.28

for allx, y∈X.

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Proof. We can obtain F1, F2 as in Theorem 2.4and F3 as in Theorem 2.2. Hence,F is a bi- Jensen mapping satisfying2.27, whereFis given by

Fx, y F₁x, y F₂x, y F₃x, y f0,0. 2.29

Theorem 2.6. Letϕ, ψ :X×X×X → 0,∞be two functions satisfying2.2and2.26for all x, y, z∈X. Letf:X×X → Y be a mapping satisfying2.4for allx, y, z∈X. Then, there exists a bi-Jensen mappingF:X×X → Y satisfying

fx, y−Fx, y≤^∞

j1

Φ

2^jx,2^jy

4^j ^∞

j0

2^jψ

0,0, y 2^j

^∞

j1

ϕ

2^jx,0,0

2^j 2.30

for allx, y∈X, where the mappingF:X×X → Y is given by Fx, y: lim

j→ ∞

1 4^j

f

2^jx,2^jy

−f 2^jx,0

−f 0,2^jy

lim

j→ ∞

1 2^jf

2^jx,0 2^jf

0,y

2^j

− 2^j−1

f0,0

2.31

for allx, y∈X.

Theorem 2.7. Letϕ, ψ :X×X×X → 0,∞be two functions satisfying2.3and2.25for all x, y, z∈X. Letf:X×X → Y be a mapping satisfying2.4for allx, y, z∈X. Then, there exists a bi-Jensen mappingF:X×X → Y satisfying

fx, y−Fx, y≤^∞

j1

Φ

2^jx,2^jy

4^j ^∞

j0

2^jϕ x

2^j,0,0

2.32

for allx, y∈X, where the mappingF:X×X → Y is given by Fx, y: lim

j→ ∞

1 4^j

f

2^jx,2^jy

−f 2^jx,0

−f 0,2^jy

lim

j→ ∞

2^jf

x 2^j,0

1

2^jf 0,2^jy

− 2^j−1

f0,0

2.33

for allx, y∈X.

Applying Theorems2.2–2.7, we easily get the following corollaries.

Corollary 2.8. Let 0< p/1,2andε >0. Letf:X×X → Y be a mapping such that J₁fx, y, z≤ε

x^py^pz^p , J2fx, y, z≤ε

x^py^pz^p 2.34

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for allx, y, z∈X. Then, there exists a unique bi-Jensen mappingF :X×X → Ysuch that fx, y−Fx, y≤ε

7 · 2^p2

24−2^p 2^p 2−2^p

x^py^p

2.35

for allx, y∈X.

Proof. ApplyingTheorem 2.2Theorems2.4and2.5, resp.for the case 0< p <12 < pand 1< p <2, resp., we obtain the desired result.

Corollary 2.9. Let 0< p, q <2 (p, q /1), andε >0. Letf:X×X → Y be a mapping such that J₁fx, y, z≤ε

x^py^pz^p , J2fx, y, z≤ε

x^qy^qz^q 2.36

for allx, y, z∈X. Then, there exists a unique bi-Jensen mappingF :X×X → Ysuch that fx, y−Fx, y

≤ε 3·2^p

4−2^px^p 2^q2 2

4−2^qx^q 2^p

2−2^px^p 2^p2

24−2^py^p3·2^q

4−2^qy^q 2^q 2−2^qy^q

2.37

for allx, y∈X.

Proof. Applying Theorems2.6and2.7, we obtain the desired result.

References

1 S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1968.

2 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, no. 4, pp. 222–224, 1941.

3 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.

4 P. G˘avruta, “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.

5 J.-H. Bae and W.-G. Park, “On the solution of a bi-Jensen functional equation and its stability,” Bulletin of the Korean Mathematical Society, vol. 43, no. 3, pp. 499–507, 2006.

6 K.-W. Jun, I.-S. Jung, and Y.-H. Lee, “Stability of a bi-Jensen functional equation,” preprint.

7 K.-W. Jun, Y.-H. Lee, and J.-H. Oh, “On the Rassias stability of a bi-Jensen functional equation,” Journal of Mathematical Inequalities, vol. 2, no. 3, pp. 363–375, 2008.