Stability of the Pexiderized Lobacevski Equation

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Volume 2011, Article ID 540274,10pages doi:10.1155/2011/540274

Research Article

Stability of the Pexiderized Lobacevski Equation

Gwang Hui Kim

Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Correspondence should be addressed to Gwang Hui Kim,[email protected] Received 16 April 2011; Accepted 11 June 2011

Academic Editor: Junjie Wei

Copyrightq2011 Gwang Hui Kim. 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.

The aim of this paper is to investigate the solution and the superstability of the Pexiderized Lobacevski equationfxy/2² gxhy, wheref,g,h:G² → Care unknown functions on an Abelian semigroupG,. The obtained result is a generalization of Gˇavrut¸a’s result in 1994 and Kim’s result in 2010.

1. Introduction

The stability problem of the functional equation was conjectured by Ulam 1 during the conference in the University of Wisconsin in 1940. In the next year, it was solved by Hyers 2in the case of additive mapping, which is called the Hyers-Ulam stability. Thereafter, this problem was improved by Bourgin3, Aoki4, Rassias5, Ger6, and Gˇavrut¸a et al.7,8 in which Rassias’ result is called the Hyers-Ulam-Rassias stability.

In 1979, Baker et al.9developed the superstability, which is that iff is a function from a vector space toRsatisfying

f xy

−fxf

y≤ε 1.1

for some fixedε >0, then eitherfis bounded or satisfies the exponential functional equation f

xy

fxf y

. E

In 1983, the superstability bounded by a constant for the sine functional equation

f xy

2 ₂

−f x−y

2 ₂

fxf y

S

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was investigated by Cholewa10and was improved by Badora and Ger11. Recently, the superstability bounded by some function for the Pexider type sine functional equation

f xy

2 2

−f x−y

2 2

gxh y

1.2

has been investigated by Kim12,13.

In 1994, Gˇavrut¸a14proved the superstability of the Lobacevski equation

f xy

2 ₂

fxf y

L

under the condition bounded by a constant.

Kim15improved his result under the condition bounded by an unknown function.

In there, author conjectured through an example that the Lobacevski equationLwill have a solution as an exponential function. Namely, for a simple example of this equation, we can find the functional equatione^xy/2²e^xe^y.

The aim of this paper is to investigate the solution and the superstability of the Pexiderized Lobacevski equation

f xy

2 2

gxh y

PL

under the condition bounded by a function. Namely, this has improved in the Pexider type for the results of Gˇavrut¸a and Kim.

Furthermore, the range of the function in all results is expanded to the Banach space.

The solution of PL will be represented as an exponential, namely, for a simple example of this equation, it will be considered as a geometric mean

fx

αβe^x αe^x

βe^x

gxhx, where α, β >0. 1.3

In this paper, letG,be a uniquely 2-divisible Abelian semigroupi.e., for eachx∈ G, there exists a uniquey∈Gsuch thatyy x: suchywill be denoted byx/2,Cis the field of complex numbers,Rthe field of real numbers, andR the set of positive reals. We assume thatf, g, h :G → Care nonzero and nonconstant functions,εis a nonnegative real constant, andϕ:G → Ris a mapping.

2. Stability of the Pexiderized Lobacevski Equation PL

We will investigate the solution and the superstability of the Pexiderized Lobacevski equation PL.

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Theorem 2.1. Suppose thatf, g, h:G → Rsatisfy the inequality

f xy

2 ₂

−gxh y

≤ε 2.1

for allx, y∈G.

Then, either there existC1, C2, C3>0 such that

gx≤C₁, |hx| ≤C₂, fx≤C₃, 2.2

for allx∈G, or else each functiongandhsatisfiesL. Heregandhare represented by

gx g0ex, hx h0ex, 2.3

fx²−g0h0·ex²≤ε, 2.4

whereexis an exponential function.

Proof. Replacingxwithyin2.1, and then subtracting them and using triangle inequality, we have

gxh y

−g y

hx≤2ε ∀x, y∈G. 2.5

It follows from the inequality2.5that there exist constantsc₁, c₂, d₁, d₂≥0 such that gx≤c1|hx|d1, 2.6

|hx| ≤c2gxd2 2.7

for allx∈G. It follows from2.6and2.7thatgis bounded if and only ifhis bounded. If either ofgorhis bounded, then we obtain2.2from2.1.

Now if hx is unbounded, then we can choose yn ∈ G so that |hyn| → ∞ as n → ∞. Lettingyynin2.1, dividing by|hyn|, and lettingn → ∞, we have

gx lim

n→ ∞

f xy_n

/22

h

y_n , ∀x∈G. 2.8

It follows from2.1and2.8that

g xy

gz lim

n→ ∞

f

xyy_n /2₂

gz h

y_n lim

n→ ∞

gxh yy_n

gz R₁ h

y_n lim

n→ ∞

gxf

yzyn

/2₂

R1R2

h

y_n gxg

yz lim

n→ ∞

R1R2

h y_n ,

2.9

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where|R1| ≤ε|gz|, |R2| ≤ε|gx|, which implies

g xy

gz gxg yz

2.10

for allx, y, z∈G.

Lettingz0 in2.10, we get g

xy

g0 gxg y

2.11

for allx, y∈G, which implies that

gx g0e1x, 2.12

whereg0/0sincegxis a nonzero and nonconstant function, ande1is an exponential.

Exchanging the roles ofgandh, by the same proceeding, we have

hx h0e2x, 2.13

whereh0/0, ande2is an exponential.

Putting2.12and2.13in2.5, it implies e₁xe2

y

−e₁ y

e₂x≤ 2ε

g0h0 M ∀x, y∈G. 2.14

Letx0 in2.14. Sincee1ande2are exponentials, this implies that|e1y−e2y| ≤M for ally∈G. Hence, from this and2.14, we have

e1

y

|e1x−e2x|e1x e1

y

−e2

y e1xe2

y

−e1

y e2x

≤e₁xMM, 2.15

which is

|e1x−e2x| ≤ e₁xMM e₁

y , 2.16

for allx, y∈G.

Sincegis unbounded from2.2, we can chooseyn∈Gso thatgyn g0e1yn →

∞asn → ∞. Lettingy y_nin2.16, we get thate₁x e₂x. Let it be denoted byex.

Then2.12and2.13state nothing but2.3. Putting2.3with x yin2.1, we get the inequality2.4.

Finally, it is immediate thatgandhin2.3satisfyL, respectively.

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Corollary 2.2. Suppose thatf, g:G → Rsatisfy the inequality

f xy

2 ₂

−gxg y

≤ε 2.17

for allx, y∈G.

Then, eithergis bounded orgsatisfiesL. In particular,gis represented by

gx g0ex, 2.18

whereeis exponential.

Corollary 2.3. Suppose thatf, g:G → Rsatisfy the inequality

f xy

2 ₂

−gxf y

≤ε 2.19

for allx, y∈G.

Then either there existC₁, C₂>0 such that

gx≤C1, fx≤C2 2.20

for allx∈G, or else each functionfandgsatisfiesL. In particular,fandgare represented by

fx f0ex, gx g0ex, 2.21

wheree:G → Ris exponential.

Corollary 2.4. Suppose thatf:G → Rsatisfy the inequality

f xy

2 ₂

−fxf y

≤ε 2.22

for allx, y∈G.

Then eitherfis bounded orfsatisfiesL. In particular,fis represented by

fx f0ex, 2.23

wheree:G → Ris exponential.

InCorollary 2.4, it is founded in papers14,15thatfsatisfiesL.

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f xy

2 2

−gxh y

≤ϕx 2.24

for allx, y∈G.

Then, eitherhis bounded orgis an exponential by the multiplying of a scalarg0and satisfies L.

Proof. Suppose thathxis unbounded. Then we can chooseyn∈Gsuch that|hyn| → ∞ asn → ∞. Lettingyynin2.24, dividing by|hyn|, and lettingn → ∞, we have

gx lim

n→ ∞

f xy_n h

y_n ∀x∈G. 2.25

Thus, it follows from2.24and2.25that

g xy

gz lim

n→ ∞

f

xyy_n /2₂

gz h

yn

lim

n→ ∞

gxh yy_n

gz R₁ h

yn

lim

n→ ∞

gxf

yzyn

/2₂

R1R2

h

y_n gxg

yz lim

n→ ∞

R1R2

h y_n ,

2.26

where|R1| ≤ε|gz|, |R2| ≤ε|gx|, which implies g

xy

gz gxg yz

2.27 for allx, y, z∈G.

Lettingz0 in2.27, we get g

xy

g0 gxg y

2.28 for allx, y, z∈G. Namely, it means thatgis an exponential function by the multiplying of a scalarg0and satisfiesL.

f xy

2 ₂

−gxh y

≤ϕ y

2.29

for allx, y∈G.

Then, eithergis bounded orhis an exponential by the multiplying of a scalarh0and satisfies L.

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Proof. The proof runs along a slight change in the step-by-step procedure inTheorem 2.5.

Remark 2.7. iAs Corollaries2.2–2.4ofTheorem 2.1, by replacinggandhwithfin Theorems 2.5and2.6, we can obtain more corollaries for the following functional equations:

f xy

2 2

gxg y

,

f xy

2 ₂

fxg y

,

f xy

2 ₂

fxf y

,

2.30

in which the case of2.30is found in paper15.

iiFor the results obtained from each equation of the abovei, by applyingϕy ϕx ε, we can obtain the same number of corollaries.

3. Extension to Banach Algebra

All obtained results can be extended to the stability on the Banach algebras. We will illustrate only for the case ofTheorem 2.1among them.

Theorem 3.1. LetE, · be a semisimple commutative Banach algebra. Assume thatf, g, h:G → Esatisfy the inequality

f

xy 2

2

−gxh y

≤ε 3.1

for allx, y∈G.

Then, for an arbitrary linear multiplicative functionalx^∗∈E^∗, either there existC₁, C₂, C₃>0 such that

x^∗◦g

x≤C₁, |x^∗◦hx| ≤C₂, x^∗◦f

x≤C₃ 3.2

for allx∈G, or else each functiongandhsatisfiesL. Heregandhare represented by

gx g0ex, hx h0ex, 3.3

fx²−g0h0·ex² ≤ε, 3.4 whereexis an exponential function.

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Proof. Assume that3.1holds, and fix arbitrarily a linear multiplicative functionalx^∗∈E. As well known, we havex^∗1, hence, for everyx, y∈G, we have

ε≥ f

xy 2

2

−gxh y

sup

y^∗1

y^∗

f

xy 2

2

−gxh y

≥ x^∗

f

xy 2

−x^∗ gx

x^∗ h

y,

3.5

which states that the superpositions x^∗◦ f,x^∗ ◦ g, andx^∗ ◦h satisfy the inequality 2.1 ofTheorem 2.1. Due to the same processing as from2.5to2.7, to fix arbitrarily a linear multiplicative functionalx^∗∈E, indeed, we have

x^∗◦g

xx^∗◦h y

−

x^∗◦g y

x^∗◦hx≤2ε ∀x, y∈G. 3.6

It follows from the inequality3.6that there exist constantsc1, c2, d1, d2≥0 such that x^∗◦g

x≤c1|x^∗◦hx|d1,

|x^∗◦hx| ≤c₂x^∗◦g

xd₂ 3.7 for allx ∈ G. Sincex^∗is an arbitrarily linear multiplicative functional, it follows from3.7 thatg is bounded if and only ifhis bounded. Assume that one ofg orhis bounded. From 3.1, we arrive at3.2.

By the assumption3.2, an appeal toTheorem 2.1shows that x^∗◦g

x

x^∗◦g0e1

x, 3.8

x^∗◦hx x^∗◦h0e2x, 3.9

x^∗◦f x−

x^∗◦g0h0e3

x≤ε, 3.10

wheree₁, e₂, e₃:G → Rare exponentials. In other words, bearing the linear multiplicativity ofx^∗in mind, for allx∈G, each diﬀerence derived from3.8and3.9

D3.8x:gx− g0e1

x,

D3.9x:hx−h0e2x, 3.11

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falls into the kernel ofx^∗. Therefore, in view of the unrestricted choice ofx^∗, we infer that D3.8x, D3.9x∈

kerx^∗:x^∗ is a multiplicative member of E^∗

3.12 for allx ∈ G. Since the algebra Ehas been assumed to be semisimple, the last term of the previous formula coincides with the singleton{0}, that is,

gx−g0e1x 0, hx−h0e2x 0, x∈G. 3.13

Putting3.13in3.6, following the same proceeding as after2.13inTheorem 2.1, then we arrive thate1x e2x. Indeed, we have

x^∗◦g0e1

xx^∗◦h0e2 y

−

x^∗◦g0e1

y

x^∗◦h0e2x≤2ε 3.14

for allx, y∈G. This implies that x^∗◦e1xx^∗◦e2

y

−x^∗◦e1 y

x^∗◦e2x≤ 2ε

g0h0 M ∀x, y∈G. 3.15 Lettingx 0 in3.15, it implies|x^∗◦e2y−x^∗◦e1y| ≤ M/x^∗1 Mfor all y∈G. Thus, from this and3.15, we have

x^∗◦e₁

y|x^∗◦e₁x−x^∗◦e₂x|

x^∗◦e1x

x^∗◦e1 y

−x^∗◦e2 y x^∗◦e₁xx^∗◦e₂

y

−x^∗◦e₁ y

x^∗◦e₂x

≤ |x^∗◦e₁x|MM,

3.16

which is

|x^∗◦e₁x−x^∗◦e₂x| ≤|x^∗◦e1x|MM x^∗◦e₁

y , 3.17

for allx, y∈G.

Sincex^∗◦g is unbounded from3.2, we can chooseyn∈Gso that|x^∗◦gyn|

|g0x^∗◦e₁yn| → ∞asn → ∞. Lettingyy_nin3.17, which arrive that

x^∗◦e1x x^∗◦e2x. 3.18 Using the same logic as before, that is, bearing the linear multiplicativity ofx^∗in mind, the diﬀerence derived from3.18,D3.18x:e₁x−e2x, falls into the kernel ofx^∗. Then, the semisimplicity ofEimplies thate1x e2x. Let it be denoted byex, which arrive the claimed3.3and3.4.

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Sinceex:G → Ris exponential, it is immediate from3.3that each functiongand hsatisfiesL.

Remark 3.2. All results of Section 2 containing Remark 2.7 can be extended to the Banach space asTheorem 3.1.

Acknowledgment

This work was supported by Kangnam University Research Grant in 2010.

References

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Stability of the Pexiderized Lobacevski Equation

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