On the Superstability of the Pexider Type Trigonometric Functional Equation

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Volume 2010, Article ID 897123,14pages doi:10.1155/2010/897123

Research Article

On the Superstability of the Pexider Type

Trigonometric Functional Equation

Gwang Hui Kim

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

Correspondence should be addressed to Gwang Hui Kim,[email protected]

Received 23 October 2009; Accepted 3 January 2010

Academic Editor: Yeol Je Cho

Copyrightq2010 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.

We will investigate the superstability of thehyperbolictrigonometric functional equation from the following functional equations:fxy±gx−y λfxgyandfxy±gx−y

λgxfy, which can be considered the mixed functional equations of the sine function and cosine function, the hyperbolic sine function and hyperbolic cosine function, and the exponential functions, respectively.

1. Introduction

Baker et al. in1stated the following: iff satisfies the inequality|E1f−E2f| ≤ ε, then eitherfis bounded orE1f E2f. This is frequently referred to as superstability.

The superstability of the cosine functional equation also called the d’Alembert equation

fxyfx−y2fxfy, C

and the sine functional equation

fxfyf _x_y

2

−f _x₋_y

2

, S

were investigated by Baker2and Cholewa3, respectively. Their results were improved by Badora4, Badora and Ger5, G˘avrut¸a6, and Kimsee7,8.

The superstability of the Wilson equation

fxyfx−y2fxgy, Cfg

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The superstability of the trigonometric functional equation concerned with the sine and the cosine equations

fxy−fx−y2fxfy, T

fxy−fx−y2fxgy, Tfg

fxy−fx−y2gxfy, Tgf

fxy−fx−y2gxhy, Tgh

was investigated by Kim10,11, Kim and Lee12.

The hyperbolic cosine function, hyperbolic sine function, hyperbolic trigonometric function, and some exponential functions also satisfy the above mentioned equations; thus they can be called by the hyperbolic cosine sine, trigonometric, exponential functional equations, respectively.

For example,

coshxycoshx−y2 coshxcoshy,

sinhxysinhx−y2 sinhxcoshy,

sinh2

_x_y

2

−sinh2

_x₋_y

2

sinhxsinhy,

caxy_cax−y₂cax

2

ay_a−y₂_cexaya−y

2 ,

exy_ex−y₂ex

2

ey_e−y₂_ex_cosh_y_,

nxycnx−yc2nxc : Jensen equation,forfx nxc,

1.1

whereaandcare constants.

The aim of this paper is to investigate the superstability of the hyperbolic sine functional equationSfrom the following functional equations:

fxygx−yλfxgy, Cfgfg

fxygx−yλgxfy, Cfggf

fxy−gx−yλfxgy, Tfgfg

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on the abelian group. As corollaries, we obtain the superstability ofSfrom the following functional equations:

fxygx−yλfxfy, Cfgff

fxygx−yλgxgy, Cfggg

fxy−gx−yλfxfy, Tfgff

fxy−gx−yλgxgy. Tfggg

Furthermore, the obtained results can be extended to the Banach space.

In this paper, letG,be a uniquely 2-divisible Abelian group,Cthe field of complex numbers, and Rthe field of real numbers. Whenever we deal withC, G,only needs Abelian which is not 2-divisibility.

We may assume thatfandgare nonzero functions,λ, εis a nonnegative real constant, andϕ : G → Ris a mapping. For simplicity, we will form the notations of the equation as follows:

fxyfx−yλfxfy, Cλ

fxyfx−yλfxgy, Cλ_fg

gxygx−yλgxgy, Cλ_g

gxygx−yλgxfy. Cλ_gf

2. Superstability of the Functional Equations

In this section, we will investigate the superstability of the hyperbolic sine functional equationSfrom the functional equationsCfgfg,Cfggf,Tfgfg, andTfggfunder the conditions from which the diﬀerences of each equation are bounded byϕxandϕy.

Theorem 2.1. Suppose thatf, g:G → Csatisfy the inequality _f

xygx−y−λfxgy≤ϕx ∀x, y∈G. 2.1

Then, eithergwithf0 0is bounded orfsatisfiesS. Particularly, ifgsatisfiesCλ, then fandgare the solutions of the Wilson type equationCλ_fg.

Proof. Letgbe the unbounded solution of the inequality2.1. Then, there exists a sequence

{yn}inGsuch that 0/|gyn| → ∞asn → ∞.

Takingy yn in the inequality2.1, dividing both sides by|λgyn|,and passing to

the limit asn → ∞,we obtain the following:

fx lim

n→ ∞

fxyngx−yn

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Using2.1, we have

_f

xyyngx−yyn−λfxgyyn

fx−yyngx−−yyn−λfxg−yyn≤2ϕx, 2.3

and thus,

fxyyngxy−yn λgyn

f

x−yyngx−y−yn

λgyn −λfx·

gyyng−yyn λgyn

≤ 2ϕx λgyn

2.4

for allx, y∈G.

We conclude that, for everyy∈G, there exists a limit function

k1

y: lim

n→ ∞

gyyng−yyn

λgyn , 2.5

where the functionk1:G → Csatisfies the equation

fxyfx−yλfxk1

y ∀x, y∈G. 2.6

Applying the casef0 0 in2.6, which implies thatf is odd and keeping this in mind, by means of2.6, we infer the equality

fxy2₋_f_x₋_y2_λf_x_k 1

yfxy−fx−y

fxfx2y−fx−2y

fxf2yxf2y−x

λfxf2yk1x.

2.7

Puttingyxin2.6, we obtain the equation

f2x λfxk1x, x∈G. 2.8

This, in return, leads to the equation

fxy2₋_f_x₋_y2_f

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being valid for allx, y∈G,which, in the light of the unique 2-divisibility ofG, states nothing else butS.

Particularly, ifgsatisfiesCλ, the limitk1states nothing else butg; thus,2.6validates the required equationCλ_fg.

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

fxygx−y−λfxfy≤ϕx ∀x, y∈G. 2.10

Then, eitherfwithf0 0is bounded orfsatisfiesS.

Proof. Replacinggybyfyin2.1ofTheorem 2.1, an obvious slight change in the proof steps applied inTheorem 2.1allows us to show thatfsatisfiesS.

Namely, forfbe unbounded, there exists a sequence{yn}inGsuch that 0/|fyn| →

∞as n → ∞. Takingy yn in the inequality 2.1, dividing both sides by |λfyn|,and

passing to the limit asn → ∞,we obtain

fx lim

n→ ∞

fxyngx−yn

λfyn , x∈G. 2.11

A similar procedure to that applied after formula2.2yields the required result by using of

2.6.

Theorem 2.3. Suppose thatf, g:G → Csatisfy the inequality

fxygx−y−λgxfy≤ϕx ∀x, y∈G. 2.12

Then, eitherfwithg0 0is bounded orgsatisfiesS. Particularly, iffsatisfiesCλ, then gandfare the solutions of equationCλ_gf.

Proof. For the unboundedf, we can choose a sequence{yn}inGsuch that 0/|fyn| → ∞

asn → ∞.

The same reasoning to the proof applied inTheorem 2.1for2.12withyyngives

gx lim

n→ ∞

fxyngx−yn

λfyn , x∈G. 2.13

Substitutingyynand−yynforyin2.12, and dividing by|λfyn|,then it gives

us the existence of a limit function

k2

y: lim

n→ ∞

fyynf−yyn

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gxygx−yλgxk2

y ∀x, y∈G. 2.15

Applying the caseg0 0 in2.15, it implies thatgis odd.

A similar procedure to that applied after formula 2.6 allows us to show that g satisfiesS.

Particularly, iffsatisfiesCλ, the limitk2states nothing else butf; thus, the required equationCλ_gfholds from2.15.

fxygx−y−λgxgy≤ϕx ∀x, y∈G. 2.16

Then, eithergwithg0 0is bounded orgsatisfiesS.

Proof. Substitutinggyforfyin2.12ofTheorem 2.3, the next of the proof runs along that of the above theorem.

xygx−y−λfxgy≤ϕy ∀x, y∈G. 2.17

Then, eitherfwithg0 0is bounded orgsatisfiesS. Particularly, iffsatisfiesCλ, then gandfare solutions of the Wilson type equationCλ_gf.

Proof. For the unboundedfof the inequality2.17, we can choose a sequence{xn}inGsuch

that 0/|fxn| → ∞asn → ∞.

Takingxxnin the inequality2.17, dividing both sides by|λfxn|, and passing to

the limit asn → ∞,we obtain

gy lim

n→ ∞

fxnygxn−y

λfxn , x∈G. 2.18

In2.17, replacingxbyxnyandxn−y, replacingybyx, and dividing by|λfxn|,

it then gives us the existence of a limit function

k3x:_nlim_{→ ∞}

fxnyfxn−y

λfxn , 2.19

gxygx−yλgxk3

y ∀x, y∈G. 2.20

Applying the caseg0 0 in2.20, it implies thatg is odd. Since2.20equals to

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thatg satisfiesS. Particularly, iff satisfiesCλ, then the limitk3states nothing else butf, thus, the required equationCλ_fgholds from2.20.

fxygx−y−λfxfy≤ϕy, ∀x, y∈G. 2.21

Proof. Substitutingfyforgyin2.17of Theorem 2.5, asCorollary 2.4, we then obtain the required result from the above theorem.

fxygx−y−λgxfy≤ϕy, ∀x, y∈G. 2.22

Then, eithergwithf0 0is bounded orfsatisfiesS. Particularly, ifgsatisfiesCλ, then fandgare the solutions of equationCλ_fg.

Proof. For the unboundedg, we can choose a sequence{xn}inGsuch that 0/|gxn| → ∞

asn → ∞.

For2.22withxxn, the same reasoning as the proof applied inTheorem 2.1gives

us

fy lim

n→ ∞

fxnygxn−y

λgxn , x∈G. 2.23

In2.22, replacingxbyxnyandxn−y,replacingxbyy, and dividing by|λgxn|,

it then gives us the existence of a limit function

k4

y: lim

n→ ∞

gxnygxn−y

λgxn , 2.24

fxyfx−yλfxk4

y, ∀x, y∈G. 2.25

Since2.25is the same as2.6, the next proof runs along that ofTheorem 2.1.

fxygx−y−λgxgy≤ϕy, ∀x, y∈G. 2.26

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Proof. Substitutinggyforfyin2.22ofTheorem 2.7, the next proof runs along that of the above theorem.

The casesϕx,ϕyin the following result follow the procedure applied in Theorems

2.1and2.5, respectively. In the obtained result, applying the casesλ2 andϕx ϕy ε, then they are founded in2,4–6.

Corollary 2.9. Suppose thatf:G → Csatisfy the inequality

_f

xyfx−y−λfxfy≤

⎧ ⎨ ⎩

ϕx,

ϕy, ∀x, y∈G. 2.27

Then, eitherfis bounded orfsatisfiesCλ.

Proof. Forf being unbounded, we can choose two sequences{xn}and{yn}inGsuch that

0/|fxn|and|fyn| → ∞asn → ∞.

(i) case

ϕ

x

Takingyynin inequality2.27, dividing it by|λfyn|, and passing to the limit asn → ∞,

we obtain

fx lim

n→ ∞

fxynfx−yn

λfyn , x∈G. 2.28

In2.27, replacingybyyyn and−yyn,and dividing by|λfyn|, it then gives,

with the application of2.28, thatfsatisfiesCλ.

(ii) case

ϕ

y

For the chosen sequence{xn}, the procedure asiimplies

fy lim

n→ ∞

fxnyfxn−y

λfxn , x∈G. 2.29

In2.27, replacingxbyxnyandxn−yand replacingybyx, the other procedure is

the same asi.

Since the proofs of the following results Theorems 2.10–2.16 and Corollaries 2.11–

2.17for the functional equationsTfgfg,Tfggf,Tfgff, andTfgggare, respectively, the same processes those of Theorems2.1–2.7and Corollaries2.2–2.8, as we will skip their proofs.

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Then, eithergwithf0 0is bounded orfsatisfiesS. Particularly, ifgsatisfiesCλ, then fandgare solutions of the Wilson type equationCλ_fg.

fxy−gx−y−λfxfy≤ϕx ∀x, y∈G. 2.31

_f

xy−gx−y−λgxfy≤ϕx ∀x, y∈G. 2.32

fxy−gx−y−λgxgy≤ϕx ∀x, y∈G. 2.33

fxy−gx−y−λfxgy≤ϕy ∀x, y∈G. 2.34

fxy−gx−y−λfxfy≤ϕy ∀x, y∈G. 2.35

fxy−gx−y−λgxfy≤ϕy ∀x, y∈G. 2.36

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fxy−gx−y−λgxgy≤ϕy ∀x, y∈G. 2.37

The cases ϕx and ϕy in the following result follow the procedure applied in Theorems 2.10and 2.14, respectively. In the obtained result, applying the casesλ 2 and

ϕx ϕy ε, then they are founded in10–12.

Corollary 2.18. Suppose thatf :G → Csatisfy the inequality

fxy−fx−y−λfxfy≤

⎧ ⎨ ⎩

ϕx,

ϕy, ∀x, y∈G. 2.38

Thenfis bounded.

Proof. Forf being unbounded, we can choose two sequences{xn}and{yn}inGsuch that

0/|fxn|and|fyn| → ∞asn → ∞.

(i) case

ϕ

x

First, going though the same process of the caseiofCorollary 2.9, then we obtain thatf satisfiesCλ.

Secondly, from the chosen sequence{yn}, we obtain

fx lim

n→ ∞

fxyn−fx−yn

λfyn , x∈G. 2.39

Replacingxbyxyn andx−ynin2.38, then we obtain, from their diﬀerence, the

inequality

fxyyn−fxy−yn λfyn

−f

x−yyn−fx−y−yn λfyn −λ·

fxyn−fx−yn λfyn ·f

y

≤ 2ϕx λfyn,

2.40

which gives, with the application of2.39, the equation

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Thus, since the functionfsatisfies two equationsCλandTλ, the functionf states nothing else but zero. It is a contradiction assuming thatfis nonzero. Thusfis bounded.

(ii) case

ϕ

y

For the chosen sequence {xn}, the same procedure as in the above case i gives us the

required result.

Remark 2.19. aSubstitutingf forgof the second term of the stability inequalities in all the results ofSection 2, then we obtain the same number of corollaries, which are the stability of the hyperbolic cosine type functional equations Cλ_fg, Cλ_gf, and the hyperbolic trigonometric type functional equationsfx y−fx−y λ fxgy, fx y−

fx−y λ gxfy.

bApplying the caseλ 2 in all the results ofSection 2anda’s application, then we obtain the same number of corollaries. Some of their stabilities were founded in papers

6,7,9–11.

cApplyingϕx ϕy εin all the results ofSection 2anda’s application, then we obtain the same number of corollaries. Some of their stabilities were founded in papers

6,7,9–12.

dApplyingλ2 andϕx ϕy εin all the results ofSection 2,a’s,b’s, and

c’s applications, then we obtain the same number of corollaries. Some of their stabilities were founded in papers5–7,9–12.

3. Extension to the Banach Space

In all the results presented inSection 2, the range of functions on the abelian group can be extended to the Banach space. For simplicity, we will only prove the plus case of3.1 of

Theorem 3.1. The other cases are similar to this, thus their proofs will be omitted.

Theorem 3.1. LetE, · be a semisimple commutative Banach space. Assume thatf, g :G → E

satisfy one of each inequalities

fxy±gx−y−λfxgy ≤ϕx, 3.1

fxy±gx−y−λgxfy ≤ϕx 3.2

for allx, y∈G.For an arbitrary linear multiplicative functionalx∗∈E∗, then,

icase3.1, eitherx∗◦g withf0 0is bounded orfsatisfiesS. Particularly, ifg satisfiesCλ, thenfandgare the solutions of the Wilson type equationCλ_fg.

iicase3.2, eitherfwithg0 0is bounded orgsatisfiesS. Particularly, iffsatisfies

Cλ, thengandfare the solutions of the Wilson type equationCλ_gf.

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

ϕx≥fxygx−y−λfxgy

sup

y∗₁

y∗_f_x_y_g_x₋_y₋_λf_x_g_y

≥x∗_f_x_y_x∗_g_x₋_y₋₂_x∗_f_x_x∗_g_y_,

3.3

which states that the superpositionsx∗◦fandx∗◦gyield a solution of inequality2.1. Since, by assumption, the superpositionx∗◦gwithf0 0 is unbounded, an appeal toTheorem 2.1

shows that the two results hold.

First, the functionx∗◦fsolvesS. In other words, bearing the linear multiplicativity ofx∗in mind, for allx, y∈G, the diﬀerenceDx, y:G×G → Cdefined by

DSx, y:f

_x_y

2

−f _x₋_y

2

−fxfy, 3.4

falls into the kernel ofx∗.Therefore, in view of the unrestricted choice ofx∗,we infer that

DSx, y∈kerx∗:x∗ is a multiplicative member of E∗ 3.5

for allx, y∈G.Since the algebraEhas been assumed to be semisimple, the last term of the above formula coincides with the singleton{0},that is,

DSx, y0, ∀x, y∈G, 3.6

as claimed.

Second, in particular, ifx∗◦gsatisfiesCλ, thenx∗◦fandx∗◦gare solutions of the Wilson type equationCλ_fg. This means that

DCλ fg

x, y:fxyfx−y−λfxgy, 3.7

falls into the kernel ofx∗. Through the above process, we obtain

DCλ fg

x, y0, ∀x, y∈G, 3.8

as claimed. The minus case is also similar.

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Corollary 3.2. LetE, · be a semisimple commutative Banach space. Assume thatf, g:G → E

fxy±gx−y−λfxfy≤ϕx, 3.9

_f

xy±gx−y−λgxgy≤ϕx 3.10

for allx, y∈G.For an arbitrary linear multiplicative functionalx∗∈E∗,

iin case3.9, eitherx∗◦fwithf0 0is bounded orfsatisfiesS.

iiin case3.10, eitherx∗◦gwithg0 0is bounded orgsatisfiesS.

Theorem 3.3. LetE, · be a semisimple commutative Banach space. Assume thatf, g :G → E

fxy±gx−y−λfxgy≤ϕy, 3.11

fxy±gx−y−λgxfy≤ϕy 3.12

for allx, y∈G.For an arbitrary linear multiplicative functionalx∗∈E∗, then,

iin case3.11, eitherx∗◦fwithg0 0is bounded orgsatisfiesS. Particularly, if fsatisfiesCλ, thengandfare solutions of the Wilson type equationCλ_gf.

iiin case3.12, eitherg withf0 0is bounded orfsatisfiesS. Particularly, ifg satisfiesCλ, thenfandgare solutions of the Wilson type equationCλ_fg.

Corollary 3.4. LetE, · be a semisimple commutative Banach space. Assume thatf, g:G → E

fxy±gx−y−λfxfy≤ϕy, 3.13

_f

xy±gx−y−λgxgy≤ϕy 3.14

for allx, y∈G.For an arbitrary linear multiplicative functionalx∗∈E∗,

iin case3.13, eitherx∗◦fwithf0 0is bounded orfsatisfiesS.

iiin case3.14, eitherx∗◦gwithg0 0is bounded orgsatisfiesS.

Remark 3.5. We obtain the same number of corollaries on the Banach space for all the theorems mentioned inSection 2and all the results obtained by applying ofa,b,c, andd in

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