Superstability of Some Pexider Type Functional Equation

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Volume 2010, Article ID 985348,16pages doi:10.1155/2010/985348

Research Article

Superstability of Some Pexider-Type

Functional Equation

Gwang Hui Kim

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

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

Received 27 August 2010; Revised 18 October 2010; Accepted 19 October 2010

Academic Editor: Andrei Volodin

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 the sine functional equation from the following Pexider-type functional equationfxygx−y λ·hxky λ: constant, which can be considered the mixed functional equation of the sine and cosine functions, the mixed functional equation of the hyperbolic sine and hyperbolic cosine functions, and the exponential-type functional equations.

1. Introduction

In 1940, Ulam1conjectured the stability problem. Next year, this problem was aﬃrmatively solved by Hyers2, which is through the following.

LetXandYbe Banach spaces with norm · , respectively. Iff:X → Ysatisfies _f

xy−fx−fy≤ε, ∀x, y∈X, 1.1

then there exists a unique additive mappingA:X → Y such that

_f_x₋_A_x_≤_ε, _∀_x_∈_X. _1.2

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In 1979, Baker et al.8showed that iffis a function from a vector space toRsatisfying

_f

xy−fxfy≤ε, 1.3

then eitherfis bounded or satisfies the exponential functional equation

fxyfxfy. 1.4

This method is referred to as the superstability of the functional equation1.4. In this paper, letG,be a uniquely 2 divisible Abelian group, the field of complex numbers, andthe field of real numbers,the set of positive reals. Whenever we only deal withC,G,needs the Abelian which is not 2-divisible.

We may assume that f, g, hand k are nonzero functions, λ, ε is a nonnegative real constant, andϕ:G → is a mapping.

In 1980, the superstability of the cosine functional equation also referred the d’Alembert functional equation

fxyfx−y2fxfy, C

was investigated by Baker9with the following result: letε >0. Iff :G → Csatisfies

_f

xyfx−y−2fxfy≤ε, 1.5

then either|fx| ≤1√12ε/2 for allx∈Gorfis a solution ofC. Badora10in 1998, and Badora and Ger11in 2002 under the condition|fxyfx−y−2fxfy| ≤ε, ϕx orϕy, respectively. Also the stability of the d’Alembert functional equation is founded in papers12–16.

In the present work, the stability question regarding a Pexider-type trigonometric functional equation as a generalization of the cosine equationCis investigated.

To be systematic, we first list all functional equations that are of interest here.

fxygx−yλhxhy Pλ f ghh

fxygx−yλfxhy, Pλ f gf h

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fxygx−yλgxhy, P_{f ggh}λ

fxygx−yλhxgy, Pλ f ghg

fxygx−yλfxgy, Pλ f gf g

fxygx−yλgxfy, Pλ f ggf

fxygx−yλfxfy, Pλ f gf f

fxygx−yλgxgy, Pλ f ggg

fxyfx−yλgxhy, P_{f f gh}λ

fxyfx−yλgxgy, Pλ f f gg

fxyfx−yλfxgy, Cλ_{f g}

fxyfx−yλgxfy, Cλ_gf

fxyfx−yλfxfy, Cλ

fxygx−y2hxky, Pf ghk

fxygx−y2hxhy, Pf ghh

fxygx−y2fxhy, Pf gf h

fxygx−y2hxfy, Pf ghf

fxygx−y2gxhy, Pf ggh

fxygx−y2hxgy, Pf ghg

fxygx−y2fxgy, Pf gf g

fxygx−y2gxfy, Pf ggf

fxygx−y2fxfy, Pf gf f

fxygx−y2gxgy, Pf ggg

fxyfx−y2fxgy, Cf g

fxyfx−y2gxfy, Cgf

fxyfx−y2gxgy, Cgg

fxyfx−y2gxhy, Cgh

fxyfx−y2fx. Jx

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equations; therefore, they can also be called the hyperbolic cosine sine, trigonometric functional equation, exponential functional equation, and Jensen equation, respectively.

For example,

coshxycoshx−y2 coshxcoshy,

coshxy−coshx−y2 sinhxsinhy,

sinhxysinhx−y2 sinhxcoshy,

sinhxy−sinhx−y2 coshxsinhy,

sinh2

xy 2

−sinh2

x−y 2

sinhxsinhy,

caxycax−y2ca x

2

aya−y2cexa

y_a−y

2 ,

exyex−y2e x

2

eye−y2excoshy,

nxycnx−yc2nxc: forfx nxc,

1.6

whereaandcare constants.

The equationCf gis referred to as the Wilson equation. In 2001, Kim and Kannappan

13investigated the superstability related to the d’AlembertCand the Wilson functional equationsCf g,Cgfunder the condition bounded by constant. Kim has also improved the superstability of the generalized cosine type-functional equationsCgg, andPf gf g,Pf ggf in papers14,15,17.

In particular, author Kim and Lee18investigated the superstability ofSfrom the functional equationCghunder the condition bounded by function, that is

1iff, g, h:G → Csatisfies

fxyfx−y−2gxhy≤ϕx, 1.7

then eitherhis bounded orgsatisfiesS;

2iff, g, h:G → Csatisfies

_f

xyfx−y−2gxhy≤ϕy, 1.8

then eithergis bounded orhsatisfiesS.

In 1983, Cholewa19investigated the superstability of the sine functional equation

fxfyf

_x_y

2 2

−f

_x₋_y

2 2

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under the condition bounded by constant. Namely, if an unbounded functionf : G → C satisfies

fxf

y−f

_x_y

2 2

f

_x₋_y

2 2

≤ε, 1.9

then it satisfiesS.

In Kim’s work20,21, the superstability of sine functional equation from the general-ized sine functional equations

fxgyf

_x_y

2 2

−f

_x₋_y

2 2

, Sf g

gxfyf

xy 2

2

−f

x−y 2

2

, Sgf

gxhyf

_x_y

2 2

−f

_x₋_y

2 2

Sgh

was treated under the conditions bounded by constant and functions.

The aim of this paper is to investigate the transferred superstability for the sine functional equation from the following Pexider type functional equations:

fxygx−yλ·hxky, λ: constant Pλ f ghk

on the abelian group. Furthermore, the obtained results can be extended to the Banach space. Consequently, as corollaries, we can obtain 29× 4 stability results concerned with the sine functional equation S and the Wilson-type equations Cλ

f g from 29 functional equations of the Pλ_{, C}λ_{, P}_{, and}_C _{types from a selection of functions} _{f, g, h, k} _{in the order} of variablesxy, x−y, x, y.

2. Superstability of the Sine Functional Equation from

the Equation

P

λ

fghk

In this section, we will investigate the superstability related to the d’Alembert-type equation

Cλ _{and Wilson-type equation}_Cλ

f g, of the sine functional equationSfrom the Pexider type functional equationP_{f ghk}λ .

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

_f

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Ifkfails to be bounded, then

ihsatisfiesSunder one of the casesh0 0orf−x −gx; and

iiIn addition, ifksatisfiesCλ_{, then}_h_and_k_{are solutions of}_Cλ

f g:hxy hx−y

λhxky.

Proof. Letkbe unbounded solution of the inequality3.12. Then, there exists a sequence{yn} inGsuch that 0/|kyn| → ∞asn → ∞.

iTakingyynin the inequality3.12, dividing both sides by|λkyn|, and passing to the limit asn → ∞, we obtain

hx lim n→ ∞

fxyn

gx−yn

λ·kyn

, x∈G. 2.2

Replaceybyyynand−yynin3.12, we have _f

xyyn

gx−yyn

−λ·hxkyyn

fx−yyn

gx−−yyn

−λ·hxk−yyn≤2ϕx

2.3

so that

fxyyn

gxy−yn

λ·kyn

f

x−yyn

gx−y−yn

λ·kyn

−λ·hx·k

yyn

k−yyn

λ·kyn

≤ 2ϕx

λ·kyn

2.4

for allx, y, yn∈G.

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

lk

y: lim n→ ∞

kyyn

k−yyn

λ·kyn

, 2.5

where the functionlk:G → satisfies the equation

hxyhx−yλ·hxlk

y, ∀x, y∈G. 2.6

Applying the caseh0 0 in2.6, it implies thathis odd. Keeping this in mind, by means of2.6, we infer the equality

hxy2−hx−y2λ·hxlk

yhxy−hx−y

hxhx2y−hx−2y

hxh2yxh2y−x

λ·hxh2ylkx.

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Puttingyxin2.6, we get the equation

h2x λ·hxlkx, x∈G. 2.8

This, in return, leads to the equation

hxy2−hx−y2h2xh2y 2.9

valid for allx, y ∈ Gwhich, in the light of the unique 2divisibility ofG, states nothing else butS.

In the particular casef−x −gx, it is enough to show thath0 0. Suppose that this is not the case.

Puttingx0 in3.12, due toh0/0 andf−x −gx, we obtain the inequality

ky≤ ϕ0

λ· |h0|, y∈G. 2.10

This inequality means thatkis globally bounded, which is a contradiction. Thus, since the claimedh0 0 holds, we know thathsatisfiesS.

iiIn the casek satisfiesCλ_{, the limit}_l

k states nothing else butk, so, from2.6,h andkvalidateCλ_{f g}.

Theorem 2.2. Suppose thatf, g, h, k:G → satisfy the inequality

_f

xygx−y−λ·hxky≤ϕy ∀x, y∈G. 2.11

Ifhfails to be bounded, then

iksatisfiesSunder one of the casesk0 0orfx −gx

iiin addition, if h satisfies Cλ, then k and h are solutions of the equation of C_gfλ : kxy kx−y λhxky.

Proof. iTakingxxnin the inequality2.11, dividing both sides by|λ·hxn|, and passing to the limit asn → ∞, we obtain that

ky lim n→ ∞

fxny

gxn−y

λ·hxn

, x∈G. 2.12

Replacexbyxnxandxn−xin2.11divide byλ·hxn; then it gives us the existence of the limit function

lhx: lim n→ ∞

hxnx hxn−x

λ·hxn

, 2.13

where the functionlh:G → satisfies the equation

kxyk−xyλ·lhxk

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Applying the casek0 0 in2.14, it implies thatkis odd.

A similar procedure to that applied after2.6 ofTheorem 2.1in 2.14allows us to show thatksatisfiesS.

The casefx −gxis also the same as the reason forTheorem 2.1.

iiIn the casehsatisfiesCλ, the limitlh states nothing else buth, so, from2.14,

kandhvalidateCλ f g.

The following corollaries followly immediate from the Theorems2.1and2.2.

Corollary 2.3. Suppose thatf, g, h, k:G → satisfy the inequality

_f

xygx−y−λ·hxky≤minφx, φy, ∀x, y∈G. 2.15

aIfkfails to be bounded, then

ihsatisfiesSunder one of the casesh0 0orf−x −gx, and

iiin addition, ifhsatisfiesCλ_{, then}_h_and_k_{are solutions of}_Cλ

f g:hxy hx−

y λhxky.

bIfhfails to be bounded, then

iiiksatisfiesSunder one of the casesk0 0orfx −gx, and

ivin addition, ifhsatisfiesCλ_{, then}_h_and_k_{are solutions of}_Cλ

gf:kxykx−

y λhxky.

fxygx−y−λ·hxky≤ε, ∀x, y∈G. 2.16

iiin addition, ifksatisfiesCλ_{, then}_h_and_k_{are solutions of}_Cλ

f g:hxyhx−

y λhxky.

iiiksatisfiesSunder one of the casesk0 0orfx −gx, and

ivin addition, ifhsatisfiesCλ_{, then}_h_and_k_{are solutions of}_Cλ

gf:kxykx−

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3. Applications in the Reduced Equations

3.1. Corollaries of the Equations Reduced to Three Unknown Functions

Replacingk by one of the functionsf, g, hin all the results of theSection 2and exchanging each functionsf, g, hin the above equations, we then obtainPλ_{, C}λ_{types 14 equations.}

We will only illustrate the results for the cases of P_{f ghh}λ , P_{f gf h}λ in the obtained equations. The other cases are similar to these; thus their illustrations will be omitted.

Corollary 3.1. Suppose thatf, g, h:G → satisfy the inequality

_f

xygx−y−λ·hxhy≤

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ϕx or

ϕy or

minϕx, ϕy or

ε

∀x, y∈G. 3.1

Ifhfails to be bounded, then, under one of the casesh0 0orf−x −gx,hsatisfies

S.

_f

xygx−y−λ·fxhy≤ϕx, ∀x, y∈G. 3.2

ifsatisfiesSunder one of the casesf0 0orf−x −gx, and

iiin addition, ifhsatisfiesCλ_{, then}_f_and_h_{are solutions of}_C

f g:fxyfx−y

λ·fxhy.

fxygx−y−λ·fxhy≤ϕy, ∀x, y∈G. 3.3

Ifffails to be bounded, then

iiin addition, iffsatisfiesCλ_{, then}_h_and_f_{are solutions of} _Cλ

gf:hxyhx−y

λ·fxhy.

_f

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aIfhfails to be bounded, then

iiin addition, ifh satisfiesCλ_{, then}_f _and _h_{are solutions of} _C

f g: fxy

fx−y λ·fxhy.

bIfffails to be bounded, then

iiin addition, iff satisfiesCλ_{, then}_h_and _f _{are solutions of} _Cλ

gf: hxy

hx−y λ·fxhy.

_f

xygx−y−λ·fxhy≤ε, ∀x, y∈G. 3.5

aIfhfails to be bounded, then

iiin addition, ifhsatisfies Cλ_{, then}_f _and _h_{are solutions of} _C

f g: fxy

fx−y λ·fxhy.

bIfffails to be bounded, then

iiin addition, iff satisfiesCλ_{, then}_h_and _f _{are solutions of} _Cλ

gf: hxy

hx−y λ·fxhy.

Remark 3.6. As the above corollaries, we obtain the stability results of 12 × 4ϕx, ϕy,min{ϕx, ϕy}, εnumbers for 12 equations by choosingf, g, h, andλ, namely, which are the following:P_{f ghf}λ ,P_{f ggh}λ ,P_{f ghg}λ ,P_{f gf g}λ ,P_{f ggf}λ ,P_{f gf f}λ ,P_{f ggg}λ ,P_{f f gh}λ ,P_{f f gg}λ ,

C_{f g}λ ,Cλ_gf, andCλ.

3.2. Applications of the Case

λ

2 in

P

_fghkλ

Let us apply the caseλ2 inP_{f ghk}λ and allPλ_{-type equations considered in the Sections 2} and Sec3.1. Then, we obtain theP-type equations

fxygx−y2·hxky, Pf ghk

and Pλ

f ghh,Pf gf hλ ,Pf ghfλ ,Pf gghλ ,Pf ghgλ ,Pf gf gλ ,Pf ggfλ ,Pf gf fλ ,Pf gggλ , andC- and

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In papersAcz´el 22, Acz´el and Dhombres 23, Kannappan24,25, and Kim and Kannappan 13, we can find that the Wilson equation and the sine equations can be represented by the composition of a homomorphism. By applying these results, we also obtain, additionally, the explicit solutions of the considered functional equations.

_f

xygx−y−2hxky≤ϕx ∀x, y∈G. 3.6

Ifkfails to be bounded, then

ihsatisfiesSunder one of the casesh0 0orf−x −gx, andhis of the form

hx Ax or hx cEx−E∗x, 3.7

whereA: G → Cis an additive function,c ∈ ,E:G → ∗ is a homomorphism and

E∗1/Ex,

iiin addition, ifksatisfiesC, thenhandkare solutions of Cf gandh, kare given by

kx Ex E∗x

2 , hx cEx−E

∗_x dEx E∗x

2 , 3.8

wherec, d∈ ,EandE∗are as in (i).

Proof. The proof of the Corollary is enough from Theorem 2.1 except for the solution. However, they are immediate from the following:

iappealing to the solutions of S in2, page 153 see also 24, 25, the explicit shapes ofhare as stated in the statement of the theorem. This completes the proof of the Corollary,

iithe given explicit solutions are taken from24,25 page 148 see also22,23.

_f

xygx−y−2hxky≤ϕy ∀x, y∈G. 3.9

iksatisfiesSunder one of the casesh0 0orf−x −gx, andkis of the form

kx Ax or kx cEx−E∗x, 3.10

whereA: G → Cis an additive function,c ∈ ,E:G → ∗ is a homomorphism and

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iiin addition, ifhsatisfiesC, thenkandhare solutions of Cf gandk, hare given by

hx Ex E∗x

2 , kx cEx−E

∗_x dEx E∗x

2 , 3.11

_f

xygx−y−2hxky≤

⎧ ⎨ ⎩

minϕx, ϕy or

ε ∀

x, y∈G. 3.12

ihsatisfiesSunder one of the casesh0 0 orf−x −gx, andhis of the formhx Ax or hx cEx−E∗x, and whereA : G → C is an additive function,c∈ ,E:G → ∗ is a homomorphism andE∗1/Ex.

iiin addition, ifksatisfiesC, thenhandkare solutions of Cf gandh, kare given

by

kx Ex E

∗_x

2 , hx cEx−E

∗_x dEx E∗x

2 , 3.13

iksatisfiesSunder one of the casesh0 0orf−x −gx, andkis of the form

kx Ax or kx cEx−E∗x, 3.14

whereA:G → Cis an additive function,c∈ ,E:G → ∗ is a homomorphism andE∗1/Ex, and

iiin addition, ifhsatisfiesC, thenkandhare solutions of Cf gandk, hare given

by

hx Ex E

∗_x

2 , kx cEx−E

∗_x dEx E∗x

2 , 3.15

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4. Extension to the Banach Space

In all the results presented in Sections2and3, the range of functions on the Abelian group can be extended to the semisimple commutative Banach space. We will represent just for the main equationPλ

f ghk.

Theorem 4.1. LetE,·be a semisimple commutative Banach space. Assume thatf, g, h, k:G →

Esatisfy one of each inequalities

fxygx−y−λ·hxky≤ϕx, 4.1 _f

xygx−y−λ·hxky≤ϕy 4.2

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

(a) case

4.1

Suppose thatx∗◦kfails to be bounded, then

ihsatisfiesSunder one of the casesx∗◦h0 0orx∗◦f−x −x∗◦gx, and

iiin addition, ifksatisfiesCλ_{, then}_h_and_k_{are solutions of} _Cλ f g.

(b) Case

4.2

Suppose thatx∗◦hfails to be bounded, then

iiiksatisfiesSunder one of the casesx∗◦k0 0orx∗◦f−x −x∗◦gx, and

ivin addition, ifx∗◦hsatisfiesCλ_{, then}_h_and_k_{are solutions of} _Cλ f g.

Proof. For i of a, assume that 4.1 holds and arbitrarily fixes a linear multiplicative functional x∗ ∈ E∗. As is well known, we havex∗ 1; hence, for every x, y ∈ G, we have

ϕx≥fxygx−y−λ·hxky

sup

y∗₁

_y∗_f_x_y_g_x₋_y₋_λ_·_h_x_k_y

≥x∗fxyx∗gx−y−λ·x∗hxx∗ky,

4.3

which states that the superpositionsx∗◦f,x∗◦g,x∗◦h, andx∗◦kyield a solution of inequality

2.1in Theorem 2.1. Since, by assumption, the superposition x∗ ◦k with x∗ ◦h0 0 is unbounded, an appeal toTheorem 2.1shows that the two results hold.

First, the superpositionx∗◦hsolvesS, that is

x∗◦h

_x_y

2 2

−x∗◦h

_x₋_y

2 2

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Sincex∗is a linear multiplicative functional, we get

x∗

h

_x_y

2 2

−h

_x₋_y

2 2

−hxhy

0. 4.5

Hence an unrestricted choice ofx∗implies that

h

xy 2

2

−h

x−y 2

2

−hxhy∈{kerx∗:x∗∈E∗}. 4.6

Since the spaceEis semisimple,{kerx∗ :x∗∈E∗}0, which means thathsatisfies the claimed equationS.

For second casex∗◦f−x −x∗◦gx, it is enough to show thatx∗◦h0 0, which can be easily check asTheorem 2.1. Hence, the proofiofais completed.

Foriiofa, asiofa, an appeal toTheorem 2.1shows that ifx∗◦ksatisfiesCλ_, thenx∗◦handx∗◦kare solutions of the Wilson-type equation

x∗◦hxy x∗◦hx−yλx∗◦hxx∗◦ky. 4.7

This means by a linear multiplicativity ofx∗that

DC_hkλ x, y:hxyhx−y−λhxky 4.8

falls into the kernel ofx∗. As the above process, sincex∗is a linear multiplicative, we obtain

DC_hkλ x, y0, ∀x, y∈G 4.9

as claimed.

bthe case4.2also runs along the proof of case4.1.

Theorem 4.2. LetE,·be a semisimple commutative Banach space. Assume thatf, g, h, k:G →

Esatisfy one of each inequalities

_f

xygx−y−λ·hxky≤

⎧ ⎨ ⎩

minϕx, ϕy or

ε

∀x, y∈G. 4.10

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

asuppose thatx∗◦kfails to be bounded, then

ihsatisfiesSunder one of the casesx∗◦h0 0orx∗◦f−x −x∗◦gx, and

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bSuppose thatx∗◦hfails to be bounded, then

iiiksatisfiesSunder one of the casesx∗◦k0 0orx∗◦f−x −x∗◦gx, and

ivin addition, ifx∗◦hsatisfiesCλ_{, then}_h_and_k_{are solutions of} _Cλ f g.

Remark 4.3. As in theRemark 3.10, we can apply all results of the Sections2 and3 to the Banach space.

Namely, we obtain the stability results of 14 × 4ϕx, ϕy,min{ϕx, ϕy}, ε numbers for the other 14 equations except for P_{f ghk}λ . Some of them are found in papers

7,11,13–15,17,18.

Acknowledgment

This work was supported by a Kangnam University research grant in 2009.

References

1 S. M. Ulam,Problems in Modern Mathematics, Science Editions John Wiley & Sons, New York, NY, USA, 1964.

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, pp. 222–224, 1941.

3 D. G. Bourgin, “Approximately isometric and multiplicative transformations on continuous function rings,”Duke Mathematical Journal, vol. 16, pp. 385–397, 1949.

4 T. Aoki, “On the stability of the linear transformation in Banach spaces,”Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.

5 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.

6 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.

7 P. G˘avrut¸a, “On the stability of some functional equations,” inStability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles, pp. 93–98, Hadronic Press, Palm Harbor, Fla, USA, 1994.

8 J. Baker, J. Lawrence, and F. Zorzitto, “The stability of the equationfxy fxfy,”Proceedings of the American Mathematical Society, vol. 74, no. 2, pp. 242–246, 1979.

9 J. A. Baker, “The stability of the cosine equation,”Proceedings of the American Mathematical Society, vol. 80, no. 3, pp. 411–416, 1980.

10 R. Badora, “On the stability of the cosine functional equation,”Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, no. 15, pp. 5–14, 1998.

11 R. Badora and R. Ger, “On some trigonometric functional inequalities,” inFunctional Equations— Results and Advances, vol. 3, pp. 3–15, Kluwer Academic Publishers, Dodrecht, The Netherlands, 2002.

12 B. Bouikhalene, E. Elqorachi, and J. M. Rassias, “The superstability of d’Alembert’s functional equation on the Heisenberg group,”Applied Mathematics Letters, vol. 23, no. 1, pp. 105–109, 2010.

13 G. H. Kim and Pl. Kannappan, “On the stability of the generalized cosine functional equations,”

Annales Acadedmiae Paedagogicae Cracoviensis—Studia Mathematica, vol. 1, pp. 49–58, 2001.

14 G. H. Kim, “The stability of pexiderized cosine functional equations,”Korean Journal of Mathematics, vol. 16, no. 1, pp. 103–114, 2008.

15 G. H. Kim, “The stability of d’Alembert and Jensen type functional equations,”Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 237–248, 2007.

16 L. Sz´ekelyhidi, “The stability of d’Alembert-type functional equations,”Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum, vol. 44, no. 3-4, pp. 313–320, 1982.

17 G. H. Kim, “On the superstability of the Pexider type trigonometric functional equation,”Journal of Inequalities and Applications, vol. 2010, Article ID 897123, 14 pages, 2010.

18 G. H. Kim and Y. H. Lee, “The superstability of the Pexider type trigonometric functional equation,”

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19 P. W. Cholewa, “The stability of the sine equation,”Proceedings of the American Mathematical Society, vol. 88, no. 4, pp. 631–634, 1983.

20 G. H. Kim, “A stability of the generalized sine functional equations,”Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 886–894, 2007.

21 G. H. Kim, “On the stability of the generalized sine functional equations,”Acta Mathematica Sinica, vol. 25, no. 1, pp. 29–38, 2009.

22 J. Acz´el, Lectures on Functional Equations and Their Applications, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York, NY, USA, 1966.

23 J. Acz´el and J. Dhombres,Functional Equations in Several Variables, vol. 31 ofEncyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.

24 Pl. Kannappan, “The functional equationfxy fxy−1 ₂_f_x_f_y_{for groups,”}_{Proceedings of the}

American Mathematical Society, vol. 19, pp. 69–74, 1968.