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Volume 2010, Article ID 985348, 16 pagesdoi:10.1155/2010/985348 Research Article Superstability of Some Pexider-Type Functional Equation Gwang Hui Kim Department of Mathematics, Kangnam

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Volume 2010, Article ID 985348, 16 pages

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,ghkim@kangnam.ac.kr

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

Academic Editor: Andrei Volodin

Copyrightq 2010 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 equation fxygx−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, Ulam1 conjectured the stability problem Next year, this problem was aﬃrmatively solved by Hyers2, which is through the following

Let X and Y be Banach spaces with norm · , respectively If f : X → Y satisfies

f

x y− fx − fy ≤ ε, ∀x,y ∈ X, 1.1

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

f x − Ax ≤ ε, ∀x ∈ X. 1.2

The above result was generalized by Bourgin3 and Aoki 4 in 1949 and 1950 In

1978 and 1982, Hyers’ result was improved by Th M Rassias5 and J M Rassias 6 which

is that the condition bounded by the constant is replaced to the condition bounded by two variables, and thereafter it was improved moreover by Gˇavrut¸a7 to the condition bounded

by the function

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In 1979, Baker et al.8 showed that if f is a function from a vector space to R satisfying

f

x y− fxfy ≤ ε, 1.3

then either f is bounded or satisfies the exponential functional equation

f

x y fxfy

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, h and k are nonzero functions, λ, ε is a nonnegative real constant, and ϕ : G → is a mapping

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

f

x y fx − y 2fxfy

was investigated by Baker9 with the following result: let ε > 0 If f : G → C satisfies

f

x y fx − y− 2fxfy ≤ ε, 1.5

then either|fx| ≤ 1 √1 2ε/2 for all x ∈ G or f is a solution of C Badora 10 in 1998, and Badora and Ger11 in 2002 under the condition |fxyfx−y−2fxfy| ≤ ε, ϕx

or ϕy, 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 equationC is investigated

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

f

x y gx − y λhxhy

P λ

f ghh

f

x y gx − y λfxhy

f gf h

f

x y gx − y λhxfy

f ghf

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x y gx − y λgxhy

f ggh

f

x y gx − y λhxgy

f ghg

f

x y gx − y λfxgy

f gf g

f

x y gx − y λgxfy

f ggf

f

x y gx − y λfxfy

f gf f

f

x y gx − y λgxgy

f ggg

f

x y fx − y λgxhy

f f gh

f

x y fx − y λgxgy

f f gg

f

x y fx − y λfxgy

f g

f

x y fx − y λgxfy

gf

f

x y fx − y λfxfy

f

x y gx − y 2hxky

, P f ghk

f

x y gx − y 2hxhy

, P f ghh

f

x y gx − y 2fxhy

, P f gf h

f

x y gx − y 2hxfy

, P f ghf

f

x y gx − y 2gxhy

, P f ggh

f

x y gx − y 2hxgy

, P f ghg

f

x y gx − y 2fxgy

, P f gf g

f

x y gx − y 2gxfy

, P f ggf

f

x y gx − y 2fxfy

, P f gf f

f

x y gx − y 2gxgy

, P f ggg

f

x y fx − y 2fxgy

f

x y fx − y 2gxfy

f

x y fx − y 2gxgy

f

x y fx − y 2gxhy

f

x y fx − y 2fx J x The hyperbolic cosine function, hyperbolic sine function, hyperbolic trigonometric function, some exponential functions, and Jensen equation satisfy the above mentioned

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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,

cosh

x y coshx − y 2 coshx coshy

,

cosh

x y− coshx − y 2 sinhx sinhy

,

sinh

x y sinhx − y 2 sinhx coshy

,

sinh

x y− sinhx − y 2 coshx sinhy

,

sinh2

x y

2

− sinh2

x − y

2

 sinhx sinhy

,

ca x y ca x −y 2ca x

2

a y a −y

 2ce x a y a −y

2 ,

e x y e x −y 2e x

2

e y e −y

 2e xcosh

y

,

n

x y cn

x − y c 2nx c : for fx nx c,

1.6

where a and c are constants.

The equationC f g is referred to as the Wilson equation In 2001, Kim and Kannappan

13 investigated the superstability related to the d’Alembert C and the Wilson functional equationsC f g, C gf under the condition bounded by constant Kim has also improved the superstability of the generalized cosine type-functional equationsC gg, and P f gf g, P f ggf

in papers14,15,17

In particular, author Kim and Lee18 investigated the superstability of S from the functional equationC gh under the condition bounded by function, that is

1 if f, g, h : G → C satisfies

f

x y fx − y− 2gxhy ≤ ϕx, 1.7

then either h is bounded or g satisfiesS;

2 if f, g, h : G → C satisfies

f

x y fx − y− 2gxhy ≤ ϕy, 1.8

then either g is bounded or h satisfiesS

In 1983, Cholewa19 investigated the superstability of the sine functional equation

f xfy

 f

x y 2

2

− f

x − y 2

2

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under the condition bounded by constant Namely, if an unbounded function f : G → C

satisfies

f xf

y

− fx y

2

fx − y

2

then it satisfiesS

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

f xgy

 f

x y 2

2

− f

x − y 2

2

g xfy

 f

x y

2

− f

x − y

2

g xhy

 f

x y 2

2

− f

x − y 2

2

S gh

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:

f

x y gx − y λ · hxky

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 C λ

f g from 29 functional

equations of the P λ , C λ , P , and C types from a selection of functions f, g, h, k in the order

of variables x y, x − y, x, y.

2 Superstability of the Sine Functional Equation from

the Equation Pλ

fghk

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

C λ and Wilson-type equation C λ f g, of the sine functional equation S from the Pexider type functional equationP f ghk λ

Theorem 2.1 Suppose that f, g, h, k : G → satisfy the inequality

f

x y gx − y− λ · hxky ≤ ϕx, ∀x,y ∈ G. 2.1

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If k fails to be bounded, then

i h satisfies S under one of the cases h0 0 or f−x −gx; and

ii In addition, if k satisfies C λ , then h and k are solutions of C λ

f g : hx y hx − y 

λh xky.

Proof Let k be unbounded solution of the inequality3.12 Then, there exists a sequence {y n}

in G such that 0 / |ky n | → ∞ as n → ∞.

i Taking y y nin the inequality3.12, dividing both sides by |λky n|, and passing

to the limit as n → ∞, we obtain

h x lim

n→ ∞

f

x y n

gx − y n

λ · ky n

Replace y by y y nand−y y nin3.12, we have

f

xy y n

gx−y y n

− λ · hxky y n

f

x−y y n

gx−−y y n

− λ · hxk−y y n ≤ 2ϕx 2.3

so that

f

x y y n

gx y− y n

λ · ky n

f

x − y y n

gx − y− y n

λ · ky n

− λ · hx · k

y y n

k−yy n

λ · ky n

≤

2ϕx

λ·k

y n

2.4

for all x, y, y n ∈ G.

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

l k

y : lim

n→ ∞

k

y y n

k−y y n

λ · ky n

where the function l k : G → satisfies the equation

h

x y hx − y λ · hxl k

y

, ∀x, y ∈ G. 2.6

Applying the case h0 0 in 2.6, it implies that h is odd Keeping this in mind, by

means of2.6, we infer the equality

h

x y2− hx − y2

 λ · hxl k

y

h

x y− hx − y

 hxh

x 2y− hx − 2y

 hxh

2y x h2y − x

 λ · hxh2y

l k x.

2.7

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Putting y x in 2.6, we get the equation

h 2x λ · hxl k x, x ∈ G. 2.8 This, in return, leads to the equation

h

x y2− hx − y2

 h2xh2y

2.9

valid for all x, y ∈ G which, in the light of the unique 2divisibility of G, states nothing else

butS

In the particular case f−x −gx, it is enough to show that h0 0 Suppose that

this is not the case

Putting x 0 in 3.12, due to h0 / 0 and f−x −gx, we obtain the inequality

k

y ≤ ϕ0

λ · |h0| , y ∈ G. 2.10

This inequality means that k is globally bounded, which is a contradiction Thus, since the claimed h0 0 holds, we know that h satisfies S

ii In the case k satisfies C λ , the limit l k states nothing else but k, so, from2.6, h and k validateC λ f g

Theorem 2.2 Suppose that f, g, h, k : G → satisfy the inequality

f

x y gx − y− λ · hxky ≤ ϕy ∀x,y ∈ G. 2.11

If h fails to be bounded, then

i k satisfies S under one of the cases k0 0 or fx −gx

ii in addition, if h satisfies C λ , then k and h are solutions of the equation of C gf λ :

k x y kx − y λhxky.

Proof i Taking x x nin the inequality2.11, dividing both sides by |λ · hx n|, and passing

to the limit as n → ∞, we obtain that

k

y

lim

n→ ∞

f

x n y gx n − y

λ · hx n , x ∈ G. 2.12

Replace x by x n x and x n −x in 2.11 divide by λ·hx n; then it gives us the existence

of the limit function

l h x : lim

n→ ∞

h x n x hx n − x

where the function l h : G → satisfies the equation

k

x y k−x y λ · l h xky

, ∀x, y ∈ G. 2.14

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Applying the case k0 0 in 2.14, it implies that k is odd.

A similar procedure to that applied after2.6 of Theorem 2.1in 2.14 allows us to

show that k satisfiesS

The case fx −gx is also the same as the reason forTheorem 2.1

ii In the case h satisfies C λ , the limit l h states nothing else but h, so, from2.14,

k and h validateC λ

f g

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

Corollary 2.3 Suppose that f, g, h, k : G → satisfy the inequality

f

x y gx − y− λ · hxky 2.15

a If k fails to be bounded, then

i h satisfies S under one of the cases h0 0 or f−x −gx, and

ii in addition, if h satisfies C λ , then h and k are solutions of C λ

f g : hx y hx −

y λhxky.

b If h fails to be bounded, then

iii k satisfies S under one of the cases k0 0 or fx −gx, and

iv in addition, if h satisfies C λ , then h and k are solutions of C λ

gf : kxykx−

y λhxky.

Corollary 2.4 Suppose that f, g, h, k : G → satisfy the inequality

f

x y gx − y− λ · hxky ≤ ε, ∀x,y ∈ G. 2.16

a If k fails to be bounded, then

ii in addition, if k satisfies C λ , then h and k are solutions of C λ f g : hxyhx−

y λhxky.

b If h fails to be bounded, then

iii k satisfies S under one of the cases k0 0 or fx −gx, and

iv in addition, if h satisfies C λ , then h and k are solutions of C λ gf : kxykx−

y λhxky.

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3 Applications in the Reduced Equations

3.1 Corollaries of the Equations Reduced to Three Unknown Functions

Replacing k by one of the functions f, g, h in all the results of theSection 2and exchanging

each functions f, g, h in the above equations, we then obtain P λ , C λtypes 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 that f, g, h : G → satisfy the inequality

f

x y gx − y− λ · hxhy ≤

⎧

⎪

ϕ

y

or

min ϕ x, ϕy

or ε

∀x, y ∈ G. 3.1

If h fails to be bounded, then, under one of the cases h 0 0 or f−x −gx, h satisfies

S .

f

x y gx − y− λ · fxhy ≤ ϕx, ∀x,y ∈ G. 3.2

If h fails to be bounded, then

i f satisfies S under one of the cases f0 0 or f−x −gx, and

ii in addition, if h satisfies C λ , then f and h are solutions of C f g : fxyfx−y 

λ · fxhy.

f

x y gx − y− λ · fxhy ≤ ϕy, ∀x,y ∈ G. 3.3

If f fails to be bounded, then

ii in addition, if f satisfies C λ , then h and f are solutions of C gf λ : hx yhx −y 

λ · fxhy.

f

x y gx − y− λ · fxhy 3.4

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a If h fails to be bounded, then

ii in addition, if h satisfies C λ , then f and h are solutions of C f g : fx y

f x − y λ · fxhy.

b If f fails to be bounded, then

ii in addition, if f satisfies C λ , then h and f are solutions of C λ

gf : hx y

h x − y λ · fxhy.

f

x y gx − y− λ · fxhy ≤ ε, ∀x,y ∈ G. 3.5

a If h fails to be bounded, then

ii in addition, if h satisfies C λ , then f and h are solutions of C f g : fx y

f x − y λ · fxhy.

b If f fails to be bounded, then

ii in addition, if f satisfies C λ , then h and f are solutions of C λ

gf : hx y

h x − y λ · fxhy.

Remark 3.6 As the above corollaries, we obtain the stability results of 12 × 4ϕx,

ϕ y, min{ϕx, ϕy}, ε numbers for 12 equations by choosing f, g, h, and λ, 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 , C λ gf, and C λ

3.2 Applications of the Case λ 2 in Pfghk λ

Let us apply the case λ 2 in P f ghk λ and all P λ-type equations considered in the Sections 2 and Sec3.1 Then, we obtain the P -type equations

f

x y gx − y 2 · hxky

, P f ghk

and P λ

f ghh, P λ

f gf h, P λ

f ghf, P λ

f ggh, P λ

f ghg, P λ

f gf g, P λ

f ggf, P λ

f gf f, P λ

f ggg , and C- and

J-typeC f g, C gf, C gg, C gh, C, and J x, which are concerned with the hyperbolic cosine, sine, exponential functions, and Jensen equation

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