A note on the Hyers-Ulam-Rassias stability of Pexider equation

A NOTE ON THE HYERS-ULAM-RASSIAS STABILITY OF PEXIDER EQUATION

YANG-HI LEE AND KIL-WOUNG JUN

ABSTRACT. In this paper we obtain the Hyers-Ulam-Rassias stabil- ity of the Pexider equation f(x+ y) = ^g(x)+h(y) in the spirit of Hyers, Ulam, Rassias and Gavruta.

1. Introduction

In 1940, S. M. Ulam [13] posed the following question concerning the stability of homomorphisms:

LetCl be a group and letC₂ be a metric group with a metricd(., .).

Given E> 0, does there exists a 6> 0 such that if a mapping h :Cl ^---?

C2satisfies the inequality d(h(xy),h(x)h(y)) < 6 forallx,y E Cl then a homomorphism H : Cl ^---? C2 exists with d(h(x),H(x)) < ^E for all x E Cl?

The case of approximately additive mappings was solved by D. H.

Hyers [2] under the assumption that Cl and C2 are Banach spaces.

Throughout this paper, we denote by X a Banach space. In 1978, Th. M. Rassias [11] gave a generalization of the Hyers' result in the following way:

Let V be a normed space and let f :V ^---? X be a mapping such that f(tx) is continuous int for each fixed x. Assume that there exist () 2:: 0 andp i= 1 such that

Ilf(x+y) - f(x) - f(y)11 ~ (}(llxIIP+IlylIP)

for all x,y ^E V(for all x,y ^E V \ {O} if^p < 0). Then there exists a unique linear mappingT : V ^---? X such that

Received September20, 1999.

1991Mathematics Subject Classification: Primary39B72, 47H15.

Key words and phrases: Hyers-Ulam-Rassias stability, Jensen functional equation.

for all x E V (for all x E V \ {O} if p < 0). However, it was showed that a similar result for the case p = 1 does not hold (see [12]). Recently, Gavruta [1] also obtained a further generalization of the Hyers-Rassias theorem (see also [3,4,7,9]).

According to Theorem 6 in [10], a mapping f :V ^---t X satisfying f(O) = 0 is a solution of the Jensen's functional equation

(x+Y)

2f - 2 - = f(x) + f(y)

if and only if it satisfies the additive Cauchy equation f(x +y) f(x) +f(y)·

Inthis paper, using the idea from the papers ofD. H. Hyers [2], Th.

M. Rassias [11] and Gavruta [1], we obtain the Hyers-Ulam-Rassias stability of the Jensen equation and the Pexider equation:

f(x+y) =g(x) +h(y).

The following result follows from Lemma 2.1 and Lemma 3.l.

THEOREM 1.1. Let V be a normed space and let f :V ---t X be a mapping. Assume that there exist 0 2:0 andp E [0,00) \ {I} such that

for allx,y EV. Then" there exists a unique additive mapping T : V ^---t X such that

IIT(x) - f(x) +j(O)11 ::; 12

~~PlllxllP

for allx E V.

We obtain.the following theorem from Corollary 2.5 and Corollary 3.4.

THEOREM 1.2. Let V be a normed space and let f,g,h : V ^---t X be mappings. Assume that there exist ()2: 0 andpE [0,00)\{1} such that

IIf(x+y) - g(x) - h(y)11 ::;O(lIxll^P +lIyllP)

for allx,y E V. Then there exists a unique additive mapping T: V ^---t X such that

IIT(x) - f(x) +f(D)11 ::; 12P_ 2111xllP4B ^+BM (4+^2P)O

IIT(x) - g(x) +g(D)II::; 12P_ 21 IjxllP^+BM

IIT(x) - hex)+h(D)11 ::;

(~~~P1IB

for allx E V where M = Ilf(D) - g(D) - h(D)11 (if1<p then M = ^D).

2. Stability in the case p < 1

We denote byC an abelian group. We also denote by c.p : C x C ---t

[D,(0) a mapping such that

00

(1) cj;(x, y) :=

L

k=O

for all x, y E C. It is easy to show that cj;(D, D) =^{2c.p(D, D)} and c.p(D, D) can be replaced by an arbitrary nonnegative real number without the loss of property (1). The following lemma for the stability of Jensen's equation is well known (see [6,8]).

LEMMA 2.1. Letf :C ---t X be a mapping such that (2)

112f

for all x, yE 2C. Then there exists a unique mapping T : C ^--tX such that

T(x+y) =T(x) +T(y) for all x,yEC,

(3) and

1 _

IIT(x) - f(x) +feD)11 ::; 2c.p(2x, D) for all x E C

T(x) = tim f(2ⁿx) for x E C.

n-too 2ⁿ

Proof Let g(x) = f(x) - f(O). Then ⁹ satisfies (2) and <p satisfies (1) and (2) . From this we can assume that f(O) =0 without the loss of generality.

Replacing x by 2ⁿ+¹x and y by 0 and dividing 2ⁿ+l on the both sides in (2), we have

Hence

n-l

112-^mf(2^mx) - 2-ⁿf(2ⁿx)II ::;

L

i=m

(5)

::; L

for n > m. From (1) and (5), we obtain the sequ~nce{2-ⁿf(2ⁿx)}

is a Cauchy sequence. Because X is a Banach space, the sequence {2-ⁿf(2ⁿx)}converges. Denote

T(x) = tim 2-ⁿf(2ⁿx)

n-+oo

for all x inG. By the definition of T and (2)

( 2X+2Y)

2T(x+y) = 2T 2 = T(2x) +T(2y) = 2(T(x)+T(y)) for all x,yE G. This proves that

T(x+y) = T(x) +T(y) for all x,yE G.

From (4), we obtain

n-l

!If(x) - ~t...^nJ(2ftx)1l :5:

L

i=O

::; L

::; 2-¹<ji(2x, 0).

Taking the limit in the above inequality, we obtain (3). Now we prove the uniqueness ofT. Let S : G - X be another additive mapping satisfying (3). Then

IIS(x) - T(x)11 ~ ¹¹S(~:x) ^{_ 1(2nx;: 1(0)11}

11

1(2nx) - 1(0) _ T(2n x) 11

+ 2ⁿ 2ⁿ

< cp(2ⁿ+1x ,0).

- 2ⁿ

Taking the limit in the above inequality as n - 00, we obtain Sex) =^T(x).

o

From Lemma 2.1, we can modify the results of [5] in the following theorem.

THEOREM 2.2. Let I, g, h :G - X be mappings such that (6) 11/(x+y) - g(x) - h(y)1I ~ rp(x,y)

for all x,yE G. Then there existsa unique additive mapping T :G - X such that

(7) IIT(x) - I(x)+1(0)11 ~ ~[cp(x,O) ^{+cp(O, x) +cp(x,x)] +M,}

(8) IIT(x) - g(x)+^g(O)11 ~ ~[<p(x, ^-x)+^cp(x,0) +cp(2x, -x)]+M,

(9) jIT(x) - hex)+^h(O)11 ~ ~[cp( -x, x) +^{cp(O, x)} + cp( -x, 2x)] +^M

where M = 11/(0) - g(O) - h(O)11 and (10)

T(x) = lim 1(2n

x) = lim g(2n

x) = lim h(2n

x) for x EG.

n--+oo 2n n--+oo 2n n--+oo 2n

Proof.

<p(0,0)

Ilf(O) - g(O) - h(O)

11

2f(X; Y) - f(x) - f(y)11

~

Ilf(X; Y) -

-

IIf(X; Y) -

-

+II f(x) -

-

Ilf(Y) -

-

~ <p(~,~) ⁺<p(~,~) ⁺<p(~,~) ⁺<p(~,~)

for allx,yE 2G. Let

<Pl(X,y)

for allx,Y E 2G. Applying Lemma 2.1, there exists a unique mapping T : G ---+X satisfying (7) and

(11)

From (6), we get

T(x)

f(2

x)

x

n-+oo 2ⁿ

112g(X; Y) - g(x) - g(y)l\

~ Ilf(~)

- g(X; Y) _h(-;X)

- g(X; Y) -

+11-f(~) ⁺

g(x)

g(y)

h(

(x+y X) (x+y Y) ( X) ( Y)

~<p -2-'-"2

+<p

+<p x'-"2 +<p Y'-"2

for allx,yE 2G. Let

<P2(X,y)

4?(X; Y,

+<P(X;Y,

cp(Y'

for all x,y E 2G. Applying Lemma 2.1 again, there exists a unique mapping T_{1 :} G --+X satisfying (8) and

(12)

From (6), we get

112h(X; Y) - h(x) - h(Y)11

~

/If(¥) -

- h(X; Y)

-

- h(X; Y)

+

11-

h(x)11

11- f(¥)

g( - ¥)

^h(Y)11

~ ~(_~,

X;Y)

¥, X;Y)

for all x,yE 2G. Let

~3

(x,

i' x; Y)

¥, x; Y)

x)

for all x,yE 2G. Similarly, there exists a unique mapping T_{2 :}G^---+ X satisfying (9) and

(13)

Replacing x by 2ⁿx and Y by 0 in (6), we get

(14)

Taking the limit in (14), we obtain

T(x)

Tl(X)

x

By the similar method we have T =T_{2 •} From (11), (12) and (13), we

obtain (10). 0

COROLLARY 2.3. Let V be a vector space. Letf,g, h : V ^---+ X be mappings such that

Ilf(x

g(x) -

for all x,yE V. Then there existsa unique additive mapping T: V ^---+

X satisfying (7), (8), (9) and (10).

The following corollary is a generalization of Theorem 1 in [4]

COROLLARY 2.4. Let V be a normed space. Let 'lj; : [0,00) ---+ R+

bea function such that

(i) 'lj;(ts) ::; 'lj;(t)'lj;(s) for all t,^S >0 and (ii) 'lj;(2)/2 < ^l.

Let f,g,h : V ---+ X bemappings such that

IIf(x+y) - g(x) - h(y)11 ::; 'lj;(llxll) +'lj;(llyll) for x=1= 0 or y=1= O.

Then there exists a unique additive mapping T : V ---+ X such that IIT(x) - f(x) +f(O)11 < 4'lj;(lIxll) +2'lj;(0) +M

- 2 - 'lj;(2)

(15) IIT(x) - g(x)+^g(O)11 < (4+'lj;(2))'ljJ(lIxll) +^'lj;(0) +M - 2 - 'lj;(2)

IIT(x) - h(x)+h(O)1I < (4+'ljJ(2))'lj;(llxll) +'lj;(0) +^M

- 2 - 'ljJ(2)

for allx E V where M = Ilf(O) - g(O) - h(O)¹¹ andT satisfies (10).

Proof. Define cp : V x V ---+ [0,00) by

(x )= { 'ljJ(llxll) +'ljJ(lIyll)) if x=1= 0 or y=1= 0 cp ,y Ilf(O) - g(O) - h(O)11 if x =0 andy =O.

Then we get

00

ep(x, y) = L 2-ncp(2nx, 2ny)

n=O 00

if x=O and y=l=O if x ⁼¹⁼ 0 and y = 0 (16)

::; L('lj;(2)/2)n('ljJ(lI x ll)+'ljJ(llyll))

n=O

'lj;(lIxll)+'lj;(llyID - 1 - 'ljJ(2)/2

for all x,y E V\{O} from (i) and (ii). By the similar method as (16), we obtain

, { 2'ljJ(0) +;~~(~]

ep(x, y) = ;~~(J} +2'ljJ(0)

2I1f(0) - g(O) - h(O)¹¹ if x=0 and y=o.

Applying Corollary 2.3, there exists a unique additive mappingT :

V ^---+ X satisfying (15) for all x =1= O. 0

COROLLARY 2.5. LetV be a normed space and let f,g,h : V -+ X bemappings. Assume that there exist 0> 0andp E [0, 1) such that

IIf(x+y) - g(x) - h(y)11 ::;O(lIxlI^P+IlyllP) for x=I: 0 ory=I: O.

Then there existsa unique additive mappingT : V -+ X such that IIT(x) - f(x) +f(O)11 ::; 2 _402PIlxll^P +OM

(4+2^P)0

IIT(x) - g(x) +g(O)II::; 2 _ 2P IIxll^P +OM IIT(x)-h(x)+h(O)II::; (~~2;;0IlxIIP+OM

for all x EV, where M = IIf(O) - g(O) - h(O)II·

Proof Define mappings fl,gl,hl : V -+ X by h(x) = !f(x), gl(X) =!g(x), h1(x) =^{!h(x) for all x E V. Define 1/J : [0,(0) -+ R+}

by'Ij;(t) = t^P and apply Corollary 2.4. 0

Itis easy to know that if2f(X!Y)- f(x)- f(y) = 0, thenf(x)- f(O) is an additive mapping. The Pexider equation satisfying the similar result is shown in the following corollary.

COROLLARY 2.6. Let V be a normed space. Let I,g,h: V ^-+ X be mappingssuch that

f (x+y) - g(x) - h(y) = 0 for x, y E V.

Then f(x) - f(O), g(x) - g(O) and h(x) - h(O) are additive mappings such that

f(x) - f(O) =g(x) - g(O) =h(x) - h(O) for all x EV.

Proof. Define<p(x,y): V x V -+ [0,00) by<p(x,y) =0 for all x,y E

V, and apply Theorem 2.2. 0

3. Stability in the case p >1

Let r/J :V x V ^--t [0,00) be a mapping such that

(17)

<Xl

4>(x,y):= L2kr/J(2-kx,2-ky) <00.

k=O

It is easy to show that 4>(0,0) =r/J(O,O) = O. Then we follow a similar approach as the above arguments and obtain the results from Lemma 3.1 to Corollary 3.4.

LEMMA 3.1. Let V be a vector space. Letf :V ^--t X be amapping such that

(18) 112f(X;y) - f(x) - f(y)11 ~ ^r/J(x,y)

forallx,y^E V. Then there exists a unique additive mappingT:V ^--t X such that

(19) IIT(x) - f(x) +f(O)¹¹ ~ 4>(x,O) for all x ^E V and

for allx E F.

Proof. We can assume that f(O) = 0 without the loss of generality.

We obtain the sequence {2ⁿf(2-ⁿx)} is a Cauchy sequence. Denote

for all x in V. Then it is easy to show that T is a unique additive

mapping satisfYing (19). .. .. . . 0

THEOREM 3.2. Let V be a vector space. Let f,^g,h : V ^--t X be mappings such that

(20) IIf(x+y) - g(x) - h(y)11 ~ r/J(x, y)

for allx, y EV. Then there existsa unique additive mapping T : V -+

X such that

IIT(x) - f(x) +f(O)11 ~ ~(~,

0)

IIT(x) - g(x)+g(O)11 _{~ ~(~, -~)} +_~(~,O) +_{~(x, -~)}

IIT(x) - h(x) +h(O)11 ~ ~( -~,~) +_~(O,~) +_~(- _~,x)

for all x EV and

(21)

{

limn-.-+oo2ⁿ[f(2-ⁿx) - f(O)]

T(x) = limn-.-+oo2ⁿ[g(2-ⁿx) - g(O)]

limn-.-+oo2ⁿ[h(2-ⁿx) - h(O)]

for all x E V.

Proof. As the result in the proof of Theorem 2.2, we get

for allx,y E V. From the above and Lemma 3.1, there existT, nand T2 such that

IIT(x) - f(x) +f(O)11 ~ ~(~,

0)

IITl(x) - g(x) +g(O)11 ~ ~(~, -~) +_~(~,

0)

Replacing x by 2-ⁿx and y by 0 in (20) and multiplying 2ⁿ on both sides of (20), we get

112ⁿ[/(2-ⁿx) - 1(0)] - 2ⁿ[g(2-ⁿx) - g(O)]11

(22) = 2ⁿll/(2-ⁿx) - g(2-ⁿx) - h(O)11

~2ⁿ</J(2-ⁿx,0).

Taking the limitin (22), we obtain

T(x) = T1(x) for all x E V.

Similarly we haveT = T2. 0

COROLLARY 3.3. Let V be anormed space. Let afunction 'l/J : [0, a) ^{- t} [0,00) satisfy

(i) 'l/J(ts) 2 'l/J(t)'l/J(s) >0 for all 0<t, s, (ii) 'l/J(2)/2 > 1 and

(iii) 'l/J(O) = O.

Let I, g, h : V ^{- t}X be mappings such that

Il/(x+y) - g(x) - h(y)11 ~ 'l/J(lIxll)+'l/J(lIyll)

for all x,yE V. Then there existsa unique additive mapping T : V ^{- t} X such that

IIT(x) - I(x)+1(0)11

~ :f~~'~';

II T (x) - g(x)+g(O)11 ~ ⁽⁴+2^P)'l/J(l\xl\)

, 'l/J(2) - 2

IIT(x) - h(x)+h(O)11 < (4+2^P)'l/J(llxll) - 'l/J(2) - 2 for allx E V and T satisfies (21).

Proof Let </>(x, y) =^?jJ(lIxll)+?jJ(lIyll)) for allx,yEV. We get

00

~(x,y) = L 2n</>(2-nx, 2- ny)

n=O

00

= L 2n(?jJ(112- nxll) +?jJ(112-ⁿyll))

n=O

00

~ L(2/?jJ(2))n(?jJ(ll xll) +?jJ(llylJ))

n=O

= ?jJ(lI xll) +?jJ(llyll) < ⁰⁰

1 - 2/?jJ(2)

from (i) , (ii) and (iii). Applying Theorem 3.2, the proof is completed.D

COROLLARY 3.4. Let V be a nonned space and let I,^g,h : V ----+X be mappings. Assume that there exist () >0 andp > 1 such that

II/(x+y) - g(x) - h(y)11 _~ ()(lIxIlP+lIyllP) for all x,yE V.

Then there exists a unique additive mappingT :V ^-+X such that 11/(x) - T(x) - 1(0)11 _~ 2P _ 211xllP4()

IIg(x) - T(x) - g(O)11 ~ (~;~P~()^IIxllP

Ilh(x) - T(x) - h(O)11 ~ (~:~P~()^IlxllP

for all x E V and T satisfies (21).

References

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Math. Soc., to appear.

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Amer. Math. Soc. 72 (1978),297-300.

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Yang-Hi Lee

~epartmentof Mathematics Education Kongju National University of Education Kongju 314-060, Korea

E-mail: [email protected] Kil-Woung Jun

Department of Mathematics Chungnam National University Taejon 305-764, Korea

E-mail: [email protected]