Stability of isometries on restricted domains

STABILITY OF ISOMETRlES ON RESTRICTED DOMAINS

SOON-Mo JUNG AND BYUNGBAE KIM

ABSTRACT. In the present paper, the classical results of the sta- bility of isometries obtained by some authors will be generalized;

More precisely, the stability of isometries on restricted (unbounded or bounded) domains will be investigated.

o.

Let E and F be real Banach spaces. A function I : E ^{- t} F is called

anisometry ifI satisfies the equality

III(x) - I(y)11 = Ilx -

yll

for all x, YE E.

Following D. H. Hyers and S. M. Ulam [8], a function 1 :E ^{- t} F is called a cS-isometry if1satisfies the inequality

111/(x) - l(y)11 - IIx - yll I ::;

for all x,y E E. Ifin this case there exist an isometry I : E ^{- t} F and a constant k 2: 0 such that II/(x) - I(x)11 ::; k<5 for all x in E, then we say that the isometry fromE into F is stable (in the sense of Hyers and Ulam).

Hyers and Ulam proved in the same paper the stability of isome- tries between real Hilbert spaces. More precisely, they proved that if a surjective function 1 :^{E - t E,} where E is a real Hilbert space, satis- fies 1(0) = 0 as well as the inequality (*) for some <5 2: 0 and for all

Received October1, 1999.

1991 Mathematics Subject Classification: Primary 39B52; secondary 39B72.

Key words and phrases: stability, isometry.

This work was financially supported by KOSEF(1998); project no. 981-0102-007- 1.

x,Y E E, then there exists a surjective isometry I : E ^---7 E such that Ilf(x) - I(x)11 ::; 108 for any x in E.

This result of Hyers and Ulam was the first one concerning the sta- bility of isometries and was further generalized by D. G. Bourgin [lJ.

Indeed, Bourgin proved the following: Assume that E is a Banach space and that F belongs to a class of uniformly convex spaces which includes the spaces Lp(0,1) for 1 < P < 00. If a function f : E ^---7 F satis- fies f(0) = 0 as well as the inequality (*) for some 8 2: 0 and for all x,y in E, then there exists a linear isometry I : E ^---7 F such that IIf(x) - I(x)11 ::; 128 for eachx in E.

Subsequently, D. H. Hyers and S. M. Ulam [9J studied the stability problem for spaces of continuous functions: Let 81 and 82 be compact metric spaces andC(8_{i )}denote the space ofreal-valued continuous func- tions on 8i equipped with the metric topology with 11 ·1100. Ifa homeo- morphismT : C(8₁₎ ^---7C(8_{2 )} satisfies the inequality

IIIT(f) - T(g)1100 -

IIf - glloo I ::;

for some 82: 0 and for allf,9E C(81),then there exists an isometryU : C(81) ^---7 C(8'l) such that IIT(f) - U(f)lIoo ::; 218 for every f E C(81).

This result of Hyers and Ulam was significantly generalized by D. G.

Bourgin again (see [2]): Let 81 and 82 be completely regular Hausdorff spaces and letT :C(81 ) ^---7C(82 ) be a surjective function satisfying the inequality (**)for some 8 2: 0 and for allf,gEe(81). Then there exists a linear isometry U :C(81 ) ^---7 C(82 ) such that IIT(f) - U(f)lIoo ::; 108 for anyf ^E C(81 ).

The study of stability problems for b0metries on finite dimensional Banach spaces was continued byR. D. Bourgin [3J.

In 1978, P. M. Gruber [6] obtained an elegant result as follows: Let E and F be real normed spaces. Suppose that f :E ^---7 F is a surjective function and it satisfies the inequality (*) for .some8 2: 0 and for all x,y E E. Furthermore, assume that I : E ^---7 F is an isometry with f(p) = I(p) for somep.E E. If IIf(x) - I(x)11 = 0(111:11) as

IIxll

uniformly, then I is a surjective linear isometry and IIf(x) - I(x)11 ~58 for all x E E. Ifin addition f is continuous, then Ilf(x) - I(x)11 ~ 38 for all x E E.

J. Gevirtz [5J established the stability ofisometries between arbitrary Banach spaces: Given real Banach spaces E and F, let f: E ^---7 F be a surjective function satisfying the inequality (*) for some 8 2: 0 and for

all x,y E E. Then there exists a surjective isometry I : E -+ F such that Ilf(x) - I(x)11 ::; 58 for each xin E.

On the other hand, R. L. Swain [14] considered the stability of isome- tries on bounded metric spaces and proved the following result: Let M be a subset of a compact metric space (E, d) and let c > 0 be given.

Then there exists a 8>0 such that iff :M ^-+ E satisfies the inequality (*) for all x,y EM, then there exists an isometry I : M -+ E with d(f(x) , I(x)) ::;c for any x ^E M.

The stability problem of isometries on bounded subsets of Rn was studied byJ. W. Fickett [4]: Fort 2:: 0, let us define Ko(t) = K1(t) = t, K2(t) = 3v'3t, Ki(t) = 27t^{m(i) ,} where m(i) = 2^1-i _{for i 2:: 3. Let} S be a bounded subset of Rn with diameter d(S), and suppose that 3Kn(8jd(S)) ::; 1 for some 8 2:: O. Ifa function f :S -+ Rn satisfies the inequality (*) for all x,YES, then there exists an isometry I : S ^-+Rn such that If(x) - I(x)1 ::; d(S)Kn+1(8jd(S)) for each x E S.

For more detailed information on the stability of isometries and re- lated topics, one can refer to [11] and [12] (see also [10] and [13]).

In this paper, the results mentioned above will be generalized by studying the stability problems of isometries on restricted (bounded or unbounded) domains.

1. Stability on Unbounded Domains

In the following theorem, we will prove the stability of isometries (in the sense of Hyers, Ulam and Rassias) on restricted (unbounded) domains by applying the direct method. The direct method was first devised by D. H. Hyers to construct a true additive function from an approximate additive function (see [7]).

THEOREM 1. Let E be a real Hilbert space with the associated inner product (-, .). Let d 2:: 1, 82:: 0, 0::; 0 ::; 16 and 0 < p < 1 be constants with 2d - 20d^{P -} 8 2:: O. Denote byB the punctured sphere defined by B = {x E E : 0 <

Ilxll ::;

a

I

/Ix -

+

/Ix -

for all x,yE E\B, then there existsa unique linear isometry I : E ~E such that

(1) Ilf(x) - lex) - f(O)11 < 28

+

+

Il

ll(1+

- V2-J2P

for all x E E\B.

Proof. Ifwe define a function9 : E ~E byg(x) =f(x) - f(O), then we have

(2) IlIg(x) - g(y)11 -lIx -

ylll ::;

+ ()

Let x E E\ (BU {O}) be given. Puttingy= 0 and y= 2x separately in (2), we get

(3) IlIg(x)II-lIxlll ::; ⁸+ () IIxIl^P, Illg(x) - g(2x)II-lIxlll ::; 8

+ ()

Furthermore, replacingx and yby 2x and 0 in (2), respectively, yields (4)

(5)

We will now prove that there is a constant C >0 such that Ilg(x) - (1/2)g(2x)^{11 ::;} 8

+

Ilxll(1+

Itfollows from (3) that

(6) IIg(x)1l^2'::;

(11xll +

Ilxll

+

IIg(x) - g(2x)11² = Ilg(x)1I²

+

- 2(g(x),g(2x))

(7) ::;

(lIxll + ()

+

The inequalities (6), (7) and (4), together with the assumption for d, 8, () and p, yield

21Ig(x)

=21Ig(x)1I²

+

=

Ilg(x)11

+

11

+

11

::; (11xll + () IlxllP +

+

+ e

+

::; (3j2)l5²

+ 8e Ilxlll+P +

IlxllP +

Ilxll

::; 2 (l5

+ J4e +

Ilxll(l+P)/2f.

Hence, if we put C :=

J 4e +

Next, we use induction to prove

n-l n-l

(8) Ilg(x) - T ng(2nx)

II ::;

L

⁺

Ilxll(l+p)/2 L

i=O i=O

for all x E E \ (B U{a}) and n E N. The inequality (5) implies the validity of (8) for n = 1. Assume that the inequality (8) holds for some n 2: 2. Ifwe replace x by 2x in (8) and then divide by 2 the resulting inequality, we can conclude by considering (5) that (8) holds for n

+

Ifwe substitute2^mx andn-m (n >m) forx andnin(8), respectively, and then divide the resulting inequality by 2^m,we get

n-l n-l

IIT^mg(2^mx) - 2-ng(2nx)

II ::;

+

Ilxll(l+p)/2

i=m z=m

which implies th~t the sequence {2-ng(2nx)} is a Cauchy sequence for each x E E\B. Since E is assumed to be complete, we can define a function I : E ---.. E by

(9) I(x) := 2-n(x) lim 2-ng(2n+n(x)x), n--->ClO

wheren(x) =min{n E No :2nx

rt

(10) I(x) = lim 2-ng(2nx)

n--->ClO

for each x E E\B. Since if x E B then 2n(x)x E E\B, the function I : E ---.. E is well defined.

Let x,yE E\B be given. Then 2nx,2ny also belong to E\B for all n E N. Hence, by (2) we obtain

1112-ng(2nx) - 2-ng(2ny)II-llx - ylt

I::;

o

^{Ilx -}

for every n EN. Taking the limit as n ---.. 00 in the last inequality and considering (10), we see that IIE\B is an isometry.

Assume that x ^E Band y ^E E\B. Since 2ⁿ(x)x and 2ⁿ(x)y belong to E\B, it follows from (9) and (10) that

2n(x)1I1(x) - l(y)11 = 111(2n(x)x) - 1(2n(x)y)II = 2n(x)lIx - yll.

Assume that both x andy belong to B. By (9) and (10) again, we have

2n(x)+n(Y)lI1(x) - l(y)11 = 1I1(2n(x)+n(y)x) - 1(2n(x)+n(y)y)II

= 2n(x)+n(y)

IIx -

Altogether, we conclude that 1 :E ^--7 E is an isometry.

We will now prove the uniqueness of 1. According to Lemma 6.2 in [12], every inner product preserving function between Hilbert spaces is linear. We can easily see that the isometry1is inner product preserving because of. the fact 1(0) = O. Therefore, 1 is a linear isometry. Assume that there is another linear isometry1* :E ^--7 E satisfying the inequality (1) for all x inE\B instead of1. Since1and 1* are linear, we obtain

1I1(x) - 1*(x)11 = T ⁿ 111(2ⁿx) - I*(2ⁿx)11

< ^2-(n-2}8+ 21-(1-p)n/2 y=68""""+""""'-::::S"f}

IIxll(1+

v'2

- t 0 as n--7OO

for every x inE.

Inequality (S), together with (10), yields the validity of the inequality

(D. 0

In the case when^f) = 0 in the inequality given in Theorem 1, we will improve the modulus in the inequality (1) by using the method presented in the paper [SJ of Hyers and Ulam.

First, we will prove in the following lemma that if x and u are orthog- onal, then the inner product off(x) and l(u) is not so large.

LEMMA 2. Let E, B, d and 8 be those stated in Theorem 1, let a function f :E ~ E satisfy the inequality

(11) Illf(x) - f(y)lI-

Ilx -

for all x,y E E\B, and let1bethe isometry given in Theorem 1. Assume that x ^E E\ (B U{a}) and that u ^E E satisfies

lIull

I

I:::;

whenever (x, u) = O.

Proof. As was in the proof of Theorem 1, if we define a function 9 : E ^{- t} E by g(x) = f(x) - f(O), then 9 itself satisfies the inequality (11) instead off.

Assume that x and u are orthogonal. We can choose an integer n 2:

n(u) such that

(13) Y=x

+

where r = 2n - J22n

-llxl1

(14) Ily -

zl12

+

/lz11 2.

Since9satisfies the inequality (11) for allx,yE E\B, it follows from (11) and (14) that

(15)

/lg(y) - g(z)/I =((y, z)

+ /ly - z/l

+ Ilz/l

+ /lg(z)ll,

where ( and TJ are appropriate correction factors with

I((y,

s

ITJ(y, z)1 S 28.

Squaring both sides of (15), we get

2(g(y),g(z») = (g(y),g(y») - 2TJ(y,

z)llg(z)11 -

and dividing by 2ⁿ+¹ both sides of the last equality allows us to obtain (16)

(g(y),2- ng(2nu») = 2-(n+l)((g(y),g(y») - TJ(y, z)2) - TJ(y, z)IITng(2nu)ll.

The relation(13) yields

(17)

lIy - xii

Furthermore, we see by (9) that

(18) lim 2-ng(2nu) = Tn(u) lirn 2-(n-n(u))g(2n- n(u)2n(u)u) = l(u).

n-+oo n-+oo

Since1is an isometry with1(0) = 0, Ilull = 1 implies 111(u)1I =1.

Ilf(x) - I(x) - f(O)11 ::; 5£5

Inview of (16), (17) and (18), we can choose a sufficiently large integer n such that the following inequalities hold for ^an arbitraryf3 >0:

1(g(x), I(u») 1 < ^I(g(x), I(u) - _Tng(2n~») 1

+

+

< IIg(x)III1I(u) - 2-ng(2nu)¹¹

+

+

+

< ,6 +,6 + 25(1 +,6) + (0+ ,6)(1 +f3).

Hence, taking the limit as ,6 ~ 0, we see that the inequality (12)

follows. 0

Inthe following theorem, we will improve the result of Theorem 1 in the case when () =0in the relevant inequality.

THEOREM 3. Let E beareal Hilbert space with the associated inner product (., .). Let d ~ 1 and 0::;0 ::; (1j20)d be fixed, and by B denote the punctured sphere defined in Theorem 1. If a function f :E ~ E satisfies the inequality(11) for all x,yE E\B, then there existsa unique linear isometry I : E ~ E such that

(19)

for all x E E\B.

Proof. Let us define a function 9 : E ~ E by g(x) = f(x) - f(O).

Then, 9 satisfies the inequality (11) for all x,y E E\B. Furthermore, define the isometry I : E ~ E by (9). By Lemma 6.2 in [12J, I is a linear isometry.

Assume that x E E \ (BU {o}) is given and that M is the (linear) subspace orth0gonal to x. Then, I(M) is the subspace orthogonal to I(x). Letw be the projection ofg(x) on I(M). Define

t = {

~jlIWII

From Lemma 2 it follows that

for W = 0, for W 0:1

o.

(20) ¹(g(x),t) I ::; 35,

since we can choose a u EM such that (x, u) = 0,

lIull

and is coplanar withg(x) and t. By the Pythagorean theorem, we have (21) I/g(x) -1(x)1/ 2= (g(x), t}2

+

Let Zn =2nx for a non-negative integer n. By Wndenote the projec- tion ofg(zn) onI(M) and define

{

0 for ^Wn =0,

tn=

w

/llw

ll

o.

We then obtain (tn ,v) =0 and by Lemma 2 we have (22)

since there is a Un E M such that (zn,un)=0, Ilunll =1 and l(un)=_tn

(for t_n I:- 0). By (22), we can obtain

o

~

(~I~~~)tllt)) ~

since Ilg(zn)/1 ~ Ilznll - <5 > d - <5 ~ 19<5. The fact Ilznll ~ IIg(Zn)

II +

(23)

In the case of (g(x),v) ~ 0, we put n = 0 in (23) and use (21) and (20) to obtain

Ilg(x) - l(x)11 ~ (3/2)V5<5.

In the case of (g(x),v) < 0, there exists an integer m ~ 0 such that (g(zm),v) < 0 and (g(Zm+1),v) ~ 0 because (l(x),v} is positive and lex) = limn->oo2-ng(zn). Hence, (23) yields

IIg(zm+1) - g(zm)11 ~ (g(Zm+1), v) - (g(zm), v) ~ 311zmll - 3<5.

Since IIg(zm+1) - g(zm)11 ~ Ilzmll +<5, we get from the last inequality that Ilxll ~ Ilzmll ~ 2<5, and hence

Ilg(x) - l(x)11 ~ Ilg(x)11

+

+

+

The uniqueness of the linear isometry ^I can be easily proved in the

same way as in the proof of Theorem 1. 0

By a slight modification of the proof of Theorem 3 we may easily prove the following corollary. Hence, we omit the proof.

COROLLARY 4. Let E, B, d and f be tbose defined in Theorem 3.

For eacb E > 0 tbere exists a sufficiently small 8 > 0 sucb tbat ifa function f :E - t E satisfies tbe inequality(11) for all x,yE E\B, tben tbere exists a unique linear isometry I : E ^{- t} E witb

IIf(x) - I(x) - f(O)11 ~48

+

for anyx ^E E\B.

2. Stability on Bounded Domains

In the previous section, we have investigated the stability problems in connection with the unbounded domains. We will now prove the stability of isometries (in the sense of Hyers, Ulam and Rassias) on bounded domains by using the direct method.

THEOREM 5. Let E beareal Hilbert space witb tbe associated inner product (', .). Given0< d ::; 1, denote by D tbe spbere of radius d and center

at

IIxll ::;

a

Illf(x) -

\Ix - ylll

Ilx -

for some 0 ~

e

2(1+p)/2

(24) Ilf(x) - I(x)

-/(0)11

1...;e

Proof. Ifwe define a function9 : E - t E byg(x) =f(x) - f(O), then 9 satisfies

(25) Ijlg{x) -g(y)II-lIx.~

ylll

e

for allx,y ED.

Let xEDbegiven. Putting y =0 in(25) we get (26) ^IIIg(x)^{11 - IIx}

lll

Ilg(x) - g((1/2)x)11² IIg(x)\I²

+

(27) ::; ((1/2)lIxll

+

Using (26) and (27) we obtain

(lj2)llg(x) - 2g((1/2)x)11²

=

(1/2)llg(x)11

+

:::; - (1/2)

+ ((1/2)llxll +

Ilx11

+ ((1/2)llxll +

:::; 2()Ilxl/ HP . Hence, we have

(28) I/g(x) - 2g((1/2)x)

11 :::;

Ilx ll

We will now use induction on n to prove

n-l

(29) I/g(x) - 2ng(Tnx)11 :::; 2..;0Ilxl/ eHP)/2L 2-(P-l)i/2

i=O

for all x E D and for every n E N. The inequality (28) implies the validity of (29) for n = 1. Assume that the inequality (29) holds for some n 2:: 2. Ifwe replace x by (1/2)x in (29) and then multiply by 2 the resulting inequality, we may conclude by considering (28) that (29) holds for n

+

Ifwe substitute2-^mx andn - m (n >m) for x and n in (29), respec- tively, and then multiply the resulting inequality by 2^m, we obtain

n-l

112mg(Tmx) - 2ng(2-nx)11 :::; 2..;0

Ilxll(1+p)/2

i=m

which implies that the sequence {2ng(2-n_x)} is a Cauchy sequence.

Hence, we can define a function I : E ^---t E by I(x) = lim 2n+mex)g(Tn-mex)x)

n->oo

for each x E E, where we set

m(x) = min{n E No :2-ⁿx E D}.

It is not difficult to prove that I is an isometry (cf. proof of Theorem 1).

The inequality (24) immediately follows from the inequality (29) be- cause m(x) = 0 holds for each xin D.

Due to Lemma 6.2 in [12], the isometry I is linear (ef. the proof of Theorem 1 or 3). If1* :E ^---7 E is another linear isometry satisfying the inequality (24), then

III(x) - I*(x)11 2n III(2-nx) - I*(Tnx)11

2(3+p)/2

<

...;e

2(P-l)/2 - 1

---7 0 as n ^---7 00

for every x in E. 0

References

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[10] K.-W. Jun and D.-W. Park, Almost linearity of c-bi-Lipschitz maps betweenreal Banach spaces, Proc. Amer. Math. Soc. 124(1996), 217-225.

[11] Th. M. Rassias, Isometries and approximate isometries, To appear.

[12J Th, M..Rassias.and C.S.Sharma, Properties of isometries, J. Natural Geom. 3 (1993), 1-38.

[13] F. Skof, Sulle o-isometrie negli spazi normati, Rendiconti Di Mat. Ser. VII, Roma 10(1990), 853-866.

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Mathematics Section / Physics Section College of Science& Technology Hong-Ik University

Chochiwon 339-800, Korea

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