Some properties oî the set oî solutions of an operator equation in a Banach space

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PEACE MATEMATYCZNE X X (1978)

Bo g d a n Bz e p e c k i

(Poznan)

Some properties oî the set oî solutions of an operator equation in a Banach space

Let E be an arbitrary Banach space with the norm || • || and let I be a compact interval on the real line. Denote by 0(1 , E) the space of all continuous functions from an interval I to E, with the usual supremum norm HI-|||.

Let us consider the equation

(+ ) У(’) = ® + {Fy){-),

where y(-) denotes an unknown function and F: G ( I , E)->G{I, E), x

E, are known.

In this paper we characterize some topological properties of the set of solutions of the equation ( + ). In particular we give theorems on continuous dependency of the solutions of (+ ) on the element x.

We introduce the following definitions:

A continuous functions у : I->E is said to be the solution of equation (+ ) if y(t) = a?-f(jFy)(t) for every t e l .

Let x e E and A с E. By the emission of a point x we understand the set 8(x) consisting of all solutions of ( + ). An emission $(M) of a set A is said to be the set U {$ (ж) : ® e A}.

By the region of the emission of a point x, or of the set of A, we mean the set of all those points in I x E by which pass the curves generated by the functions belonging to the corresponding emission. These regions will be denoted by e(x), в (A), respectively.

Let s0 e l . Denote by eSQ(A) the intersection of the region of the emission e(A) of a set A with the hyperplane t = s0. If J is any set in I , let ej{A) = *e<7}.

Moreover,

will always denote a metric of the product I x E . We assume th a t the investigated emissions are non-empty sets. This assumption remains valid throughout all the paper and will not be re­

peated in formulations of particular theorems.

The following properties are easily established:

(a) I f F is a continuous operator on S(A) and A is compact, then the set S(A) is closed.

(b) I f the emission of a set A is compact (conditionally compact), then its region is compact (conditionally compact).

(c) I f a set e(A) is closed, then the set et(A) is closed.

(d) I f e(A) is compact (conditionaly compact), then et ( A) is compact (conditionaly compact).

(e) I f a set A is closed, $ (A) is compact and F is continuous on $(A), then the set ej(A) is closed.

0. Let (A n) be a sequence of subsets of a metric space X . We asso­

ciate with the sequence (A n) two subsets Li A n and Ls A n of X as follows:

n - > C Q П -Х Э О

x e Li A n if and only if there is a sequence (xn) such th a t xn e A n

71-+CQ

(n — 1 , 2 , . . . ) and xn->x as n-^oo;

x e Ls A n if and only if there exist indices kx < kz < ... and a sequence

t t - * o o

(xn) such th a t xn e A kn (n = 1 , 2 , . . . ) and xn->x as n->oo.

If Li A n — A — Ls A n , we write A = Lim A n.

П - У О О ? i- - > o o n - > CO

For the properties of this operations of Li, Ls and Lim we refer to [3].

Let (X , d) be an arbitrary metric space. If X x, X 2 are non-empty subsets of X , then let

D ( X x, X z) = sup{d(#, X z): x e X J and

dist(X x, X z) = m ax(D (X 1, X z), D ( X Z, X x)), where d(z, Z) = inf {d(z, x): x e Z } . The function dist defines a metric (called the Itausdorff metric) in the family of non-empty, bounded, closed subsets of X .

Let us denote by X * the system of all compact subsets of X with the exception of the set 0 . I t is well known (see, for instance, [3]) th a t:

1 . Let A m e X* for m = 0 ,1 , 2 , ... If the space X is compact, then lim d ist(An, A 0) = 0 if and only if L im X w = A 0.

0 0 n - > o o

2. If X is a compact space, then X* is also compact (with respect to the distance function dist).

Let {X, a) and ( Y , r) be metric spaces and let 9 0: X - ^ Y be a surjec­

tion. Denote by q>_x [{y}] an inverse image of y e Y by the function y.

Consider the map y: Y->2X defined as y{y) = y - x[{y}]. The following theorem holds:

0.1.

Let the mapping у satisfy the following conditions

(i) an inverse image of any compact set by у is a compact set;

(ii) <p is a closed mapping. (x)

Then for any y Qe Y and for any e > 0 there exists a number Ô > 0 such that i f y e Y, r(y, y 0) < <3, then

<p(y) c= {x e X : inf a(x, z) < e}.

zev(yQ)

P r o o f . Let y Q e Y and let e > 0. We have to prove the existence of such a number <5 > 0 th a t conditions y e Y, r(y, y Q) < <5 imply cp{y)

cz

K e1 where K 8 denotes the generalized open ball with its center in ф(у0) and radius e.

Suppose th a t th e theorem is false. Then for any i = 1, 2, ... there exist y{ e Y and xi e p ( y i) such th a t

r{yify o X » " 1,

oo

Let Z — {y^: j = 0 , 1 , 2 , ...}. We have <р_г[£] = U <P(Vj)' Because о

<p-i[Z] is a compact set and x t e (p_1 [Z], (x{) has a convergent subse­

quence (xn). Let xn-+x. We prove th a t xe<p(y0).

We have yn~>y0 and q>(xn) — yn for n — 1 , 2 , ... Prom this it follows th a t cp{x) = y 0 and x e <p(y0).

Since inf {o(xn, z): z еф(у0)} ^ o(xn, x), f o r e there exists N (e) such th a t xn e K B, where n > N ( e ) . This contradiction concludes the proof.

From the above theorem we obtain as the corollary:

0. 2. Let the assumptions from Theorem 0.1 be satisfied and let mi inverse image of every one-element set by cp be a one-element set. Then ф is continuous on Y.

Theorem 0.1 enables us to investigate emissions and regions of emis­

sion for equation ( + ) . In particular it allows us to investigate the con­

tinuity of the solution of (+ ) with respect to the element x , right-hand side of the equation, and the parameter.

1. On the basis of the Theorem 0.1 we shall now investigate emis­

sions and their regions (cf. [ 1 ]).

1.1.

Let A

E and let F be a continuous operator on 8(A).

Assume, moreover, that is compact for every compact set A x

A . Then for any x e A and for any e > 0 there exists a number ô > 0 such that i f z e A , \\z — x\\ < 6, then

8(z) c {v e G(I, E): inf |||« - y ||| < «}

y eS (x )

and

e(z) cz {($, r) e l x E : inf [\t — tx\ + ||r — га||: (tx, г*) e e(x)'\< e}.

p) I.e. for any sequence (xn) in X such that xn->x and <p(xn)-+y we have 99 (ж) = у.

P r o o f . Let (p be a function which assigns to every solution у e S(A) an element æ e l such th a t y(-) — x-\-(Fy)( •). To prove the first part of our theorem it suffices to verify for q> conditions (i), (ii) of Theorem 0.1.

Condition (i) is satisfied. We now prove th a t condition (ii) is also satisfied. Let yn e S(xn) (n = 1 , 2 , . . . ) a n d l e t \\xn - x 0\\^0 , |||y„ —y 0 lll->0 as n->oo. We prove th a t y 0 e S ( x 0).

Since F is continuous and

i

\\yo(t) -

№У

) II

(i )

о (t) Il

IK

®oll

II

( * ) - (F Vo) (t) II

we have \\y0(t) — oi>Q~-(Fy0)(t)\\ — 0 for each t e l . This completes the proof of the first p a rt of our theorem.

Fix e > 0 and x e A. There exists such a number <5 > 0 th a t from z e A and \\z — x\\ < ô it follows th a t

8(z) cz {v e C ( I , E): inf |||« — y\\\ < e}.

yeS(x)

Let q0 = (sQ, t 0) e e(z). Then there exists an y 0 e8( z) such th a t Уо Ы = ro and inf{||| 2 / 0 - 2 /ll|: y e S ( x ) } < e . Since

infflSo-^l + H t o - r J : e e(x)} ^ mî{\\y0(s0) - y ( s 0)\\: y e 8(x)}

< m f { | | | y 0 - y l l | : y e S ( x ) } < e , we have q0 eK{e(x), e), where K[e(x),e) denotes the generalized open ball with center in e(x) and radius e.

1.2.

Let the assumptions of Theorem 1.1 be satisfied and let \\xn — a?0||->0 as n-^-oo, where xm e A for m = 0,1, 2, ... Suppose that equation ( + ) has exactly one solution y 0 for x = x 0, and yn is a solution of (+ ) for x = x n. Then |||yn- y 0IIKO as n-+oo.

2. Presently we characterize the sets ej(A). First, we have the fol­

lowing

2.1.

Let the assumptions of Theorem 1.1 be satisfied and let

|Ja?n — a?0||->0 as n-+oo, where xm e A for m = 0 , 1 , 2 , . . . Suppose that equation ( -f ) has exactly one solution y 0 for x = x 0. Then

supdist (et (xn), e*(a ?0))->0

t e l

as n —

P r o o f . Fix t e l and n > 1. As et (x0) = ( t , y Q(t)), so

et (xn)) = inf {$((«, y 0(t)), (t, y(t))): y e #(®n)} < D(et(xn), et(x0)), whence

dist(e,(a?ft), et (x0)) = L>(et(xn), et (x0)).

Since the set et(xn) is compact, we can easily show th a t there exists q0n e et (xn) snch th a t

8 Щ»{е( 2 »«|(®0)): q e e t {son)) = e(g0», et {x0)).

From this it follows th a t there exists y0n e 8 (xn) snch th a t d . i s t (я?я) , в,(я?0)) = 11 Уо »(0 —Уо(ОП- Since y0ne 8 (xn),

sup dist (< 9 j(a?w), et {x 0)) = Ц|у0» - УоШ for n ^ l

t e l

and \\xn — a?0||->0 as n->oo, we infer from Corollary 1.2 th a t lim snp dist (et (xn), et (x 0)) = 0 . This completes the proof.

M-*oo t e l

We introduce the following notations:

9 = {ej(A): J is a non-empty subset in I}, X = {J : J is a compact subset in 1}.

The sets 5) and X are considered with the Hausdorff metric generated by, respectively, a product metric

of the space I x E and an'Euclidean metric.

2.2.

Let A

E and let e(A) be a compact set in I x E . Then the map J\-^ej{A) is a homeomorphism between X and 3).

P r o o f . Let us denote the map ej(A)v->J by <p. First we show th a t in fact the map q> satisfies the assumptions by Corollary 0.2.

Since e(A) is compact, e(A)* is a compact space and therefore con­

dition (i) is satisfied. To prove condition (ii) let us suppose th a t lim dist [ej (A), ea (A)) = 0 and lim d ist(Jn, J 02) = 0,

where J 01, J n are non-empty subsets in I and J 02 e X. Since the set e{A) is compact, we obtain

Lim 6 j ( A ) = 6 j ( A ) and Lim J n = J 02,

We shall prove th a t J 02 = J 01. Let t e J oz — Li J n = Li J n. Then

o o со

there exists tn e J n such th a t tn->t as n-*oo. Let y 0 e $(JL). Let us put now V = (h Уо(t)) and p n = (tn, y 0{tn)) for n = 1 , 2 , ...

Then p n e e tn(A) and p n->p as n->oo, whence p e L i e tn(A). Since Li operation is a monotonie, we have n~*°°

Li ^ U ) <= Li et (A)

We have p e e j (A), whence t e J 01. Consequently, J 02 c J 01.

To prove th a t J 0i c Ла) let t e Joi and let t'm->t as m^-oo, where 4 e J 01. Let y 0 e S ( A ) . Since 0 , {A) c so К = ( 4 , 2Л>(4))

g

Li 6j (JL) for m = 1, 2, and p m-+p as m ^ô o , where p = (t, y Q(t)).

ft-* 00

Consequently, p e Li eJn(A). From this it follows th a t there exist t'n e J n

ft- * oo

such th a t t'n-+t as n->oo, whence t e Li J n, i.e., t e J 0Z. Thus we get the

ft-*

00

inclusion J Ql c J 02 and therefore we have proved condition (ii).

On account of Corollary 0.2 the map p_I is continuous on X. I t re­

mains to verify th a t 9 ? is continuous on 5).

Let limdist(et 7 n(J.), ejQ(A)) = 0 . To prove th a t lim d ist(J№, J 0) = 0

f t - * 00 f t - * 0 0

it is enough to show th a t J 0 a Li J n and Ls J n c J 0. In fact, then

f t - * 0O f t - * 00

Li J n = U J n = J 0 = Ls J n = Ls J n

f t - * OO f t —*0 0 f t - * 0 0 f t - * O0

and therefore Lim J n — J 0, w hat implies d is t(J n, / 0)->0 as м-+оo.

f t —* 0Q

Let t

J 0 and let t'm->t as

where t'm e j 0. We have e , (^i) c ejQ{A) = Lim eJn(A) = Li eJn(A) for m > 1.

lm n-+oo n->-co

Let y 0

Then

Pm = (C , Уо(О) e L i

for m = 1 , 2 , . . . ,

f t - * 00

and p m->p as m -> 00 , where p = [t, y 0{t)). Since the lower limit of the sequence of sets is a closed set, we have th a t p

Li ej (A). This simply

ft-*oo

means th a t there exists a sequence (pn), p n

eJn(A) convergent to p.

Consequently, there exist tn

J n and yn e 8( A) such th a t p n = (tn, yn(tn)) for n = 1 , 2 , ... Hence tn-+t as n-> 00 , and therefore t

Li J n.

_ n-*oo

Now, we will prove Ls J n cz J 0. Let

Ls J n. Then there exists

f t —*0 0 f t —*00

sequence (tkn) such th a t Tcx< lcz < ... , tk n e J kn ( n ^ l ) and h n~*t as n->oo. Taking y 0 e S ( A ) and setting

P = [h VS) ) and p n = (tkn, y 0(tkn)), we obtain

P n e e Ju (-4.) л = 1 , 2 , . . . and p n->p as n-+oo.

«ft Hence, because

Lim ej k (A) = Lim ej (A) = ^ o(^L),

n -*o o n n->oo

it follows p e e Jo(A). From this it follows th a t there exist yn e 8(A), t'n e J Q and lim [t'n, y'n(t'n))

p. This implies as

whence t e J 0.

71—>00

Consequently, Ls J n <=. J 0. This completes the proof.

n —> c o

Let us denote by Ж the denumerable Cartesian product of closed intervals [ — 1 , 1 ] with product topology. Each topological space homeomorphic with Ж is called a Hilbert cube.

I t is known [4] th a t the space of all closed subsets of the interval [0, 1] (with topology induced by Hausdorff’s metric) is homeomorphic with Ж. From the above and from Theorem 2.2 follows

2.3.

Let the assumptions of Theorem 2.2 be satisfied.

Then the space $) is homeomorphic with Hilbert cube.

The restriction of Ji-^ej(A) to I (I is regarded as a subspace of X) defines another mapping from I onto (et (A): t e l } . The following theorem holds :

2.4.

Let the assumptions of Theorem 2.2 be satisfied. Then the function t\~^et(A) is a homeomorphism of the interval I into the metric space of non-empty compact subsets of I x E .

3. In the sequel we admit the following assumptions, valid through out all this section, without mentioning.

Suppose th a t A is a subset of E such th a t 8(x) Ф 0 for every æ e A, 8(A) is compact and F is continuous on 8(A).

3.1.

Let A n

A , J n

I,

Ф 0 , J n

0 (n

2 ,

Ls А п Ф 0 , Ls J n Ф 0 and let Ls A n с A 0 <= A, Ls J n cz J 0 с: I. Then

П —> 0 0 7 l~ > 0 O n - > 0 0 71—> 0 0

\im.D(ejn(An) , ejQ( A0)) = 0 .

n - > o o

P r o o f . Assume the existence of e > 0, a subsequence (J{) of sequence (Jn), and a subsequence (Af) of the sequence (An) such th a t

su p [Q(q,eJo( AQ)): о е е ^ ( А {) \ ^ е for i = l , 2 , . . .

Fix an index i. Then there exists a sequence of points (q^ ) , qÿ e е ^ ( А{) (m = l , 2 , ...) such th a t

f i - — <

for m = 1, 2, ...

Since the set Cj^Af) is compact, we obtain ->& as

q{ е е ^ ( А {), and therefore Q(q{, eJo( A0)) ^ e for each i ^ 1 .

Similarly, we obtain qt->q0 e e(A) as i->oo; this implies q(q0, Cj ( Ай))

> e, whence qQ$eJo(Af).

We now prove th a t q0 e eJ(j{A0), what gives the required contradic­

tion ending the proof. Since e ej^Af), there exist t$ e J i9 oc$ e A iy y{£ e S ( x (J}) (n = 1 , 2 , ...) such th a t

4 ? ~ 4 - e

J i

e A {

y® -+yt e 8 {A{) (as oo)

and

• П т (tf, у1]{ Щ = &.

n->oo

On the other hand, for t e l

IIyt {t) - - (Fy{) (t) |K 11\У{п - Vi 111 + l|a<? - Il + 11 № yÿ -^ У г 111 this implies у{ е 8 { х ^ . Consequently, for given i (i = 1 , 2 , . . . ) there exist ti e J î , x i e  i9 and yi e 8 { x i) such th a t & — (^> 2/i(^))- W ithout loss of generality (Z is compact) we may assume th a t f - ^ t , xt->x9 and Vi-*y (as i->oo). Hence q0 = (t,y(t)).

I t can easily be proved th a t t e j 0, x e A 0, and y e 8 (x), what means th a t q 0 e e Jo(A).

3.2. C

. Let tn e l , xn e E (n = 1 , 2 , . . . ) and let tn->t as n

and xn-+x as n->o

Then

\ i mD(etn{xn), et {x)) = 0 .

»->oo

P r o o f . Let A n == {xn} and J n = {in} for n = 1, 2, ... Then Lim A n

n - > O O

= {x}, Lim J n = {t} and the thesis follows from Theorem 3.1.

n-> OO

The following theorem partly generalizes Theorem 2.1.

3.3.

Let (An) be a sequence such as in the Theorem 3.1.

Then for any e > 0 there exists a natural number N such that L{ei(An), fy(Z0)) <C s,

where n > N and t e l .

P r o o f . Let us assume the existence of e0 > 0? a sequence (f), tt e I , and subsequence {Af) of sequence (An) such th a t D (е1{(Аг), eti{AQ)) > г 0 for i = 1, 2 , . . . Proceeding similarly as in the proof of Theorem 2.1 we conclude th a t there exist x{ e A { 1yi e 8 (x{) such th a t

Q (to, %{Ao)) > £o, where qt = (ti9 у fa )) as well as

%->t e I , x{->x e A 0, y ^ y e 8(x)

as i->o

Let q = (t,y(t)). Then q ^ q as i-+

and q e e t( A0). How we

will prove th a t g(q, p) > e0 for any e et{A0).

Let p e e t{A0). Then p = (t,y(t)j, where y e S ( A Q) and p ^ p as i->oo, where p t = (tit у(Щ e et.{A0) for i > 1 . Hence, 0 < g(p, et{{A0))

< Q(p,Pi) and therefore lim o{p, et.(A0)) = 0. Moreover,

i-y oo

« о < £ > ( & > % ( A o)) < Q { 4 i , P ) + Q { P , % ( Ao ) )

and consequently e„ < ç(q, p).

Since for any s > 0 there exists p e e et(A0) such th a t g(q, et(A9)) -f + e > g(q, p e), we have g{q, et {Af)) + s > e0, and this, because of q e et { A 0), is a contradiction and this completes the proof of the theorem.

3.4.

Let (A n) be a sequence of non-empty and, dosed sub­

sets of the E such that A n+1 c- A n (n = 1 , 2 , . . . ) and A x = A . Then for any e > 0 there exists a natural number N for which

O O

dist(e<(Mn), et ( f } A n) ) < e

71 = 1

when n > N and t e l .

OO

Pr oof . Setting A 0 = A n, by virtue of the Cantor theorem, we obtain A 0 Ф 0 . Let us note th a t

sup{e(g,e,(Mn)): q eet (An)\ < s u p { e (g, et {A0)): q e e t {An)}, whence

dist(e{(M J, et( A0)) = sup{e(g, e,(A0)): q e e t{An)}

D{et{An) , et {A0))

O O

for t e l and n = 1, 2 , ... Since Q A n = Lim A n = Ls JLn, we infer from

n = I n-> oo n->- oo

Theorem 3.3 th a t dist [et(An), e*(M0)) < e for n > N(e) and t e l .

3.5.

Let the assumptions of Theorem 3.1 be satisfied and let ej 0(Ao) c Li eJn{An). Then

dist (ej J A n), eJo(A 0))-+0 as n-^-oo.

P r o o f . Note th a t eJn{An)-+ejQ{A0) if and only if Lim ej (An) = ejQ{AQ).

__________________ n-»oo

To show th a t Lim Cj (An) = e j ^ Af ) it suffices to note th a t

ej 0 Uo) <= Li eJn{An) and Ls eJn{An) a ejQ{AQ)

— we shall only show th a t Ls eJn(An) <= eJo(A0).

16 — Roczniki PTM Prace Mat. XX.2

Let q0 e L

e

{An). Then there exist indices lix< Jcz < ... and points

П—ЮО

qk e€j. (A k ) (n = 1 , 2 , ...) such th at qk -+q0 as n-^oo. We have ïï /См Й б(Якп, eJo{AQ)) < sup{e(g, eJo( A0)): q e e JkJ A kn)}

for n = 1 , 2 , . . . On account of the Theorem 3.1 it follows th a t lim I) (ej (An), eJo(d.0)) = 0 . This implies \im g(qk , e Jo( A0)) = 0 , and consequently,

_____ n-voo

2o e«j0(^o)-

3.6.

Let {An) be a sequence of non-empty closed subsets of the E such that A x = A , A n+l <= A n (n = 1 , 2 , . . . ) and let (J n) be a sequence of non-empty closed subsets of the I such that </,n + l (n = 1 , 2 , ...). Setting J 0 = О J n and A 0 = П A n, we obtain

n = 1 n = 1

limdist(ejrn(An) , e Jo(A0)) = 0 .

«-► 00 P r o o f . Note th a t

L im Jn = J 0, Limd.n = A { and Lim eJn(An) = П eJn(An).

n->-oo n — 1

I t is obvious th a t Т о Ф 0 and А п Ф 0 . Since the set S { A X) is compact and A 0 = A 0, we obtain eJm(Am) = eJm(Am) for m = 0 , 1 , 2 , ...

We prove now th a t eJo( A0) c Li ed {An). To prove this, suppose

П-ИЭО oo

th a t q e e Jo( A0). Then there exist t e J Q and у e S ( A 0) <= П S ( A n) such

CO n = l

th a t q = (t,y(t)j, whence (An). Thus we get the inclusion fl—l

eJo( A0) cz Li eJn(An).

n-> 00

Finally, the assertion follows from Theorem 3.5.

3.7. T

. Let the assumptions of Theorem 2.1 be satisfied. Then limdist (e(a?n), e(x0)) = 0 .

n-их)

P r o o f . Let A n = {xQ, xn, xn+lJ ...}, J n = I for n = 1 , 2, ... Since

OO

= {^0}> we infer from Corollary 3.6 th a t Li me( J. n) = e{x0).

п = 1 «-►OO

Let q = (t,x) e Ls ej(xn). Then there exist indices Jcx< &2< ... and

П-+00

points qkn = (hn, y kn(tkn)) e e(ot>kn) such th a t t ^ t as n->

and ykn(tkn)->x

as n

>■

, Since xkn~+x0 as n-+oo and yk e ) for n = 1 , 2 , . . . , we

infer from Corollary 1.2 th a t ykn-^ y 0 as n->oo. This implies x = y0(t),

whence q = (t, y 0{t)).

Let y'n e S ( x n), t'n e I (n = 1 , 2 , . . . ) and let t'n->t as n->oo. Then Уп~+Уо and, since yn( O - * y 0{t), we obtain lim (t'n, y'n{t'n)) = {t, y 0(t)) what

П-^OQ

means th a t q е Ы e{xn). Thus we get the inclnsion Lse(#n) c L ie(xn).

n -+OO n-*- 00 n-VOO

This implies Li e(xn) = Ls e(xn), i.e., Lim e(xn) exists.

n-*oo n-> oo n->co

Since Lim e(xn) a Lim e(An), we have Lim e(xn) c: e(xf). I t is easily

П-+СО n-+

00

verified th a t e(x0) c Li e(#n) = Lim e(a?n). Consequently, Lim e{xn) = e{x0)

n—>-oo n->00 n-> 00

what completes the proof.

4. By means of the function (/, we can generalize the classi­

cal theorem on continuous dependence of the differential equation solu­

tion on initial condition.

Denote by:

g — the space of all compact subsets of I x E , except the set 0 , with the Hausdorff metric dist;

& — the space of all homeomorphisms of the interval I into g, with the usual metric a{fx, f 2) = sup {dist (fx(t), f z(t)): t e l J.

For each x e E we now define

Jix{t) = et {x) and H(x) = hx .

Denote by Z the subset of E consisting of these elements x for which 8 (x) Ф 0 and e(x) is compact. I t follows from Theorem 2.4 th a t hx e &

if a? e Z. I t is clear th a t the mapping H : Z->SF is one-to-one. Consequently, H ~l exists. I t can easily be seen th a t the mapping H ~l is continuous.

Indeed, let gm e S [ Z ] (m = 0 , 1 , 2 , ...) and let gn->g0- Then there exist xm e Z such th a t H ~ 1{gn) = xn, H ~ 1(g0) = a?0,

d is t(«?, (<»„), etQ{x0)) = supdist((JLrn)(£), (Hot>0)(t))

te l

= tt(gn,9o) for n = 1 , 2 , ..., and therefore lim dist ( et {xn), e* (a?0)) = 0 . Since

dist(e,0(®n), e/Q(a?o)) = dist ((<0> ®n), {t0, ®0))

= ll®n-®oll f o r w ^ l ,

we obtain \\H~1(gn) — -H’“ 1 (ÿ 0)||->0 as n ^ o o . We

prove

4.1.

Denote by P the subset of E consisting of these elements x for which equation ( + ) has exactly one solution. Suppose that S(P) is a com­

pact set and let F be continuous on S(P).

Then, the mapping H is a homeomorphism of the subspace P into ZF.

P r o o f . I t suffices to show th a t the mapping H is continuous. To prove this suppose th a t \\xn — ® 0 ||-> 0 , where xm e P for m = 0 , 1 , 2 , . . . Since et(x) = et{x) for each t e l and x e P, we infer from Theorem 2.1 th a t lim sup dist (et (xn) , et {x0)) = 0 , what completes the proof.

» - » oo t e l

E e m a r k . Let I

[t0, t0 + a], x 0 e ( —

В

e (

|a? — л?01 &}? b ^ b 0 and let / : I x [ x 0 — b, х 0 + Ь]->( — оо, oo) be con­

tinuous function. Assume th a t for every x e В there exists exactly one solution yx of the problem

У' У(*о)=я>

and let the constant b be sufficiently large so th a t

\yx {t) — x Q\ < b for t e l and x e B .

Obviously, the operator (Pyx)( •) ==/(•, yx( •)) is continuous for every x e В and the emission of the set B is a compact set. Therefore, by Theo­

rem 4.1, we deduce th a t

{t0, a?)»-» ($o + «, yx{t0 + a)) is homeomorphism (cf. [ 2 ]).

R e f e r e n c e s

[1 ] A. B ie le c k i, Differential equations and some of their generalizations [in Polish], Warszawa 1961.

[2] E. A. C o d d in g to n and N. L e v in s o n , Theory of ordinary differential equations, New York 1955.

[3] K. K u ratow sk i,'~ T o2>oZo02/, Warszawa 1966.

f4 ] R. M. S c h o r i and J. E. W e st, 21 is homeomorphic to the Hilbert cube, Bull.

Amer. Math. Soc. 78 (1972), p. 402-406.