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Topological properties of the sets of solutions for functional differential inclusions
Nonlinear Analysis, Theory, Printed
in Great
Methods & Appiicotmnr. Vol. 5, No. 12, pp. 1349-1366. 1981
0362-546X/81/121349-18 0 19X1 Pergamon
Britain.
TOPOLOGICAL FOR
SOZ.oO/O Prc*s Ltd
PROPERTIES THE SETS SOLUTIONS DIFFERENTIAL INCLUSIONS GEORGES HADDAD
CEREMADE, Universite Paris IX Dauphine, Place Marechal de lattre de Tassigny. 75016 Paris, France (Receiced
18Februar~~ 1981)
Key words: Correspondences, upper semi continuity, paracompacity, functional differential inclusions, set of solutions, acyclicity.
locally finite partition of unity,
0. INTRODUCTION THE PURPOSE of this paper is the study of the topological properties of the sets of solutions for functional differential inclusions. A first approach was given by Arronszajn [0] in the case of ordinary differential equations. Then Iasry and Robert [l] have studied the topological properties of the sets of solutions for a large class of differential inclusions including differential difference inclusions. The present paper is then a continuation of their paper for the most general class of functional differential inclusions. In the first part this paper gives an approximation theorem for upper semi continuous convex compact valued correspondences defined on a metric space. As a first immediate application of this theorem we show that some fixed point theorems in their infinite dimensional version or in their set valued version (Kakutani’s fixed point theorem) can be directly deduced from Brouwer’s fHed point theorem in the finite dimensional version. The second part of the paper more specifically uses the approximation theorem to study the existence of solutions to functional differential inclusions (with memory) and to give an approximation result for the sets of solutions. In the third part we study the topological properties of the sets of solutions. We mainly prove that the sets of solutions are acyclic, which, in particular, implies that the sets of solutions are connected. The notion of acyclicity has been used as an extension of the notion of convexity in fixed point theorems: first results were given by Eilenberg and Montgomery [2], and there has been a wide literature on various generalisations (see Lasry-Robert’s paper and references in it). Acyclicity is also used in accessibility problems in optimal control as shown in a paper by Cornet [3] in the accessibility of Pareto optima. PART
A. APPROXIMATION
OF UPPER
SEMI
CONTINUOUS
CORRESPONDENCES
We shall state here the main approximation theorem for upper semi-continuous ences with convex compact values. We first give some preliminary definitions and properties. 1349
correspond-
1350
GHADDAD 1. DEFINITIONS
AND
USEFUL
PROPERTIES
Let X and Y be two metric spaces and I : X -+ Y a given correspondence. We say that I is upper semi continuous (u.s.c.) at xg E X if we can associate to any neighborhood w of I(.~,,) in Y a neighborhood V(x,) of x0 in X such that: T(x) c co for all x E I/(x,,). If I is U.S.C.at every point x E X, it is then said to be U.S.C.on X. We recall the following properties that we shall need. If Y is compact then a correspondence I is U.S.C.on X if and only if its graph G(I) = ((.Y,y) c X x Y/y E I(x)) is a closed subset of X x I: Furthermore if I,, : X -+ Y IZE RJ. is a sequence of U.S.C.correspondences then the correspondence l- : X ---f Y defined by I(x) = n I,Jx) for all x E X, is U.S.C.too. Now we recall that by a locally finite open covering (wJiEI of X we mean an open covering of X such that for any xt X there exists a neighborhood l’(x) of Y in X such that (it I;tuir V(x) # @) is finite. 3. PRELIMINARIES For the approximation
theorem
we shall need some preliminary
results we give in this section.
LEMMA A. Let X be a metric space. To any locally finite open covering (fii)ic_I of X. we can associate a locally Lipschitz partition of unity.
D&&ion. By a locally finite partition a locally finite covering of X.
of unity in X, we mean a partition
of unity associated
to
Proc$ From Bourbaki [4] we first know that there exists an open covering (w~)~.~ of X such that Oi c Ri for any i E I. We notice that since (Ri)iE, is locally finite, the same is true for (oi)iE ,. Let us now define for any i L I the function ‘pi : X -+ Rf by vi(x) = d(x, X - oi) where d(. , X - oi) denotes the distance function to the subset X - wi = {x E X/x 4:wi). Then each cp, is obviously Lipschitz and verifies supp ‘pi = {x E X&,(x) We now define for any i E I the function yi(x) = $)
# 0) = oi c ni.
‘Pi : S + W+ such that for any x E X.
We first verify that for any x c X. 1 cpj(x) is well defined since the covering (uJiE, is locally jet finite and that c cpj(x) > 0 since there exists at least one index j E J such that x E oj which gives jE.l qj(x) = d(x, X - wi) > 0. Furthermore each Yi is continuous on X and takes its values in the interval [0, 11. Moreover we obviously verify that supp ‘Pi = supp ‘pi = Wi c CJ and that 1 Y’i(x) = 1 it1 for any x E X. To complete the proof we shall now prove that each function ‘Pi is locally Lipschitz on X.
Topological
properties
of the sets of solutions
for functional
differential
1351
inclusions
Indeed let x t x’ be given. Then there exists an open neighborhood V(x) of x which meets only a finite number of oi, i E I. we easily verify that Let us denote them oil, oi,, . . . , oi, for example. Then by construction foranyyEV(X),Yi(y)=OifI$ji ,,..., !P}. Then those ‘Pi, i $ {i,, . . . , ip> are obviously Lipschitz on V(x). We now consider Yir with i, E {i,, . . . , i,}. Since &Vi(X) by continuity
= i,,il.,,i
there exists a neighborhood
icpi(x) “9
W(s) c i’(x) of x and m > 0. M > 0 such that:
m < ,&,cPi(Y)= in!; ,,.,(, C i,qgY)
foranyyEW(X).
”
Then for any y, z E W(x) we get:
(Pi(‘)- cPi,(‘) 1
C
vi,(Y)
i_Q,,...,in
li_Il ..,.,I
=
c
ie!il,.__, i,,jvi(J‘) ic:i~C,.i,~]rpi(z)
< $
’ $
’
f
cPi,(Y)
I~i,(Y)cPi(‘) C ~c!~~,...,i~) C IS!1 I,....
i,,l
+ $
’ $,
1
ie!il.....ip}
telli
C
211Ip < ~ 4Y, m”
-
(cPi(z)cPi,(z) -
izi, < >
i,)
cPi(‘) - Vi,(‘)
Vi,(‘)
’
Cpib)
cPi(Y)l
cPi,(z)Vi(Y)j
!c(‘Mik(Y)
IVik(‘)l. IV,(‘) -
C
ic-!ii ,..._ i,,)
Vi(Y)
-
cPi(2)cPik(z)l
cPi(Y)J+ 5
jPi(z)l’ C IE{L,,...,~~}
(Pi,(Y)
-
cPi,(‘)l
2)
by Iv1 on W(x) and Lipschitz with since each ‘pi for i E {i,, . . . . ip} is obviously upper-bounded constant equal to 1. We have thus effectively proved that each Yik with i, E {iI, . . . , ip} is Lipschitz on W(x). We shall now recall a very useful result on metric spaces. PROPOSITION A. Let A’ be a metric space. To any open covering (Ua)aeA of x’, we can associate a locally finite open covering (QJiEI such that: for any iE I there exists CIE A such that Ri c U,. Moreover if X is separable then I is countable and if x’ is compact then I can be taken finite. Prooj: The property
is due to the paracompactness
of metric spaces as defined in Bourbaki
[4].
1352
G. HADDAD
Now if X is separable, let {x~}“~~ be a dense sequence in X and let I,, = {i E Zjx, E 0). Then In # 0 since (Ri)iE, is a covering of X and In is finite since (Ri)iEI is locally finite. Furthermore since {x,JnGN is dense in X and since (Qi)iGI is an open covering ofX we obviously have I = U Zn nI 1 which proves that I is countable. If X is compact the proof is then really obvious. 3.
THE
APPROXIMATION
THEOREM
We can now state the main theorem of this part. For a given normed by i!I the closed unit ball centered at the origin.
space E we shall denote
THEOREM
A. Let X be a metric space, E a normed space, K c E a convex compact subset and I : X --+K an U.S.C. correspondence with non-empty convex compact values from X into K. Then there exists a sequence In : X + K, n E N, of U.S.C. correspondences with non-empty convex compact values such that: (1) (2) (3) n3N (4)
for any x E X for any x E X for any x E X . I%h I, is of
and for any n E N ; T(x) c F,,(x), and for any n E N, I, + ,(.x) c m(x), and for any E > 0 there exists N,,. I E N such that In(x) c I(x) + i:B for all the following
type:
rn(x)=
1
Y~"'(x)C!"' for any XE X,
i-r<,, finite, locally Lipschitz
where (‘yf“‘)iEI(“) is a locally associated family of non-empty
convex compact
partition
subsets
of unity in X and (Cin))ia r,,,, is an
of K.
Remark A,. If the space E is finite dimensional then instead of assuming that I takes its values in a convex compact subset of E, we could equivalently have supposed that I is bounded. In that case we choose K = conv I(X) which is closed convex hull of the set I(X) = U I-(x). -Yc-x ProoJ We fix p > 0. Let us then cover X with the open balls (B(x, P)}~~~. We shall now give a construction of ro. By Proposition A we can associate to the preceding covering of X by open balls, a locally finite open covering (flj”))ie I,Oj such that: for any i E Ito, there exists xi”) E X with Qj”) c B(xi’), p). We then define for any i E Ito,, Ci”’ = conv -i
w h’K h IS a non-empty u r(X) I x~B(xp’. 2p)
Now thanks to Lemma A we can associate to the open covering (R~“‘)i, ,(“). We then define for all x E X:
r,(x)=
convex compact
subset of K.
a locally Lipschitz
C
partition
of unity (‘I’i(oJi,ojE ,~“)
Y?'(X) ~10).
itI
The so defined correspondence IO is obviously a non-empty convex compact valued correspondence from X into K. well defined and U.S.C. on X since (Y(i”))iEI,O,is a locally finite. locally Lipschitz partition of unity in X. In order to define I’, we do the same as before with the open covering j&x, ~/3)}~~ x.
Topological
properties
of the sets of solutions
for functional
differential
inclusions
1353
Thus we define a locally finite open covering (Qi”).,Elcii’of X associated to {B(x~“, ~/3)}~~~~,, and an associated locally Lipschitz partition of unity (‘Pi )iGlc,‘. As before we set for all i E I(,,:
-1
u
C!” = conv
XEB[X}*),
rtw 2p 31
i
And then we can define for any x E X: I-,(x) =
1
Y!i’(x) q’.
ic-I,,,
The correspondence Ii enjoys the same properties as IO. We shall now prove that for any x E X, I,(x) c I,(x). Let us fix x E X. We define: Z$’ = {i E Z,,,/x E B(x~~‘,p)} and Z;i, = {i E I&
E B(x;“, p/3)}.
Let i(,, E Z;‘o,and i(,’ E Z;” be given. Then if y E B(xi,l,‘,,2p/3) we get: d(y, x{:,‘,)< F
with cl(x, xipd,)< p
and d(x, xi:,‘,) < p/3. Thus we have d(y, xi::) < (2p/3) + @/3) + p = 2~. And then B[x!“, 2p/3] c B[x~~$2p] for all i,,’ E Z;‘o,and all iclj E ZFly This leads to Cl::, c C!” l(1),for such ind%s. Then for all i, 1,E I ;‘,) we have :
C’icl,: = & Y’l”WC:O’= Jo, c
y;Y~)c;o)
= rotx).
this being true by convexity arguments and since (Yi”‘)iS1u) is a locally finite partition of unity associated to (Qi”‘)iB1,0’and thus also to {B(x$“, P)}~.~~,,which in particular says that Y!“(x) = 0 if i # ZFo’. And then for the same reasons we get :
r,(x) = Jl,
y;“(x) c;l) = iET yp(x) cp c r,(x). IIL
In fact when x is given, only a finite number of in I,,’ and of i E I(,) has to be considered since are locally finite. (‘J’~“)i~~(,, E I(o) and Hence we have effectively proved that I,(x) c I,(x) for all x E X. We shall now prove, for example, that for any x E X, I(x) c I’,(x). Indeed from the construction of Ci”’ we have I(x) c Ci” for all i E ZFo,. Then always for convexity reasons and since (Yi”‘)iEI,O,is a locally finite partition of unity associated to (Qi”‘)ialtO,hence to {B(x!“, P))~.~,,, we get:
(“1O’)i
r(x) c i& yjO’(~)c;O’ = i~,,,~~O~(~)c~O~ = r,(x). 101
1354
G. HADDAD
Now let us define p,, = (t,” p for any n E N. Then as for To associated to p0 = p and for T, associated to pr = p/3 we can by induction build a sequence of correspondences II,, : A’ + K, n E N, each of them being u.s.c., non-empty convex compact valued and verifying properties (I), (2) and (4) of the theorem. To finish the proof we have to show that (3) is verified. Let x E X be given. Since T is u.s.c., for any 8 > 0 there exists u~,,~ > 0 such that d(y, x) < v~,, implies r-(y) c r(x) + 8. Then there obviously exists NE, x such that for n 3 NE x we have p, < q, J3. ’ Let us define as before Z& = (i E Z,,/x E B(xj”), p,)}. For the same reasons as for To and T, we can write:
rncx) =
i
C Y",")(x)~)") E
I&,
where
U r(Y). 1 ytBLr,(“). 2p,j
C!“’ = conv -1 Then for all y c B(xy), 2p,) with i E $,,
we have
cl(y, x) Q d(y ’ x!“‘) I + d(x!“’ I ’ x) if we take n 3 N_.
< 2p, + p, = 3p, < q,,,
Thus for all n 3 NE,*_we have T(y) c T(x) + ED for all y E B[xy), 2p,] with i E I;,,,, But since T(x) + EB is closed convex we get:
Cl") c rtx)+ EB for alliEl;nr And then always for convexity
reasons
r,,(x)
=
we get:
1 Y?)(X) iE Ii%)
for all n 3 NE x. Thus (3) is verified and the proof of Theorem
cl”’
c
r(x)
+ &B
A is complete.
Remark A,. From (1) and (3) and since T(x) is compact
we deduce that for all x E X we have:
f-(x) = n m(x). II=PI Remark A,. If X is a compact metric space, then instead of locally finite partitions of unity we more simply deal with finite partitions of unity, which means that the sets of indexes I,,, are finite. Remark A,. When E is finite dimensional, if T is no longer supposed to be bounded on X, we can deduce from Theorem A a local approximation result for T by restricting it on a neighborhood of the point x E .I’. we consider, so that the correspondence will be bounded on this neighborhood. 4. APPLICATIONS
TO
FIXED
POINT
THEOREMS
As a direct application of Theorem A we shall show now that fixed point theorems in their infinite dimensional version as well as Kakutani’s fixed point theorem can be easily deduced
Topologicals
properties
of the sets of solutions
for functional
differential
1355
inclusions
from Brouwer’s fixed point theorem in the finite dimensional case. Indeed let X be a convex compact subset of a normed space E and let I : X -+ X be an U.S.C. correspondence with non-empty convex compact values. The case when I is single valued is obtained by T(x) = {f(x)} for all x E X with f : X + X a given continuous mapping. We can then approximate I by a sequence of correspondences I, : X + X, n E N, which verify (i), (2), (3) and (4) of Theorem A. It is now obvious that the existence of a futed point for I is equivalent to the existence of a fixed point x,, E X for each I,. Let us then consider I, : X + X such that (4) is verified. Since X is compact we know by Remark A, that I(“) can be taken finite. For simplicity we shall take I,,, = {1,2, . . . , k}. Let us then choose for any i E I 1,2,. . . , k}, an element ui E Ci c X. We can thus consider the mapping y, : X -+ X defined by: y,(x) = i
Yi(x)ui
for all x E X.
i=l
This mapping verifies y,(x) E I”(x) for all x E X. Furthermore we can restrict y, to the set defined by K = conv{ui,..., uk} which is convex compact finite dimensional. Thus the restriction of y, to K takes its values in K and is continuous by construction. The Brouwer’s fixed point theorem insures then the existence of a fixed point for y, (or its restriction) which is obviously also a fixed point for r,. PART
B. FUNCTIONAL
DIFFERENTIAL SETS
INCLUSIONS. SOLUTIONS
OF
APPROXIMATION
OF
THE
We show here how the preceding approximation theorem can be used both to prove the existence of solutions for functional differential inclusions and to state an useful approximation result for the sets of solutions. 1. DEFINITIONS Let T > 0 be given. We define %?= U([ - r, 01, Rp) the space of continuous functions from C-r, 0] into [WP.The topology on V will be the topology of uniform convergence. For any t E R, let A(t) denote the operator which to any continuous function x(. ), defined at least on the interval [-r + t, t], associates the function A(r)x E % such that:
[A(t)x](B) = x(t + 0)
for all 0 E [-r,
01.
Let C2be a non-empty subset of Iw x %?and I : R - lQpa given correspondence with non-empty values. We call a functional differential inclusion (with memory) the differential inclusion:
(W
$ (t)
E
r(t,
AH).
A continuous function x( .) from an interval [- r + t,, t, + T,] into IWPwith T, > 0, is said to be a solution of (h/l) if: x(. ) is absolutely continuous on [t,, t, + T’], (t, A(t)x) E Sz for all t E [t,, C, + T,], (dx/dt)(t) E r(t, A(t)x) for almost all t E [to, t, + 7J.
1356
G. HADDAD
We shall say that x( .) is a solution of (n/r) with initial time t, and initial value Q(, E %’ if x(. ) is a solution of (IU) and verities A(@ = a,. For simplicity we shall also say that x( .) is a solution of (h1) through (t,. @,). Differential inclusion (M) is a very general type of differential inclusions, particularly ordinary differential inclusions (dx/dt)(t) E T(t, x(t)) are obtained when r = 0. This is also true for differential difference inclusions: dx d$f) E s(t, x(t)).
X(t - rr(f)),
.
. X(t -
T#)),
where S is a correspondence from 0’ c iw x (tWPn+linto[WPandO
EXISTENCE
OF
SOLUTIONS
The existence of solutions for functional differential inclusions has been already studied in a previous paper [5]. We just give here another proof of the existence depending directly on the approximation theorem. PROPOSITION B. Let Q be an open subset of [w x % and let T : 0 -+ Rp be an U.S.C.correspondence with non-empty convex compact values. Then for any (t,, QO) E Sz there exists a solution of (&I) through (to, 0”).
Proof Since T is U.S.C.convex compact valued on a, there exists L, > 0 and R,, > 0 such that r is bounded by a constant 1, > 0 on the set [to, t,, + L,] x B(m),, 2R,) c R. where:
B(@,,2R,) = {@~%/j/@- (Doll < 2R,j with II.11 denoting the uniform convergence norm defined on %?. Let us then denote X = [t,, t, + L,] x B(@,, 2RJ. This set will play the role of the metric space X in Theorem A. We know then the existence of a sequence T,, : _X -+ UP, n E N, of U.S.C. non-empty convex compact valued correspondences, uniformly bounded by 3, an verifying properties (1) (2) (3) and r,(t, Q) = C Yi"'(t, @) C;"),for all (t, @) E X, (4) ier~,) where (Y!“‘). is a locally Lipschitz, locally finite partition of unity on X and (Cl”‘),E, an associated fam8y of non-empty convex compact subsets of Iwp,each of them bounded by i,!“’ Thus to each T,, n E N, can be associated the functional differential inclusion: (M.) The existence of solutions through (t,, @J for each (n/I,,) is obviously deduced from (4). Indeed it suffices to choose 211”) E Cl”) for all i E I(,,) and to consider the associated single-valued function
Topological
properties
f,(t, CD)= This leads to the retarded
of the sets of solutions
for functional
differential $
inclusions
1357
for all (t, 0) E X.
1 Yin’(t, Q) ain) E I,(t, 0) ieI(,)
functional
differential
equation:
(t) = .r;,(t. .4(t)),
It is first obvious that each solution of (E,) is a solution of (h/l,). Now since f, is obviously continuous on X there exists actually a solution of (E,) through (to, @J. (See Hale [5].) Let us denote x, such a solution. From the standard proof of the existence we know that xn is in fact defined on an interval [to - r, t, + 7J where T, does not depend on J, and I,,, but depends only on QO, L,, R, and 2. Now since In andfn, n E PV,are uniformly bounded by 2, it is obvious that the solutions x,,, n E N, are I-Lipschitz on [to, h, + T,]. Then according to Ascoli’s theorem we can extract a subsequence (again denoted x,,) which converges uniformly to a continuous function x on [to - r, t, + T,] which is A-Lipschitz on [t,, t, + T,] and verifies A(t,)x = A(t,) xn = 0,. Furthermore since X is a closed subset of R we obviously have (t, A(t)x) E .Y for all t E [t,, t, + To]. From x Lipschitz we are sure that (dx/dt)(t) exists for almost all t E [t,, t, + T,] and that dx/dt E L”([t,, t, + To], IF). We shall now prove that x is a solution of (n/f). For any s, t E [t,, t, + To] we have:
x,(t) - x,(s) = converges
to
And since dx,/dt, n E N, are equibounded by 2 on [t,, t, + To], we easily deduce that the sequence dx,/dt converges weakly to dx/dt in L”([t,, t, + To], [wp)dual of L’([t,, t, + To], [WJ’). But since L”([t,, t, + T,,], [wp)c L’([t,, t, + To], [wp),it is obvious that dx,/dt converges weakly to dx/dt in L’([t,, t, + To], Rp). Then using Mazur’s convexity theorem [7] we can build a sequence of convex combinations of the following type: with al 3 0,
all but a finite number
equal
to zero and+f a; = 1, such that CL,converges strongly to dx/dt n=l in L’([t,, t, + To], [wp)as 1 -+ + co. Then we can extract a subsequence of pl (again denoted p[ for simplicity) such that p,(t) converges to (dx/dt)(t) for almost all t E [t,, r. + To] as I-+ + w. Let t E [to, r. + To] be such a point of convergence. Then using property (3) we know that for any E > 0 there exists NE E /V such that: r,,(t, A(t)x) c r(t, A(t)x) + f B
for all n 3 NE.
1358
G. HADDAD
Let us take NE. By the continuity property of operator A(t), since xn converges uniformly to x on [to - Y,t, + T,] and since rN, is u.s.c., we are sure that there exists N: 3 NC such that for all n > N:: I-& Now by property
A(t)x,) c rr, (t, A(t)x) + ; B.
(2) we know that for all n z N:: T,(t, 4x,)
All this obviously
c r,j4
4t)x,).
implies that for all II 3 N’ : &
qt,
A(+,)
t
qt,
AX)
+ EB.
And then since .f,(t, A(t)x,) E T,(t, A(t)x,), for all n 3 Nk we have: (t) = .f,(t, AX,)
2
E qt,
AX)
+ cB.
Since r(t, A(t)x) is convex, for I 3 N: we get:
ia
dx p,(t) = C a: 2 (t) t- r(t, A(t)x) ,,=l
And since r(t, A(t)x) is compact,
+ t:B.
dt
the limit as 1 --f + co we have:
by taking
g(t) E
r (t, am)
+ CB
This being true for any E > 0, we deduce that: dx dr (t) E r(t, ox)
(W and this for almost all t E [t,, t, + T,]. The proof is then complete. 3. APPROXIMATION
OF THE
SET
OF
SOLUTIONS
From what precedes and by standard arguments on differential equations (see Hale [5]), it is obvious that there exists p,, > 0,O -C p0 Q R,, depending only on a,, L,, R, and 1 such that for any initial value @ E B(@,, p,) all the solutions of (&f) and (n/f,,) are defined at least on the same interval [t, - r, t, + To] where T, depends also only on CD,,L,, R, and 1. Moreover all these solutions will verify: (t, A(t)x) E X
for all t E [fO, t, + 7J.
This is a necessity for all solution of (IVY,)since r, is only defined on X. To make it true for (n/f) it suffices to choose T, and p0 small enough. We now denote, for any (DE B(CD,,,pO) by S,o(@) the non-empty set of the restrictions to [t,, t, + T,] of the solutions of (hf) through (to, @) and by S!l_“b(@) the corresponding restrictions for each (n/r”),n E N. We can now give the following result:
Topological properties of the sets of solutions for functional differential inclusions
1359
PROPOSITIONS B,. Let I be as in Proposition B, and let (I,JnEN be the approximated sequence verifying (l), (2), (3) and (4). Then under the preceding notations, for all CDE B(Q,, p,,) we have:
9”To+ “(CD)c SF l(Q) for all n E N, (5) S,o(O) : n $A(@). (6) IlEN Remark. For the proof we shall in fact need only properties (l), (2) and (3). The utility of (4) has already appeared for the existence of solutions and will again be obvious in the topological study of the sets of solutions. Proof The proof of (5) is obvious since from (2) we know that for all (t, @)E X: I-”+ ,(r, 0) = I”@, 0). Thus all solutions of (Mn + i) is a solution of (n/r,). Now from (1) we also easily deduce that: S,&(D) c SF:(@) for all n E N and all Q,E B(Q,, p,). Conversely let x E n $$(I)) be given. nthi Then for all n E iV we have: g (t)E r,(t,A(t)x) for almost all t E [to, t, + T,].
(M,)
Let us then consider the set: 3 = 1
t E [t,, t, + T,] g(t) :’
E r,,(t, A(t)x) for all n E N 1
This set has obviously a complementary subset of measure zero in [t,, t, + T,]. Let t E 3 be given. From (3) we deduce that for any E > 0 there exists NE E N such that: T,(t, A(t)x) c r(t, A(t)x) + EB
for all n 2 NE.
This obviously implies that (dx/dt)(t) E r(t, A(t)x) + EB. Since this is true for any E > 0 and since r(t, A(t)x) is compact, we deduce that: p(t)
Eqt, ii(+)
and this for all t E 2. Thus we have x E S,“(Q) which is the desired property and completes the proof. PART
C. TOPOLOGICAL
PROPERTIES
OF THE
SETS
OF SOLUTIONS
In this part we use the approximation theorem to study the topology of the sets ST,,(Q)defined in Part B as the restrictions to [t,, t, + To] of all the solutions through (t,, @) of the given functional differential inclusions. We first give some useful definitions and theorems. 1. PRELIMINARY
DEFINITIONS
AND
THEOREMS
We define here the notion of acyclicity and give some useful results related to that notion.
I360
G. HADDAD
We denote by H* = (H”!,.% the Tech cohomological functor with coefficients in Z, defined on the category of topologrcal pairs (Y, A). We refer to Godement [8] or Spanier [9] for the usual properties of cohomology. We define by l?’ the reduced cohomology. We shall then say that a topological space Y is acyclic if for all n E N; ifn( Y) = (0). The first very important example of acyclicity is given by convex sets which are acyclic. As a topological property of acyclic sets we know that they are connected and even simply connected. We shall need the following two theorems which can be found in [8] and [9] under more general versions. THEOREM 1. Let (K,Jnc 1 be a sequence of non-empty compact acyclic subspaces space Y such that K ,,+ , c Kn for all n E N. Then K = n Kn is acyclic. ,I’F,
of a topological
THEOREM2 (Vietoris-Begle). Let Y and Z be two paracompact topological spaces andf‘ : Y + Z a given continuous surjective mapping which is supposed to be closed. Let us suppose that for any ZE Z, ,f ‘(2) is acyclic. Then the acyclicity of Y or Z implies the acyclicity of the other one. 2. THE We can now give the main theorem
MAIN
THEOREM
of this part.
C. Let R be an open subset of R x % and let T : R + Rip be an U.S.C.correspondence with non-empty convex compact values, to which we associate the functional differential inclusion: THEOREM
d; (f) i I (1, A(r)x). Then for any (t,, @,,) E R there exists p0 > 0 and To > 0 such that for any @E B((D,, p,,) the set S,(,(@), as defined precedingly, is non-empty compact acyclic. Moreover the correspondence S,(, from B(@,,,.pO) into %([t,, t, + T,], [WP)is U.S.C.and verifies that S,(,m is a relatively compact subset of Q?([t,, t, + T,], RP). Remark. The topologies on B(O,, pJ and @([t,, t, + &I, Rp) are naturally topologies of uniform convergence.
the corresponding
Proof We recall that in Part B, for (t,, QO) E Q we have built X = [to. t, + L,] x B(Qo, 2R,,) c R such that T is bounded by EL> 0 and then we have defined a sequence Tn, n E N, of correspondences from X into [WI’.verifying properties (l), (2) (3) and (4) of Theorem A. Moreover we have pointed out the existence of pO, 0 < p0 < L, and of T,, 0 < r, < L,,, depending only on 00, L,, R, and I such that there exists solutions of (h/l) and of (M,), n E N, through any (t,, @) with @ E B(@,,, p,J. All the solutions being at least defined on [t, - r, t,, + T,] and verify (t, A(t)x) E X for all z E [t,, t, + T’,‘,]. We have then defined for any Q E B(@,, p,,) the non-empty sets S,,,(Q) and SF$@), II E N, to be the sets of the restrictions to [t,, t, + T,] of the solutions through (t,, CD). Now thanks to properties (5) and (6) of Proposition B, the proof will be divided in two steps. In a first step we shall prove the remaining conclusions of Theorem C for each S!$@), n E N. And in a second step we shall deduce all these conclusions for Sri,(@).
1361
Topological properties of the sets of solutions for functional differential inclusions
Step 1. lit
us consider
:
a correspondence I- : x = [to, t, + L,]
x B(D,,, 2R,) + W such that: (4)
r(t, 0) = 1 Yi(t, @)Ci
for all
(t,
@)
E X,
iel
where (Yi)iE I is a locally finite, locally Lipschitz partition family of non-empty convex compact subsets of KY, each Since W = W([ - Y,O]W) is separable for the topology is true for X. Thus by Proposition A we have I countable. independently of I, let us define: F = ((ri)it I/vi 6 L’([t,> to + To]> “)> On the vector space F we define the following
of unity on X and ( Ci)i E, an associated of them bounded by A > 0. of uniform convergence, then the same Now since p. and To have been defined
21~= 0, except for a finite number norm:
II(ri)iEIJIF = C IIuiJI1 = ,E r”“’ iel
it1
{u = (uJiEI; ~4~E L”([t,,
/(ci(r)// dr.
111
This norm is obviously well defined. Furthermore by its constitution and since I is countable, Its topological dual space 1;* is defined by: F* =
of i E I}.
the normed
space F is separable.
t, + T,], FF’) for all i E I and sup Ijuill =, < + a}. icl
The associated
Let
us
norm
on F* will be:
now define : Bc = {u = (u~)~~,E F*/ui(t) E Ci for almost all t E [t,, t, + To] and for all i E I>.
We then have the following LEMMA C,. Bc is a convex
result: compact
subset of F* for its weak* topology.
Proof: The convexity is obvious since each Ci, i E I, is convex. Bc is also obviously bounded in F* since the sets Ci, i E I, are uniformly bounded by 2. We shall now prove that Bc is weak* closed. Indeed let u = (ui)isl E B;*the weak* closure of Bc. We must then prove that u E B, which is equivalent to u,(t) E Ci for all i E I and for almost all t E [t,, t, + To]. Let us then fix i E I and take any (vi)iGl E F such that ui = 0 for all j # i. Then for any E > 0 there exists uE = (QEl E B, such that: f”+ T”
fn+ To ui(r) vi(s) dz ISfn
u;(r) vi(r) dz < E, s 10
where u:(t) E Ci for almost all t E [to, t, + To]. Thus ui E By* with Bc, = {UE L”([t,, t, + To], RP)/ui(t)c Ci for almost all t E [to, t, + To]} which is weak* compact. Indeed Bci is a closed convex subset of L’([t,, t, + To], W’) since Ci is convex compact. Moreover since :
1362
G. HADDAD
W[t,, t, + &I, RP)=
wt,,>t, + T”1,RP),
have I?;* = BiVthe weak closure of B in L’. But by the convexity argument we have’@‘: = EC1= Bc,. And then Bc, which is bounded, weak* closed, is effectively weak* compact. Thus we can deduce that ui E Bc, and this for all i E I. Then Bc is weak* closed and thus weak* compact since bounded. Furthermore an important remark is that since F is separable, we know that Bc is metrizable for the weak* topology. Now for any u = (u~)~~,E B, we define the mapping j; from X = [t,,, t, + 7;,] x B(4),,, 2R,,) into IV’, such that: WC
,f,(t, 0) = C Yi(f, @)ui(t)
for all (t, @) t X.
ial
From the properties of (Yi)iE, and of u E B, we easily deduce that 1; is locally Lipschitz with respect to (t, @) and thus to 0, measurable with respect to r and bounded by 3, > 0 independently of u. Thus the associated functional differential equation d$r)
(4,) verifies that for any initial value on [to - r, t, + To] for the same differential equations. (See Hale We can then denote by sTO(u, 0) Thus we have defined a mapping LEMMA C?. The mapping
= .tJr, 4)x),
0 E B(
‘s,.~’ is continuous
B(@,,, p,) as well as %([t,, 1, + T,], LWP) with their corresponding ence.
with the weak* topology topologies
of uniform
and
converg-
Proqf: Since Bc is metrizable, we can work with sequences. Let un = (u:“‘)~, ,E Bc and Q’,,F B(Do, p,), n E N, be such that lim U” = u and lim Q’, = CD. ,Im+ +X ,I .+I Then the associated sequence xn = s,r,,(u”, On) verifies for all n E N : (EU”)
li”
(l)
=
1
‘yip,
A(t)x,)
u;“‘(t)
ii-l
I
for almost all t G [to, t, + 7J. with .4(0)x, = QD,.
Since xI1has been only considered on [t,, t,, + T,,], the above writing is not completely rigorous. But for conveniency we identify here x,, to the whole solution of (EJ through (t,, Q’,) which is defined on [t, - r, t, + To]. We then easily verify that each x,, is A-Lipschitz on [to, t, + 7;,] and verifies ~~(0) = (D,,(O) with II@,,(O)- @,(O)Ij d po. Thus by Ascoli’s theorem there exists a subsequence uniformly on [t,,, r, + T,] to a A-Lipschitz mapping x.
(again
denoted
x,,) which converges
Topological
properties
of the sets of solutions
for functional
differential
We shall new prove that this limit x is equal to sT,(u, O), which is equivalent
I
(EU) 2
Iwith
1363
inclusions
to prove that:
(4 = ,sl Yi(t, A(t)x) t+(t) for almost all r E [r,, t, + T,]
A(O)x = @.
As before we identify x to the very solution of (E,) through (t,, CD),which is defined on [t, - r, t, + To]. Then we first notice that the sequence A(t)x, converges uniformly to A(t)x on the interval [t,, t, + T,] since a,, converges uniformly to @ on r, 0] and since x,, converges uniformly to x on [to, 1, + T,,]. This implies that (t, A(t)x)~ [t,, t, + T,] x B(@,, 2R,) for all t E [t,, t, + To], which is the first condition x must verily to be a solution of (EJ. Furthermore since the mapping t + A(t)x, from [to, t, + T,] into 59 = %?([-r, 01, IV), is obviously continuous we deduce that the set 3” = {(t, A(t)x)/t E [t,, t, + To]) is a compact subset of X. Thus since (YJiEI is locally finite, we are sure that on an adequate neighborhood of X in X, at most a finite number of ‘Pi, i E I, are non equal to zero. It can now easily be proven, since the functions ‘Pi, i E I, are locally Lipschitz, since only a finite number of them is to be considered on a neighborhood of X compact and since (t, A(t)x,) converges uniformly to (t, A(t)x) on [t,, t, + To], that Yi(t, A(t)x,) converges uniformly to Yi(t, A(t)x) on [t,, t, + T’], independently of i E I. But for all n E N we have:
[-
x,(t) - x&J
=
=
’ c s f0 iel zI
Yi(z, A(z)x,) L@‘(Z)dr
[
Yi(‘,
‘(‘)x,)‘~‘(‘)
dz>
fo
which is verified for all t E [t,, t, + To] and with the same i E I, in finite number, to be considered when n is large enough. Now, since the sequences ~4:) are weak* convergent to ui, being uniformly bounded by 2 and since the continuous functions Yi(t, A(t)x,) converge uniformly to Yi(t, A(t)x) on [t,, t, + To], by taking the limit as n -+ + cc we have for all t E [to, t, + To]: ’ Yi(z, A(z)x) ~~(7)dz s to
x(t) - x(q)) = c isI
=
tt s
,sI
Yi(‘,
‘(7)x)
‘i(z)
dr.
0’
And thus x = sTO(u, CD),which is the desired equality. LEMMA C,. For any @ E B(@‘,, p,) we have S,,,(Q) = sTo(Bc, a,) = {~~,(a, @)/u E B,). Proof. Let us first verify that sTO(u, @) E STO(@)for any u E B,. Indeed by definition the mapping x = sTO(u, CD)is the restriction to [t,, t, + To] of the solution through (t,, a) of the functional differential equation:
1364
G. HADDAD
g(t)
= ,fi(t, A(t)x)
f,(rj ‘(r)x)
for almost all t E [t,, t, + T,].
= iFI yi(r? A(t)x)ui(r) E C yi(t? A(t)x)
‘i
isI
=
r(t,
A(t)x)
for almost all t E [t,, t, + To].
And then we do effectively have sT,(u, @) E ST,(@). Conversely let x E ST,,(@) be given. By the very definition of a solution to a functional differential inclusion and since f is bounded, we know that x is Lipschitz in [to, t, + To] and thus the mapping b(t) = (dx/dt)(t) defined almost everywhere on [lo, t,, + T,] is a borelian mapping. Furthermore since the set X = {(t, A(t)x)/t E [t,, t, + To]} is compact, we know that at most a finite number of ‘Pi is non-zero on X. Let us denote them Y’, , Y2,. , Yn. We can then define the mapping p from [t,, t, + To] x (W)” into IWPsuch that:
Iw, a,,
.
.
3
an) = i
Yi(t, A(t)x) ui = b(t),
i=l
forany(t,a,,..., a”) E [r,, t, This mapping is obviously Let us now define A : [to, E C, x C, . . x C,,//?(t, ul,.
+ T,] x (RP)“. a borelian mapping. t, + T,] -+ C, x C, x . . . x C,, such that A(t) = {u = (~1~.. . , an) . . , a,) = O}. Since x E STO(Q) we have: b(t) = %(t)t
i
Yi(t, A(t)x) ci
i-l
for almost all t E [to, t, + To]. Thus A(t) # 0 for almost all t E [to, t, + To]. At last the graph of A is equal to ,!-‘(0) which is a borelian set. Thus from Von Neumann’s selection theorem [lo] there exists u : [t,, t, + 7J + C‘, x C2 x . . x C,, which is measurable and such that u(t) E A(t) for almost all t E [to, t, + T,,]. Then if we denote u(t) = (ul(t), . . . , u,,(t)) we get: b(t) = ~(1)
= ~ yi(t, A(t)x)ui(t) i=l
for almost all t E [t,, t, + To]. And thus obviously x = sTO(u, @) where u E Bc is defined by the ui, i = 1 . . . n, we have obtained. The other indexes having no importance since Yi(t, A(t)x) = 0 for such indexes. We can for example choose any constant mapping on the corresponding Ci. The proof of Lemma C, is then complete. LEMMA
C,. For any @ E B(Q,, p,), S,o(@) is compact
Proof: Let us consider
the mapping
acyclic.
sTo(. , @,)from Bc into S,o(@) which to any u E Bc associates
ST”64 @) E S,O(@). Then from the preceding lemmas we know that this mapping is continuous and surjective. Since Bc is compact, then S,“(Q) is compact. From the convexity of the sets Ci, i E I, we easily verify that the inverse image by S,“(. , 0) of any x E SrO(Q) is convex thus acyclic. At last sr,,(. , 0) is obviously a closed mapping since it is continuous and since Bc is compact.
Topological properties of the sets of solutions for functional differential inclusions
1365
Then all the hypotheses of Theorem 2 are satisfied. And since B, is convex thus acyclic, we know that SrO(@)is also acyclic. C,. The correspondence ST0 from B(Qo, po) into ‘%?([t,,,t, + T,], Rp) is U.S.C.and ST<,[@&-$I 1sa relatively compact subset of %?([t,, t, + T,], Rp).
LEMMA
Praof Let us first consider STo[B(@,,, p,)]. S ince the correspondence I is bounded by 1 on X, we have already noticed that all the solutions of the associated functional differential inclusion (M) with initial value Q E B(Q,,, po), are i-Lipschitz on [to, t, + T,]. Furthermore for any x E S,O[m] there exists Q E B(Qo, p,,) such that x E S,“(Q), which implies that x(0) = m(O)and so that I/x(O) - @JO)// < pO. . is relatively compact. Thus by Ascoli’s theorem SrO[m] . 0 bviously Thus ST0 takes its values in a compact subset of V([t,,, t, + T,], Rp). To prove that ST0 is U.S.C. it is now sufficient to show that its graph is closed. lim an = CDfor the corresponding Let xn E STo(@J, n E N, such that “$I~ xn = x and n-+a topologies of uniform convergence. From the surjectivity of the mapping sTO,we know the existence of u,, E Bc, n E N, such that xn = srO(u., QJ for all n E N. Since B, is compact metrizable for the weak* topology, there exists then a subsequence (again denoted ~4”)such that u, converges to u as n -+ + co. Since the mapping srO is continuous, we have x = s(u, 0) and x E SrO(Q). The proof of Step 1 is now complete. Step 2. We can now return to I in the general case. From Proposition any 0 E B(@,, p,) we have: 9”To+ l’(m) c SF:(Q) for all n E N,
S,“(Q) = f-j sy(@).
B, we know that for (5)
(6)
neN
By Step 1 we know that for any Q,E B(@,, po), and for any n E N, SF:(@) is non-empty compact acyclic. Then all the hypotheses of Theorem 1 are satisfied and thus SrO(@)is compact acyclic for any 0 E B(@,,, po). Furthermore ST0 is an U.S.C.correspondence since it is the intersection of the sequence of U.S.C.correspondences SF:, n E N. At last SrO[m] is relatively compact in %?([t,, t, + T,], Rp) for similar reasons as those given in the proof of Lemma C,. The proof of Theorem C is now complete. Remark. In fact we have proved the stronger property that for any n E N, the set S$$Q) is the image of an associated compact convex set (which is denoted by B, in the proof) by a contmuous mapping which verifies that the inverse image of any point is convex. We have then proved that S,“(a) is the intersection of such a decreasing sequence.
As an immediate consequence of Theorem C we have the single-valued version for functional differential equations. COROLLARY
C, . Let R be an open subset of IR x 5%and let S : R -+ Ripbe a continuous mapping.
1366
G. HADDAD
Then we have exactly equation:
the same conclusions
as in Theorem
C for the functional
differential
dx
ProojY We obviously
consider
the associated
r(t, CD)= { ,f(t, 0)) This correspondence
is obviously
univalued corespondance
defined by :
for all (t, 0) E R.
U.S.C. convex compact
valued.
COROLLARY C,. Let all the hypoteses of Theorem C be satisfied. Then for any CDE B(@,, po) the set R,o(O) = {x(t, + 7Jx E SrO(@)} is compact and connected. Moreover the correspondence R,(, from B(@,,, L),,) into I@’ is U.S.C. and verifies that
RrI,[mp,)l
ia relatively
compact
in UP.
Proqf. Let II : %?([t,, t, + T,], EP) + Rp be the mapping such that n(x) = x(t, + TO) for any x E %([t,, t, + TO], KY). Then R,o(@) = lIl[S,,(@)] f or all CDE B(BO, pJ. Since n is continuous and since S,“(Q) is compact and acyclic thus connected, we easily deduce that R,,,(Q) is compact connected. is obviously U.S.C. and R,,l[B(@,,, p,)] = Furthermore the correspondence RTI, = n._S, is relatively compact. W,“(B(cD,,l is relatively compact since S,.,I[e)] REFERENCES ARRONSZAJN N., Le correspondant
de I‘umcitk dans la thCorle des iquations dif(erentlelles. Amls Math. 43, No. 4 (Octobre 1442). . - . cahier de Math de la dicision No. 761 I 1. LASRYJ. M. & ROBERT R.. Analvse non 1inCaire multivoaue. Am. J. Murh 58, 214 222 2. EILENBERG S. & MONTC~MERV c., Fixed point theorem for multivalued transformations, (1946). 3. CORNET, IS., An abstract theorem for planning procedures. In C’ongrrs d’Ancr!,sr Conrcxe. Murat le Qualre (1976). Lecture Notes in Economics and Mathematical Systems. 4. BOURBAKI N., TopologirGhhdr. Hermann. 5. HALE J., Theory of Functional DiJ$rential Equations. Springer Verlag. Berlin. 6 HAIX)AII G.. Monotone viable trnlcctories for functional diftel-cnllal inclusion\. ./. t/i//. Eqns (IO appcarl I. YOSIDA K., Functional Analysis, pp. 120-123. Springer Verlag, Berhn. R.. Toooloaie alqibrique et thPorie des faisceaux. Hermann, Paris (1964). 8. GODEMENT New York (1966). 9. SPANIER E., AlgedraicUTo~ojogy.McGraw.Hill, IO. voh NI-UMANNJ.. On Rings of operators. reduction theory. Aw~/.c Moth. 50. 401-485 (lY4Y) 0
toDologiaue
in Great
Methods & Appiicotmnr. Vol. 5, No. 12, pp. 1349-1366. 1981
0362-546X/81/121349-18 0 19X1 Pergamon
Britain.
TOPOLOGICAL FOR
SOZ.oO/O Prc*s Ltd
PROPERTIES THE SETS SOLUTIONS DIFFERENTIAL INCLUSIONS GEORGES HADDAD
CEREMADE, Universite Paris IX Dauphine, Place Marechal de lattre de Tassigny. 75016 Paris, France (Receiced
18Februar~~ 1981)
Key words: Correspondences, upper semi continuity, paracompacity, functional differential inclusions, set of solutions, acyclicity.
locally finite partition of unity,
0. INTRODUCTION THE PURPOSE of this paper is the study of the topological properties of the sets of solutions for functional differential inclusions. A first approach was given by Arronszajn [0] in the case of ordinary differential equations. Then Iasry and Robert [l] have studied the topological properties of the sets of solutions for a large class of differential inclusions including differential difference inclusions. The present paper is then a continuation of their paper for the most general class of functional differential inclusions. In the first part this paper gives an approximation theorem for upper semi continuous convex compact valued correspondences defined on a metric space. As a first immediate application of this theorem we show that some fixed point theorems in their infinite dimensional version or in their set valued version (Kakutani’s fixed point theorem) can be directly deduced from Brouwer’s fHed point theorem in the finite dimensional version. The second part of the paper more specifically uses the approximation theorem to study the existence of solutions to functional differential inclusions (with memory) and to give an approximation result for the sets of solutions. In the third part we study the topological properties of the sets of solutions. We mainly prove that the sets of solutions are acyclic, which, in particular, implies that the sets of solutions are connected. The notion of acyclicity has been used as an extension of the notion of convexity in fixed point theorems: first results were given by Eilenberg and Montgomery [2], and there has been a wide literature on various generalisations (see Lasry-Robert’s paper and references in it). Acyclicity is also used in accessibility problems in optimal control as shown in a paper by Cornet [3] in the accessibility of Pareto optima. PART
A. APPROXIMATION
OF UPPER
SEMI
CONTINUOUS
CORRESPONDENCES
We shall state here the main approximation theorem for upper semi-continuous ences with convex compact values. We first give some preliminary definitions and properties. 1349
correspond-
1350
GHADDAD 1. DEFINITIONS
AND
USEFUL
PROPERTIES
Let X and Y be two metric spaces and I : X -+ Y a given correspondence. We say that I is upper semi continuous (u.s.c.) at xg E X if we can associate to any neighborhood w of I(.~,,) in Y a neighborhood V(x,) of x0 in X such that: T(x) c co for all x E I/(x,,). If I is U.S.C.at every point x E X, it is then said to be U.S.C.on X. We recall the following properties that we shall need. If Y is compact then a correspondence I is U.S.C.on X if and only if its graph G(I) = ((.Y,y) c X x Y/y E I(x)) is a closed subset of X x I: Furthermore if I,, : X -+ Y IZE RJ. is a sequence of U.S.C.correspondences then the correspondence l- : X ---f Y defined by I(x) = n I,Jx) for all x E X, is U.S.C.too. Now we recall that by a locally finite open covering (wJiEI of X we mean an open covering of X such that for any xt X there exists a neighborhood l’(x) of Y in X such that (it I;tuir V(x) # @) is finite. 3. PRELIMINARIES For the approximation
theorem
we shall need some preliminary
results we give in this section.
LEMMA A. Let X be a metric space. To any locally finite open covering (fii)ic_I of X. we can associate a locally Lipschitz partition of unity.
D&&ion. By a locally finite partition a locally finite covering of X.
of unity in X, we mean a partition
of unity associated
to
Proc$ From Bourbaki [4] we first know that there exists an open covering (w~)~.~ of X such that Oi c Ri for any i E I. We notice that since (Ri)iE, is locally finite, the same is true for (oi)iE ,. Let us now define for any i L I the function ‘pi : X -+ Rf by vi(x) = d(x, X - oi) where d(. , X - oi) denotes the distance function to the subset X - wi = {x E X/x 4:wi). Then each cp, is obviously Lipschitz and verifies supp ‘pi = {x E X&,(x) We now define for any i E I the function yi(x) = $)
# 0) = oi c ni.
‘Pi : S + W+ such that for any x E X.
We first verify that for any x c X. 1 cpj(x) is well defined since the covering (uJiE, is locally jet finite and that c cpj(x) > 0 since there exists at least one index j E J such that x E oj which gives jE.l qj(x) = d(x, X - wi) > 0. Furthermore each Yi is continuous on X and takes its values in the interval [0, 11. Moreover we obviously verify that supp ‘Pi = supp ‘pi = Wi c CJ and that 1 Y’i(x) = 1 it1 for any x E X. To complete the proof we shall now prove that each function ‘Pi is locally Lipschitz on X.
Topological
properties
of the sets of solutions
for functional
differential
1351
inclusions
Indeed let x t x’ be given. Then there exists an open neighborhood V(x) of x which meets only a finite number of oi, i E I. we easily verify that Let us denote them oil, oi,, . . . , oi, for example. Then by construction foranyyEV(X),Yi(y)=OifI$ji ,,..., !P}. Then those ‘Pi, i $ {i,, . . . , ip> are obviously Lipschitz on V(x). We now consider Yir with i, E {i,, . . . , i,}. Since &Vi(X) by continuity
= i,,il.,,i
there exists a neighborhood
icpi(x) “9
W(s) c i’(x) of x and m > 0. M > 0 such that:
m < ,&,cPi(Y)= in!; ,,.,(, C i,qgY)
foranyyEW(X).
”
Then for any y, z E W(x) we get:
(Pi(‘)- cPi,(‘) 1
C
vi,(Y)
i_Q,,...,in
li_Il ..,.,I
=
c
ie!il,.__, i,,jvi(J‘) ic:i~C,.i,~]rpi(z)
< $
’ $
’
f
cPi,(Y)
I~i,(Y)cPi(‘) C ~c!~~,...,i~) C IS!1 I,....
i,,l
+ $
’ $,
1
ie!il.....ip}
telli
C
211Ip < ~ 4Y, m”
-
(cPi(z)cPi,(z) -
izi, < >
i,)
cPi(‘) - Vi,(‘)
Vi,(‘)
’
Cpib)
cPi(Y)l
cPi,(z)Vi(Y)j
!c(‘Mik(Y)
IVik(‘)l. IV,(‘) -
C
ic-!ii ,..._ i,,)
Vi(Y)
-
cPi(2)cPik(z)l
cPi(Y)J+ 5
jPi(z)l’ C IE{L,,...,~~}
(Pi,(Y)
-
cPi,(‘)l
2)
by Iv1 on W(x) and Lipschitz with since each ‘pi for i E {i,, . . . . ip} is obviously upper-bounded constant equal to 1. We have thus effectively proved that each Yik with i, E {iI, . . . , ip} is Lipschitz on W(x). We shall now recall a very useful result on metric spaces. PROPOSITION A. Let A’ be a metric space. To any open covering (Ua)aeA of x’, we can associate a locally finite open covering (QJiEI such that: for any iE I there exists CIE A such that Ri c U,. Moreover if X is separable then I is countable and if x’ is compact then I can be taken finite. Prooj: The property
is due to the paracompactness
of metric spaces as defined in Bourbaki
[4].
1352
G. HADDAD
Now if X is separable, let {x~}“~~ be a dense sequence in X and let I,, = {i E Zjx, E 0). Then In # 0 since (Ri)iE, is a covering of X and In is finite since (Ri)iEI is locally finite. Furthermore since {x,JnGN is dense in X and since (Qi)iGI is an open covering ofX we obviously have I = U Zn nI 1 which proves that I is countable. If X is compact the proof is then really obvious. 3.
THE
APPROXIMATION
THEOREM
We can now state the main theorem of this part. For a given normed by i!I the closed unit ball centered at the origin.
space E we shall denote
THEOREM
A. Let X be a metric space, E a normed space, K c E a convex compact subset and I : X --+K an U.S.C. correspondence with non-empty convex compact values from X into K. Then there exists a sequence In : X + K, n E N, of U.S.C. correspondences with non-empty convex compact values such that: (1) (2) (3) n3N (4)
for any x E X for any x E X for any x E X . I%h I, is of
and for any n E N ; T(x) c F,,(x), and for any n E N, I, + ,(.x) c m(x), and for any E > 0 there exists N,,. I E N such that In(x) c I(x) + i:B for all the following
type:
rn(x)=
1
Y~"'(x)C!"' for any XE X,
i-r<,, finite, locally Lipschitz
where (‘yf“‘)iEI(“) is a locally associated family of non-empty
convex compact
partition
subsets
of unity in X and (Cin))ia r,,,, is an
of K.
Remark A,. If the space E is finite dimensional then instead of assuming that I takes its values in a convex compact subset of E, we could equivalently have supposed that I is bounded. In that case we choose K = conv I(X) which is closed convex hull of the set I(X) = U I-(x). -Yc-x ProoJ We fix p > 0. Let us then cover X with the open balls (B(x, P)}~~~. We shall now give a construction of ro. By Proposition A we can associate to the preceding covering of X by open balls, a locally finite open covering (flj”))ie I,Oj such that: for any i E Ito, there exists xi”) E X with Qj”) c B(xi’), p). We then define for any i E Ito,, Ci”’ = conv -i
w h’K h IS a non-empty u r(X) I x~B(xp’. 2p)
Now thanks to Lemma A we can associate to the open covering (R~“‘)i, ,(“). We then define for all x E X:
r,(x)=
convex compact
subset of K.
a locally Lipschitz
C
partition
of unity (‘I’i(oJi,ojE ,~“)
Y?'(X) ~10).
itI
The so defined correspondence IO is obviously a non-empty convex compact valued correspondence from X into K. well defined and U.S.C. on X since (Y(i”))iEI,O,is a locally finite. locally Lipschitz partition of unity in X. In order to define I’, we do the same as before with the open covering j&x, ~/3)}~~ x.
Topological
properties
of the sets of solutions
for functional
differential
inclusions
1353
Thus we define a locally finite open covering (Qi”).,Elcii’of X associated to {B(x~“, ~/3)}~~~~,, and an associated locally Lipschitz partition of unity (‘Pi )iGlc,‘. As before we set for all i E I(,,:
-1
u
C!” = conv
XEB[X}*),
rtw 2p 31
i
And then we can define for any x E X: I-,(x) =
1
Y!i’(x) q’.
ic-I,,,
The correspondence Ii enjoys the same properties as IO. We shall now prove that for any x E X, I,(x) c I,(x). Let us fix x E X. We define: Z$’ = {i E Z,,,/x E B(x~~‘,p)} and Z;i, = {i E I&
E B(x;“, p/3)}.
Let i(,, E Z;‘o,and i(,’ E Z;” be given. Then if y E B(xi,l,‘,,2p/3) we get: d(y, x{:,‘,)< F
with cl(x, xipd,)< p
and d(x, xi:,‘,) < p/3. Thus we have d(y, xi::) < (2p/3) + @/3) + p = 2~. And then B[x!“, 2p/3] c B[x~~$2p] for all i,,’ E Z;‘o,and all iclj E ZFly This leads to Cl::, c C!” l(1),for such ind%s. Then for all i, 1,E I ;‘,) we have :
C’icl,: = & Y’l”WC:O’= Jo, c
y;Y~)c;o)
= rotx).
this being true by convexity arguments and since (Yi”‘)iS1u) is a locally finite partition of unity associated to (Qi”‘)iB1,0’and thus also to {B(x$“, P)}~.~~,,which in particular says that Y!“(x) = 0 if i # ZFo’. And then for the same reasons we get :
r,(x) = Jl,
y;“(x) c;l) = iET yp(x) cp c r,(x). IIL
In fact when x is given, only a finite number of in I,,’ and of i E I(,) has to be considered since are locally finite. (‘J’~“)i~~(,, E I(o) and Hence we have effectively proved that I,(x) c I,(x) for all x E X. We shall now prove, for example, that for any x E X, I(x) c I’,(x). Indeed from the construction of Ci”’ we have I(x) c Ci” for all i E ZFo,. Then always for convexity reasons and since (Yi”‘)iEI,O,is a locally finite partition of unity associated to (Qi”‘)ialtO,hence to {B(x!“, P))~.~,,, we get:
(“1O’)i
r(x) c i& yjO’(~)c;O’ = i~,,,~~O~(~)c~O~ = r,(x). 101
1354
G. HADDAD
Now let us define p,, = (t,” p for any n E N. Then as for To associated to p0 = p and for T, associated to pr = p/3 we can by induction build a sequence of correspondences II,, : A’ + K, n E N, each of them being u.s.c., non-empty convex compact valued and verifying properties (I), (2) and (4) of the theorem. To finish the proof we have to show that (3) is verified. Let x E X be given. Since T is u.s.c., for any 8 > 0 there exists u~,,~ > 0 such that d(y, x) < v~,, implies r-(y) c r(x) + 8. Then there obviously exists NE, x such that for n 3 NE x we have p, < q, J3. ’ Let us define as before Z& = (i E Z,,/x E B(xj”), p,)}. For the same reasons as for To and T, we can write:
rncx) =
i
C Y",")(x)~)") E
I&,
where
U r(Y). 1 ytBLr,(“). 2p,j
C!“’ = conv -1 Then for all y c B(xy), 2p,) with i E $,,
we have
cl(y, x) Q d(y ’ x!“‘) I + d(x!“’ I ’ x) if we take n 3 N_.
< 2p, + p, = 3p, < q,,,
Thus for all n 3 NE,*_we have T(y) c T(x) + ED for all y E B[xy), 2p,] with i E I;,,,, But since T(x) + EB is closed convex we get:
Cl") c rtx)+ EB for alliEl;nr And then always for convexity
reasons
r,,(x)
=
we get:
1 Y?)(X) iE Ii%)
for all n 3 NE x. Thus (3) is verified and the proof of Theorem
cl”’
c
r(x)
+ &B
A is complete.
Remark A,. From (1) and (3) and since T(x) is compact
we deduce that for all x E X we have:
f-(x) = n m(x). II=PI Remark A,. If X is a compact metric space, then instead of locally finite partitions of unity we more simply deal with finite partitions of unity, which means that the sets of indexes I,,, are finite. Remark A,. When E is finite dimensional, if T is no longer supposed to be bounded on X, we can deduce from Theorem A a local approximation result for T by restricting it on a neighborhood of the point x E .I’. we consider, so that the correspondence will be bounded on this neighborhood. 4. APPLICATIONS
TO
FIXED
POINT
THEOREMS
As a direct application of Theorem A we shall show now that fixed point theorems in their infinite dimensional version as well as Kakutani’s fixed point theorem can be easily deduced
Topologicals
properties
of the sets of solutions
for functional
differential
1355
inclusions
from Brouwer’s fixed point theorem in the finite dimensional case. Indeed let X be a convex compact subset of a normed space E and let I : X -+ X be an U.S.C. correspondence with non-empty convex compact values. The case when I is single valued is obtained by T(x) = {f(x)} for all x E X with f : X + X a given continuous mapping. We can then approximate I by a sequence of correspondences I, : X + X, n E N, which verify (i), (2), (3) and (4) of Theorem A. It is now obvious that the existence of a futed point for I is equivalent to the existence of a fixed point x,, E X for each I,. Let us then consider I, : X + X such that (4) is verified. Since X is compact we know by Remark A, that I(“) can be taken finite. For simplicity we shall take I,,, = {1,2, . . . , k}. Let us then choose for any i E I 1,2,. . . , k}, an element ui E Ci c X. We can thus consider the mapping y, : X -+ X defined by: y,(x) = i
Yi(x)ui
for all x E X.
i=l
This mapping verifies y,(x) E I”(x) for all x E X. Furthermore we can restrict y, to the set defined by K = conv{ui,..., uk} which is convex compact finite dimensional. Thus the restriction of y, to K takes its values in K and is continuous by construction. The Brouwer’s fixed point theorem insures then the existence of a fixed point for y, (or its restriction) which is obviously also a fixed point for r,. PART
B. FUNCTIONAL
DIFFERENTIAL SETS
INCLUSIONS. SOLUTIONS
OF
APPROXIMATION
OF
THE
We show here how the preceding approximation theorem can be used both to prove the existence of solutions for functional differential inclusions and to state an useful approximation result for the sets of solutions. 1. DEFINITIONS Let T > 0 be given. We define %?= U([ - r, 01, Rp) the space of continuous functions from C-r, 0] into [WP.The topology on V will be the topology of uniform convergence. For any t E R, let A(t) denote the operator which to any continuous function x(. ), defined at least on the interval [-r + t, t], associates the function A(r)x E % such that:
[A(t)x](B) = x(t + 0)
for all 0 E [-r,
01.
Let C2be a non-empty subset of Iw x %?and I : R - lQpa given correspondence with non-empty values. We call a functional differential inclusion (with memory) the differential inclusion:
(W
$ (t)
E
r(t,
AH).
A continuous function x( .) from an interval [- r + t,, t, + T,] into IWPwith T, > 0, is said to be a solution of (h/l) if: x(. ) is absolutely continuous on [t,, t, + T’], (t, A(t)x) E Sz for all t E [t,, C, + T,], (dx/dt)(t) E r(t, A(t)x) for almost all t E [to, t, + 7J.
1356
G. HADDAD
We shall say that x( .) is a solution of (n/r) with initial time t, and initial value Q(, E %’ if x(. ) is a solution of (IU) and verities A(@ = a,. For simplicity we shall also say that x( .) is a solution of (h1) through (t,. @,). Differential inclusion (M) is a very general type of differential inclusions, particularly ordinary differential inclusions (dx/dt)(t) E T(t, x(t)) are obtained when r = 0. This is also true for differential difference inclusions: dx d$f) E s(t, x(t)).
X(t - rr(f)),
.
. X(t -
T#)),
where S is a correspondence from 0’ c iw x (tWPn+linto[WPandO
EXISTENCE
OF
SOLUTIONS
The existence of solutions for functional differential inclusions has been already studied in a previous paper [5]. We just give here another proof of the existence depending directly on the approximation theorem. PROPOSITION B. Let Q be an open subset of [w x % and let T : 0 -+ Rp be an U.S.C.correspondence with non-empty convex compact values. Then for any (t,, QO) E Sz there exists a solution of (&I) through (to, 0”).
Proof Since T is U.S.C.convex compact valued on a, there exists L, > 0 and R,, > 0 such that r is bounded by a constant 1, > 0 on the set [to, t,, + L,] x B(m),, 2R,) c R. where:
B(@,,2R,) = {@~%/j/@- (Doll < 2R,j with II.11 denoting the uniform convergence norm defined on %?. Let us then denote X = [t,, t, + L,] x B(@,, 2RJ. This set will play the role of the metric space X in Theorem A. We know then the existence of a sequence T,, : _X -+ UP, n E N, of U.S.C. non-empty convex compact valued correspondences, uniformly bounded by 3, an verifying properties (1) (2) (3) and r,(t, Q) = C Yi"'(t, @) C;"),for all (t, @) E X, (4) ier~,) where (Y!“‘). is a locally Lipschitz, locally finite partition of unity on X and (Cl”‘),E, an associated fam8y of non-empty convex compact subsets of Iwp,each of them bounded by i,!“’ Thus to each T,, n E N, can be associated the functional differential inclusion: (M.) The existence of solutions through (t,, @J for each (n/I,,) is obviously deduced from (4). Indeed it suffices to choose 211”) E Cl”) for all i E I(,,) and to consider the associated single-valued function
Topological
properties
f,(t, CD)= This leads to the retarded
of the sets of solutions
for functional
differential $
inclusions
1357
for all (t, 0) E X.
1 Yin’(t, Q) ain) E I,(t, 0) ieI(,)
functional
differential
equation:
(t) = .r;,(t. .4(t)),
It is first obvious that each solution of (E,) is a solution of (h/l,). Now since f, is obviously continuous on X there exists actually a solution of (E,) through (to, @J. (See Hale [5].) Let us denote x, such a solution. From the standard proof of the existence we know that xn is in fact defined on an interval [to - r, t, + 7J where T, does not depend on J, and I,,, but depends only on QO, L,, R, and 2. Now since In andfn, n E PV,are uniformly bounded by 2, it is obvious that the solutions x,,, n E N, are I-Lipschitz on [to, h, + T,]. Then according to Ascoli’s theorem we can extract a subsequence (again denoted x,,) which converges uniformly to a continuous function x on [to - r, t, + T,] which is A-Lipschitz on [t,, t, + T,] and verifies A(t,)x = A(t,) xn = 0,. Furthermore since X is a closed subset of R we obviously have (t, A(t)x) E .Y for all t E [t,, t, + To]. From x Lipschitz we are sure that (dx/dt)(t) exists for almost all t E [t,, t, + T,] and that dx/dt E L”([t,, t, + To], IF). We shall now prove that x is a solution of (n/f). For any s, t E [t,, t, + To] we have:
x,(t) - x,(s) = converges
to
And since dx,/dt, n E N, are equibounded by 2 on [t,, t, + To], we easily deduce that the sequence dx,/dt converges weakly to dx/dt in L”([t,, t, + To], [wp)dual of L’([t,, t, + To], [WJ’). But since L”([t,, t, + T,,], [wp)c L’([t,, t, + To], [wp),it is obvious that dx,/dt converges weakly to dx/dt in L’([t,, t, + To], Rp). Then using Mazur’s convexity theorem [7] we can build a sequence of convex combinations of the following type: with al 3 0,
all but a finite number
equal
to zero and+f a; = 1, such that CL,converges strongly to dx/dt n=l in L’([t,, t, + To], [wp)as 1 -+ + co. Then we can extract a subsequence of pl (again denoted p[ for simplicity) such that p,(t) converges to (dx/dt)(t) for almost all t E [t,, r. + To] as I-+ + w. Let t E [to, r. + To] be such a point of convergence. Then using property (3) we know that for any E > 0 there exists NE E /V such that: r,,(t, A(t)x) c r(t, A(t)x) + f B
for all n 3 NE.
1358
G. HADDAD
Let us take NE. By the continuity property of operator A(t), since xn converges uniformly to x on [to - Y,t, + T,] and since rN, is u.s.c., we are sure that there exists N: 3 NC such that for all n > N:: I-& Now by property
A(t)x,) c rr, (t, A(t)x) + ; B.
(2) we know that for all n z N:: T,(t, 4x,)
All this obviously
c r,j4
4t)x,).
implies that for all II 3 N’ : &
qt,
A(+,)
t
qt,
AX)
+ EB.
And then since .f,(t, A(t)x,) E T,(t, A(t)x,), for all n 3 Nk we have: (t) = .f,(t, AX,)
2
E qt,
AX)
+ cB.
Since r(t, A(t)x) is convex, for I 3 N: we get:
ia
dx p,(t) = C a: 2 (t) t- r(t, A(t)x) ,,=l
And since r(t, A(t)x) is compact,
+ t:B.
dt
the limit as 1 --f + co we have:
by taking
g(t) E
r (t, am)
+ CB
This being true for any E > 0, we deduce that: dx dr (t) E r(t, ox)
(W and this for almost all t E [t,, t, + T,]. The proof is then complete. 3. APPROXIMATION
OF THE
SET
OF
SOLUTIONS
From what precedes and by standard arguments on differential equations (see Hale [5]), it is obvious that there exists p,, > 0,O -C p0 Q R,, depending only on a,, L,, R, and 1 such that for any initial value @ E B(@,, p,) all the solutions of (&f) and (n/f,,) are defined at least on the same interval [t, - r, t, + To] where T, depends also only on CD,,L,, R, and 1. Moreover all these solutions will verify: (t, A(t)x) E X
for all t E [fO, t, + 7J.
This is a necessity for all solution of (IVY,)since r, is only defined on X. To make it true for (n/f) it suffices to choose T, and p0 small enough. We now denote, for any (DE B(CD,,,pO) by S,o(@) the non-empty set of the restrictions to [t,, t, + T,] of the solutions of (hf) through (to, @) and by S!l_“b(@) the corresponding restrictions for each (n/r”),n E N. We can now give the following result:
Topological properties of the sets of solutions for functional differential inclusions
1359
PROPOSITIONS B,. Let I be as in Proposition B, and let (I,JnEN be the approximated sequence verifying (l), (2), (3) and (4). Then under the preceding notations, for all CDE B(Q,, p,,) we have:
9”To+ “(CD)c SF l(Q) for all n E N, (5) S,o(O) : n $A(@). (6) IlEN Remark. For the proof we shall in fact need only properties (l), (2) and (3). The utility of (4) has already appeared for the existence of solutions and will again be obvious in the topological study of the sets of solutions. Proof The proof of (5) is obvious since from (2) we know that for all (t, @)E X: I-”+ ,(r, 0) = I”@, 0). Thus all solutions of (Mn + i) is a solution of (n/r,). Now from (1) we also easily deduce that: S,&(D) c SF:(@) for all n E N and all Q,E B(Q,, p,). Conversely let x E n $$(I)) be given. nthi Then for all n E iV we have: g (t)E r,(t,A(t)x) for almost all t E [to, t, + T,].
(M,)
Let us then consider the set: 3 = 1
t E [t,, t, + T,] g(t) :’
E r,,(t, A(t)x) for all n E N 1
This set has obviously a complementary subset of measure zero in [t,, t, + T,]. Let t E 3 be given. From (3) we deduce that for any E > 0 there exists NE E N such that: T,(t, A(t)x) c r(t, A(t)x) + EB
for all n 2 NE.
This obviously implies that (dx/dt)(t) E r(t, A(t)x) + EB. Since this is true for any E > 0 and since r(t, A(t)x) is compact, we deduce that: p(t)
Eqt, ii(+)
and this for all t E 2. Thus we have x E S,“(Q) which is the desired property and completes the proof. PART
C. TOPOLOGICAL
PROPERTIES
OF THE
SETS
OF SOLUTIONS
In this part we use the approximation theorem to study the topology of the sets ST,,(Q)defined in Part B as the restrictions to [t,, t, + To] of all the solutions through (t,, @) of the given functional differential inclusions. We first give some useful definitions and theorems. 1. PRELIMINARY
DEFINITIONS
AND
THEOREMS
We define here the notion of acyclicity and give some useful results related to that notion.
I360
G. HADDAD
We denote by H* = (H”!,.% the Tech cohomological functor with coefficients in Z, defined on the category of topologrcal pairs (Y, A). We refer to Godement [8] or Spanier [9] for the usual properties of cohomology. We define by l?’ the reduced cohomology. We shall then say that a topological space Y is acyclic if for all n E N; ifn( Y) = (0). The first very important example of acyclicity is given by convex sets which are acyclic. As a topological property of acyclic sets we know that they are connected and even simply connected. We shall need the following two theorems which can be found in [8] and [9] under more general versions. THEOREM 1. Let (K,Jnc 1 be a sequence of non-empty compact acyclic subspaces space Y such that K ,,+ , c Kn for all n E N. Then K = n Kn is acyclic. ,I’F,
of a topological
THEOREM2 (Vietoris-Begle). Let Y and Z be two paracompact topological spaces andf‘ : Y + Z a given continuous surjective mapping which is supposed to be closed. Let us suppose that for any ZE Z, ,f ‘(2) is acyclic. Then the acyclicity of Y or Z implies the acyclicity of the other one. 2. THE We can now give the main theorem
MAIN
THEOREM
of this part.
C. Let R be an open subset of R x % and let T : R + Rip be an U.S.C.correspondence with non-empty convex compact values, to which we associate the functional differential inclusion: THEOREM
d; (f) i I (1, A(r)x). Then for any (t,, @,,) E R there exists p0 > 0 and To > 0 such that for any @E B((D,, p,,) the set S,(,(@), as defined precedingly, is non-empty compact acyclic. Moreover the correspondence S,(, from B(@,,,.pO) into %([t,, t, + T,], [WP)is U.S.C.and verifies that S,(,m is a relatively compact subset of Q?([t,, t, + T,], RP). Remark. The topologies on B(O,, pJ and @([t,, t, + &I, Rp) are naturally topologies of uniform convergence.
the corresponding
Proof We recall that in Part B, for (t,, QO) E Q we have built X = [to. t, + L,] x B(Qo, 2R,,) c R such that T is bounded by EL> 0 and then we have defined a sequence Tn, n E N, of correspondences from X into [WI’.verifying properties (l), (2) (3) and (4) of Theorem A. Moreover we have pointed out the existence of pO, 0 < p0 < L, and of T,, 0 < r, < L,,, depending only on 00, L,, R, and I such that there exists solutions of (h/l) and of (M,), n E N, through any (t,, @) with @ E B(@,,, p,J. All the solutions being at least defined on [t, - r, t,, + T,] and verify (t, A(t)x) E X for all z E [t,, t, + T’,‘,]. We have then defined for any Q E B(@,, p,,) the non-empty sets S,,,(Q) and SF$@), II E N, to be the sets of the restrictions to [t,, t, + T,] of the solutions through (t,, CD). Now thanks to properties (5) and (6) of Proposition B, the proof will be divided in two steps. In a first step we shall prove the remaining conclusions of Theorem C for each S!$@), n E N. And in a second step we shall deduce all these conclusions for Sri,(@).
1361
Topological properties of the sets of solutions for functional differential inclusions
Step 1. lit
us consider
:
a correspondence I- : x = [to, t, + L,]
x B(D,,, 2R,) + W such that: (4)
r(t, 0) = 1 Yi(t, @)Ci
for all
(t,
@)
E X,
iel
where (Yi)iE I is a locally finite, locally Lipschitz partition family of non-empty convex compact subsets of KY, each Since W = W([ - Y,O]W) is separable for the topology is true for X. Thus by Proposition A we have I countable. independently of I, let us define: F = ((ri)it I/vi 6 L’([t,> to + To]> “)> On the vector space F we define the following
of unity on X and ( Ci)i E, an associated of them bounded by A > 0. of uniform convergence, then the same Now since p. and To have been defined
21~= 0, except for a finite number norm:
II(ri)iEIJIF = C IIuiJI1 = ,E r”“’ iel
it1
{u = (uJiEI; ~4~E L”([t,,
/(ci(r)// dr.
111
This norm is obviously well defined. Furthermore by its constitution and since I is countable, Its topological dual space 1;* is defined by: F* =
of i E I}.
the normed
space F is separable.
t, + T,], FF’) for all i E I and sup Ijuill =, < + a}. icl
The associated
Let
us
norm
on F* will be:
now define : Bc = {u = (u~)~~,E F*/ui(t) E Ci for almost all t E [t,, t, + To] and for all i E I>.
We then have the following LEMMA C,. Bc is a convex
result: compact
subset of F* for its weak* topology.
Proof: The convexity is obvious since each Ci, i E I, is convex. Bc is also obviously bounded in F* since the sets Ci, i E I, are uniformly bounded by 2. We shall now prove that Bc is weak* closed. Indeed let u = (ui)isl E B;*the weak* closure of Bc. We must then prove that u E B, which is equivalent to u,(t) E Ci for all i E I and for almost all t E [t,, t, + To]. Let us then fix i E I and take any (vi)iGl E F such that ui = 0 for all j # i. Then for any E > 0 there exists uE = (QEl E B, such that: f”+ T”
fn+ To ui(r) vi(s) dz ISfn
u;(r) vi(r) dz < E, s 10
where u:(t) E Ci for almost all t E [to, t, + To]. Thus ui E By* with Bc, = {UE L”([t,, t, + To], RP)/ui(t)c Ci for almost all t E [to, t, + To]} which is weak* compact. Indeed Bci is a closed convex subset of L’([t,, t, + To], W’) since Ci is convex compact. Moreover since :
1362
G. HADDAD
W[t,, t, + &I, RP)=
wt,,>t, + T”1,RP),
have I?;* = BiVthe weak closure of B in L’. But by the convexity argument we have’@‘: = EC1= Bc,. And then Bc, which is bounded, weak* closed, is effectively weak* compact. Thus we can deduce that ui E Bc, and this for all i E I. Then Bc is weak* closed and thus weak* compact since bounded. Furthermore an important remark is that since F is separable, we know that Bc is metrizable for the weak* topology. Now for any u = (u~)~~,E B, we define the mapping j; from X = [t,,, t, + 7;,] x B(4),,, 2R,,) into IV’, such that: WC
,f,(t, 0) = C Yi(f, @)ui(t)
for all (t, @) t X.
ial
From the properties of (Yi)iE, and of u E B, we easily deduce that 1; is locally Lipschitz with respect to (t, @) and thus to 0, measurable with respect to r and bounded by 3, > 0 independently of u. Thus the associated functional differential equation d$r)
(4,) verifies that for any initial value on [to - r, t, + To] for the same differential equations. (See Hale We can then denote by sTO(u, 0) Thus we have defined a mapping LEMMA C?. The mapping
= .tJr, 4)x),
0 E B(
‘s,.~’ is continuous
B(@,,, p,) as well as %([t,, 1, + T,], LWP) with their corresponding ence.
with the weak* topology topologies
of uniform
and
converg-
Proqf: Since Bc is metrizable, we can work with sequences. Let un = (u:“‘)~, ,E Bc and Q’,,F B(Do, p,), n E N, be such that lim U” = u and lim Q’, = CD. ,Im+ +X ,I .+I Then the associated sequence xn = s,r,,(u”, On) verifies for all n E N : (EU”)
li”
(l)
=
1
‘yip,
A(t)x,)
u;“‘(t)
ii-l
I
for almost all t G [to, t, + 7J. with .4(0)x, = QD,.
Since xI1has been only considered on [t,, t,, + T,,], the above writing is not completely rigorous. But for conveniency we identify here x,, to the whole solution of (EJ through (t,, Q’,) which is defined on [t, - r, t, + To]. We then easily verify that each x,, is A-Lipschitz on [to, t, + 7;,] and verifies ~~(0) = (D,,(O) with II@,,(O)- @,(O)Ij d po. Thus by Ascoli’s theorem there exists a subsequence uniformly on [t,,, r, + T,] to a A-Lipschitz mapping x.
(again
denoted
x,,) which converges
Topological
properties
of the sets of solutions
for functional
differential
We shall new prove that this limit x is equal to sT,(u, O), which is equivalent
I
(EU) 2
Iwith
1363
inclusions
to prove that:
(4 = ,sl Yi(t, A(t)x) t+(t) for almost all r E [r,, t, + T,]
A(O)x = @.
As before we identify x to the very solution of (E,) through (t,, CD),which is defined on [t, - r, t, + To]. Then we first notice that the sequence A(t)x, converges uniformly to A(t)x on the interval [t,, t, + T,] since a,, converges uniformly to @ on r, 0] and since x,, converges uniformly to x on [to, 1, + T,,]. This implies that (t, A(t)x)~ [t,, t, + T,] x B(@,, 2R,) for all t E [t,, t, + To], which is the first condition x must verily to be a solution of (EJ. Furthermore since the mapping t + A(t)x, from [to, t, + T,] into 59 = %?([-r, 01, IV), is obviously continuous we deduce that the set 3” = {(t, A(t)x)/t E [t,, t, + To]) is a compact subset of X. Thus since (YJiEI is locally finite, we are sure that on an adequate neighborhood of X in X, at most a finite number of ‘Pi, i E I, are non equal to zero. It can now easily be proven, since the functions ‘Pi, i E I, are locally Lipschitz, since only a finite number of them is to be considered on a neighborhood of X compact and since (t, A(t)x,) converges uniformly to (t, A(t)x) on [t,, t, + To], that Yi(t, A(t)x,) converges uniformly to Yi(t, A(t)x) on [t,, t, + T’], independently of i E I. But for all n E N we have:
[-
x,(t) - x&J
=
=
’ c s f0 iel zI
Yi(z, A(z)x,) L@‘(Z)dr
[
Yi(‘,
‘(‘)x,)‘~‘(‘)
dz>
fo
which is verified for all t E [t,, t, + To] and with the same i E I, in finite number, to be considered when n is large enough. Now, since the sequences ~4:) are weak* convergent to ui, being uniformly bounded by 2 and since the continuous functions Yi(t, A(t)x,) converge uniformly to Yi(t, A(t)x) on [t,, t, + To], by taking the limit as n -+ + cc we have for all t E [to, t, + To]: ’ Yi(z, A(z)x) ~~(7)dz s to
x(t) - x(q)) = c isI
=
tt s
,sI
Yi(‘,
‘(7)x)
‘i(z)
dr.
0’
And thus x = sTO(u, CD),which is the desired equality. LEMMA C,. For any @ E B(@‘,, p,) we have S,,,(Q) = sTo(Bc, a,) = {~~,(a, @)/u E B,). Proof. Let us first verify that sTO(u, @) E STO(@)for any u E B,. Indeed by definition the mapping x = sTO(u, CD)is the restriction to [t,, t, + To] of the solution through (t,, a) of the functional differential equation:
1364
G. HADDAD
g(t)
= ,fi(t, A(t)x)
f,(rj ‘(r)x)
for almost all t E [t,, t, + T,].
= iFI yi(r? A(t)x)ui(r) E C yi(t? A(t)x)
‘i
isI
=
r(t,
A(t)x)
for almost all t E [t,, t, + To].
And then we do effectively have sT,(u, @) E ST,(@). Conversely let x E ST,,(@) be given. By the very definition of a solution to a functional differential inclusion and since f is bounded, we know that x is Lipschitz in [to, t, + To] and thus the mapping b(t) = (dx/dt)(t) defined almost everywhere on [lo, t,, + T,] is a borelian mapping. Furthermore since the set X = {(t, A(t)x)/t E [t,, t, + To]} is compact, we know that at most a finite number of ‘Pi is non-zero on X. Let us denote them Y’, , Y2,. , Yn. We can then define the mapping p from [t,, t, + To] x (W)” into IWPsuch that:
Iw, a,,
.
.
3
an) = i
Yi(t, A(t)x) ui = b(t),
i=l
forany(t,a,,..., a”) E [r,, t, This mapping is obviously Let us now define A : [to, E C, x C, . . x C,,//?(t, ul,.
+ T,] x (RP)“. a borelian mapping. t, + T,] -+ C, x C, x . . . x C,, such that A(t) = {u = (~1~.. . , an) . . , a,) = O}. Since x E STO(Q) we have: b(t) = %(t)t
i
Yi(t, A(t)x) ci
i-l
for almost all t E [to, t, + To]. Thus A(t) # 0 for almost all t E [to, t, + To]. At last the graph of A is equal to ,!-‘(0) which is a borelian set. Thus from Von Neumann’s selection theorem [lo] there exists u : [t,, t, + 7J + C‘, x C2 x . . x C,, which is measurable and such that u(t) E A(t) for almost all t E [to, t, + T,,]. Then if we denote u(t) = (ul(t), . . . , u,,(t)) we get: b(t) = ~(1)
= ~ yi(t, A(t)x)ui(t) i=l
for almost all t E [t,, t, + To]. And thus obviously x = sTO(u, @) where u E Bc is defined by the ui, i = 1 . . . n, we have obtained. The other indexes having no importance since Yi(t, A(t)x) = 0 for such indexes. We can for example choose any constant mapping on the corresponding Ci. The proof of Lemma C, is then complete. LEMMA
C,. For any @ E B(Q,, p,), S,o(@) is compact
Proof: Let us consider
the mapping
acyclic.
sTo(. , @,)from Bc into S,o(@) which to any u E Bc associates
ST”64 @) E S,O(@). Then from the preceding lemmas we know that this mapping is continuous and surjective. Since Bc is compact, then S,“(Q) is compact. From the convexity of the sets Ci, i E I, we easily verify that the inverse image by S,“(. , 0) of any x E SrO(Q) is convex thus acyclic. At last sr,,(. , 0) is obviously a closed mapping since it is continuous and since Bc is compact.
Topological properties of the sets of solutions for functional differential inclusions
1365
Then all the hypotheses of Theorem 2 are satisfied. And since B, is convex thus acyclic, we know that SrO(@)is also acyclic. C,. The correspondence ST0 from B(Qo, po) into ‘%?([t,,,t, + T,], Rp) is U.S.C.and ST<,[@&-$I 1sa relatively compact subset of %?([t,, t, + T,], Rp).
LEMMA
Praof Let us first consider STo[B(@,,, p,)]. S ince the correspondence I is bounded by 1 on X, we have already noticed that all the solutions of the associated functional differential inclusion (M) with initial value Q E B(Q,,, po), are i-Lipschitz on [to, t, + T,]. Furthermore for any x E S,O[m] there exists Q E B(Qo, p,,) such that x E S,“(Q), which implies that x(0) = m(O)and so that I/x(O) - @JO)// < pO. . is relatively compact. Thus by Ascoli’s theorem SrO[m] . 0 bviously Thus ST0 takes its values in a compact subset of V([t,,, t, + T,], Rp). To prove that ST0 is U.S.C. it is now sufficient to show that its graph is closed. lim an = CDfor the corresponding Let xn E STo(@J, n E N, such that “$I~ xn = x and n-+a topologies of uniform convergence. From the surjectivity of the mapping sTO,we know the existence of u,, E Bc, n E N, such that xn = srO(u., QJ for all n E N. Since B, is compact metrizable for the weak* topology, there exists then a subsequence (again denoted ~4”)such that u, converges to u as n -+ + co. Since the mapping srO is continuous, we have x = s(u, 0) and x E SrO(Q). The proof of Step 1 is now complete. Step 2. We can now return to I in the general case. From Proposition any 0 E B(@,, p,) we have: 9”To+ l’(m) c SF:(Q) for all n E N,
S,“(Q) = f-j sy(@).
B, we know that for (5)
(6)
neN
By Step 1 we know that for any Q,E B(@,, po), and for any n E N, SF:(@) is non-empty compact acyclic. Then all the hypotheses of Theorem 1 are satisfied and thus SrO(@)is compact acyclic for any 0 E B(@,,, po). Furthermore ST0 is an U.S.C.correspondence since it is the intersection of the sequence of U.S.C.correspondences SF:, n E N. At last SrO[m] is relatively compact in %?([t,, t, + T,], Rp) for similar reasons as those given in the proof of Lemma C,. The proof of Theorem C is now complete. Remark. In fact we have proved the stronger property that for any n E N, the set S$$Q) is the image of an associated compact convex set (which is denoted by B, in the proof) by a contmuous mapping which verifies that the inverse image of any point is convex. We have then proved that S,“(a) is the intersection of such a decreasing sequence.
As an immediate consequence of Theorem C we have the single-valued version for functional differential equations. COROLLARY
C, . Let R be an open subset of IR x 5%and let S : R -+ Ripbe a continuous mapping.
1366
G. HADDAD
Then we have exactly equation:
the same conclusions
as in Theorem
C for the functional
differential
dx
ProojY We obviously
consider
the associated
r(t, CD)= { ,f(t, 0)) This correspondence
is obviously
univalued corespondance
defined by :
for all (t, 0) E R.
U.S.C. convex compact
valued.
COROLLARY C,. Let all the hypoteses of Theorem C be satisfied. Then for any CDE B(@,, po) the set R,o(O) = {x(t, + 7Jx E SrO(@)} is compact and connected. Moreover the correspondence R,(, from B(@,,, L),,) into I@’ is U.S.C. and verifies that
RrI,[mp,)l
ia relatively
compact
in UP.
Proqf. Let II : %?([t,, t, + T,], EP) + Rp be the mapping such that n(x) = x(t, + TO) for any x E %([t,, t, + TO], KY). Then R,o(@) = lIl[S,,(@)] f or all CDE B(BO, pJ. Since n is continuous and since S,“(Q) is compact and acyclic thus connected, we easily deduce that R,,,(Q) is compact connected. is obviously U.S.C. and R,,l[B(@,,, p,)] = Furthermore the correspondence RTI, = n._S, is relatively compact. W,“(B(cD,,l is relatively compact since S,.,I[e)] REFERENCES ARRONSZAJN N., Le correspondant
de I‘umcitk dans la thCorle des iquations dif(erentlelles. Amls Math. 43, No. 4 (Octobre 1442). . - . cahier de Math de la dicision No. 761 I 1. LASRYJ. M. & ROBERT R.. Analvse non 1inCaire multivoaue. Am. J. Murh 58, 214 222 2. EILENBERG S. & MONTC~MERV c., Fixed point theorem for multivalued transformations, (1946). 3. CORNET, IS., An abstract theorem for planning procedures. In C’ongrrs d’Ancr!,sr Conrcxe. Murat le Qualre (1976). Lecture Notes in Economics and Mathematical Systems. 4. BOURBAKI N., TopologirGhhdr. Hermann. 5. HALE J., Theory of Functional DiJ$rential Equations. Springer Verlag. Berlin. 6 HAIX)AII G.. Monotone viable trnlcctories for functional diftel-cnllal inclusion\. ./. t/i//. Eqns (IO appcarl I. YOSIDA K., Functional Analysis, pp. 120-123. Springer Verlag, Berhn. R.. Toooloaie alqibrique et thPorie des faisceaux. Hermann, Paris (1964). 8. GODEMENT New York (1966). 9. SPANIER E., AlgedraicUTo~ojogy.McGraw.Hill, IO. voh NI-UMANNJ.. On Rings of operators. reduction theory. Aw~/.c Moth. 50. 401-485 (lY4Y) 0
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