EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS WITH NONCONVEX RIGHT HAND SIDE 459

In this paper we proe some new existence theorems for differential inclusions with a nonconvex right -hand side, which is lower semicontinuous or continuous in the state variable, measurable in the time variable and takes volucs in a finite or infinite dimensional separable Pmch space. 1990 Mathemntics Subject Classifcation: 34GOZ


I. INTRODUCTION.
In the recent years there }as been an increase in interest in the investigation of systems described by differential inclusions.In ay ordinary differential equation the tangent at each point is prescribed by single valued function.In a differential inclusion the tangent is prescribed by a mu]tfunction (set valued function) which is usually called a: orientor field..'Lanyproblems of appl mathentics lead us to the study of d3q_nmical systems having velocities not miquc]y determined by the state of the system, but depending only loosely upon it.In these cases the classical equation '(t) f(t.x(t)) describing the dy,xmics of the system is replaced by a relation of the form (t) F(t,x(t)) where F(.,.) is a multifunction (the orientor field).Such a "set valued differential equation" is called "differential inclusion".The initial impetus to study diffe.rentialinclusions came from control theory.Then the subject found additional impo,'tmt applications in ma{hc,natical economics [I], nonsmooth dynamics [2], optimization [3], differential equations with a discontinuous forcing term [4] etc.
The purpose of this paper is to prove existence theorems for differential inclusions governed by nonconvex valued, lower semicontinuous orientor fields which take values in a separable Banach space.Until now, most of the existence theory for differential inclusions was developed for upper semicontinuous, convex valued orientor fields with values in n.However lower semicontJnuous, nonconvex valued orienror fields appear often in control theory in connection with the bang-bang principle.So it is important to have existence theorems for differential inclusions governed by such orientor fields.

2.PRELIMINARIF.
Let (,2) be a measurable space and let X be a separable Banach space, with X being its topological dual.We will use the following notation.
For A E 2X\c).we set [A[ sup [[x[[ and by dA(.we denote the distance xEA function from A i.e. for all x e X.dA(X inf aA A multifunction F d Pf(X) is said to be measurable if it satisfies any of the following equivalent conditions.
(i) d (x) is measurable for all x X FC) (ii) there exists a sequence {fn(.)}n of measurable functions s.t.F() cl{fn()}n for all (Castaing's representation) (iii) for all U X open F-(U) E O F() 0 U } e 2 (in the language of measurable multifunction F (U) is called the inverse image of U under F(')).
A detailed treatment of measurable multifuntions can be found n Castaing-Valadier  [5]  ad Himmelberg  We denote by S F the set of all selectors of F(') that belong to the LebesNe-Bochner sNce (n) i.e. S F {f(.) e (n) f() e F()-a.e.}.
Finally if -{A-}n are nonempty subsets of X, we define s-l__im An (x X x s-lira Xn,Xn 6 A n.n _> I}. n-)co By W(') we will denote the Hausdorff measure of noncompactness i.e. if B _C X is botmded, then (B) inf{r > 0 B can be covered by finitc.lym.nny balls of radius r}.This is equivalent to the Kuratowski measure of nonconq>actness [7] (see also Banas-(;oebel [8]).Recall that by a Farc function we mean a function w [O,TJ'xtR+ + satisfying the Caratheodory conditions (i.e. it is measurable in t and continuous in x),w(t,O) 0 a.e. and such that u(t) z 0 is the only solution of the problem u(t) (s,u(s))ds,u(O) O.

F_XISTENCETHEOREIS.
The setting is the following.e are given a fnite intervaI T [O,b].On T we consider the Lebesgue measure dr.lso let X be a separable reflexive Banach space.By X we ill denote X ith the weak topology.e Cauchy problem under consideration is the following: By a solution of ) we understand an absolutely continuous function x T X satisfying () for almost all t T.
then () admits a solution.
PROOF: Let r 1111.mnd consider Br(XO) {x X IlX-Xoll r}.Because o the reflexivity of X,Br(XO) is w-compact md metrizable for the weM: [opology (see anford-Schwartz [9], theorem 3, p. 3).In the sequel we will al,:ays consider Br__fXo) with the were topology'.Let L Br..iXo Pf(I2.(T))_Xbe the mtl] ti function defined by L(x) S(.,x).Our claim is tha L(.) is 1.s.c.From Delahaye-Deel [10] we know that it suffices to sl,ow t.hat tot any x x in B (Xo) we have n r 1 Sc x) C s-lm SF( x )" For that purpose let t'(') SF( x)" Then fCt) F(t,x) n-'n a.e.A straightforward application of Au,nknann's selection theorem can give us f .)E S( x s t d (fCt)) lit(t) fn(t)ll for all t C T. Since F(t .) is s c n F(t,Xn) from Xw into X, F(t.x) C_ s-lim___ F(t,Xn) and so lim d nn-F(t,Xn) n-o O, which by the dominated convergence theorem implies that f (.) f(.) => n 1 f(.)E s-lim SF(" Xn ).So we lmve.shown that SF(, x) C s-lim SF( which as we --,Xn) already said, implies the lower semicontinuity of L(-).Hence we can now apply theorem 3.1 of Fryszkowski [11] and deduce that there exists $ Br(XO) [,(T) continuous s.t.8Cx) 6 LCx for x e BrCXo).set f(t.x) $Cx)C t) and consider the following single valued Cauchy problem {xC-e CxCT xCt e BrCxo) W W defined by for al t T} and consider the nap (x)Ct) x 0 + fCs,xCs))ds.
So (R) O. which means that R is a relatively compact subset of Cx(T ).
RENARK.The theorem remains true if we essume that X is a separable dual space 'ith a separable predual and F(t..} is 1.s.c.from Xw into X. When X is finite dimensional we can have a more general boundedncss hypothesis.THEORF 3.2.I__f F TxX Pf(X} is a multifunction s.t. 1) for all x 6 X, F(.,x) is measurable and IF(t,:}] a(t)llxll + b(t) a.e. with a(-),b(,) 6 EI(T 2) for all t 6 T,F(t,.) is 1.s.c. then () admits a solution.
[F(t,x) for Ilxll N F(t.x)IF Mx (t II-D for llxll > N Then for every t T,F(t,') is the composition of the multiftmction F(t.') and of the M-radial retraction map r X -BN{O} {z X Ilzll (_ N} defined by r(x) j if Ilxll _< 14  []ll-Mx f Ilxll > M It is well known that r(,) is Lipschitz.So F(t,.) is 1.s.c.Clearly it is measurable in t and for all x 6 X, [F(t,x)[ a(t)M + b(t)a.e.So F(.,-) satisfies the hypotheses of theorem 3.1 (recall that X is finite dimensional) and so by that theorem there exists x T X absolutely continuous s.t. (t) (,x(t)) a.e., x(O) x O. Our goal is to show that for all t e T, IIx(t)ll _< M. We proceed by contradiction.Suppose that there exist tl,t 2 T s.t. for t 6 (tl,t2)llx(t)ll > hi.
Since for t t (tl,t2) we already have that IIx(t)ll (_ I, and get that for all t E T z(t) z(O) / 2 (a(s) + b(s))z(s)ds.
Another existence result in this direction is the following.
Again assume that we It iseasy to see that this set is closed and it is nonempty if and oniy if Assume that Y,Z are topological spaces and F Y d 21a,{}.We say that is lower semicontinuous (1.s.c.) if and only if for all