A NOTE ON MULTIVALUED DIFFERENTIAL EQUATIONS ON PROXIMATE RETRACTS

This paper discusses viable solutions for differential inclusions in Banach spaces. Existence will be established in two steps. In step 1, a nonlinear alternative of Leray-Schauder type [8] for maps with closed graphs will be used to establish a variety of existence principles for the Cauchy differential inclusion. Step 2 involves using the results in step 1 together with some tricks involving the Bouligand cone (and sometimes the Urysohn function) so that new existence criteria can be established for multivalued differential equations on proximate retracts.

In this paper we study the existence of solutions y: [O, T]--.K C_ E (so called viable solutions) to the differential inclusion y'(t) e (t,y(t)) a.e.t e [0, T] e K is a proximate retract and b" [0, T] x K--2E; here E is a real Banach space and 2 E denotes the family of all nonempty subsets of E. Using a nonlinear alternative of Leray-Schauder type, we were able in [7] to establish some general existence principles and theory for (1.1) (however in [7] we had to assume was a K-Carathfiodory map [4]).In this paper using some recent results of the author (see [6,8]) we are able to discuss a more general .
The technique to establish the existence of viable solutions to (1.1) will be in two steps.In step one we discuss the differential inclusion ' F(t,) .. o, [0,] x(0)x o E. (1.2) Our goal will be to establish some general existence principles for (1.2) which will automatically lead to new criteria for the existence of viable solutions to (1.1).We will discuss (1.2) in the introduction.The proofs of our existence principles will be elementary since all the analysis was completed in [6-8].In Section 2 (which is step 2) we will first discuss directly (1.1) when is not necessarily K-Carathfiodory.New results are presented which extend previously known results in the literature [2, 3, 5, 7. 9].Then we will examine (1.1) indirectly; the idea in this case is to examine the differential inclusion x'E expconv(t,x) a.e. on [0, T] (1.3) (0) o K.
Remark 1.1:For a set A, exp(A) is the set of extreme points of A and conv(A) is the convex hull of A.
Again new results will be obtained for (1.3) which will lead to new existence criteria for (1.1).
To conclude this section we discuss the differential inclusion (1.2) where F:[0, T] x EC(E) (here C(E) denotes the family of all nonempty, compact subsets of E).We look for solutions to (1.2) ) then u is differenti- able almost everywhere on [0, T], u' G LI([0, T], E) and u(t) u(O) f toU'(s)ds for t G [0, T].] Before we specify conditions on F we first recall some well known concepts [4].Let E 1 and E 2 be two Banach spaces, X a nonempty closed subset of E 1 and S a measurable space (respectively S-I x E, where I is a real interval, and A C_ S is (R) % measurable if A belongs to the r-algebra generated by all sets of the form N x D where N is Lebesgue measurable and D is Borel measurable).Let H: X---E 2 and G:S---E 2 be two multifunctions with nonempty closed values.The func- tion G is measurable (respectively (R) % measurable) if the set {t G S: G(t) N B :/: q)} is measurable for any closed B in E 2. The function H is lower semicontinuous (1.s.c.) (respectively upper semicontinuous (u.s.c.)) if the set {x E X:H(x)n B 7 0} is open (respectively closed) for any open (respectively closed) set B in E 2.
When we examine (1.2) we assume F:[O,T] x EC(E) satisfies some of the follow- ing conditions (to be specified later)" (i) tF(t,x)is measurable for every x E (ii) xF(t,x)is u.s.c, for a.e.t [0, T] (1.4) (i) tF(t,x)is measurable for every x E (ii) x--F( t, x) is continuous for a.e.
[0, T] (1.5) (i) (t,x)F(t,x)is(R)measurable (ii) xF(t,x)is 1.s.c. for a.e.E [0, T] (1.6) and for each r > 0 there exists h r E LI[0, T] such that I I F(t,x)II <_ hr(t) for a.e.E [0, T] and every x E with I I I I _< there exists h LI[0, T] such that I I F(t,x)II <_ h(t) for a.e.t [0, T] and all x E there exists 3' _> 0 with 27T < 1 and with c(F([0, t] x f)) _< 7c(f) for any bounded subset a of E; here c denotes the Kuratowskii measure of noncompactness. (1.7) We now present six existence principles for (1.2) which will be needed in Section 2.
Proof: The result follows from Theorem 2.1 in [6].In [6] we assumed X was bounded (here we assume N(X) is a subset of a bounded set in C([0, T,E)); however the proof is the same.

Differential Inclusions on Proximate Retracts
In this section we study the existence of viable solutions x" [0, TIcK C_ E to the differ- ential inclusion x'(t) (t,x(t)) a.e.
[O,T] x(0) Yo e K. (2.1) By a solution (viable) to (2.1) we mean a x E WI'I([O,T],E) with x' E (t,x) a.e. on [0, T], x(0) Y0 and x(t) K for t G [0, T].Throughout this section we assume K is a proximate retract.
Definition 2.1" [9] A nonempty closed subset K of E is said to be a proximate re- tract if there exists an open neighborhood U of K in E and a continuous (singlevalued) mapping r: U---,K (called a metric retraction) such that the following two con- ditions are satisfied" (i) r(x) x for all x G K; (ii) I I r(x)-x [[ dist(x, K) for all x G U. Remark 2.1: Now since we can take a sufficiently small U (for example by restrict- ing V to U fl {y E E: dist(y, K) < 6} for some given 6 > 0) we may assume (and we do so) that I[r(u)-ull <-6fralluGU" For most of this section we will assume satisfies either or (: [0, T] K-,CK(E) satisfies (1.4) and (1.7) here F is replaced by and E is replaced by K) : [0, T] x KC(E)satisfies (1.7) and either (1.5) or (1.6) (here F is replaced by and E is replaced by K).
Assume also that where (t,x) C_ TK(X for all x e K and a.e.E [O,T] v e U:liminf dist(x + tv, K) TK(X) t t-+O + -0} is the Bouligand tangent cone to K at x.We now concentrate our study on the differential inclusion x'(t)(t,x(t)) x(O) Yo K. .
Theorem 2.1" Let E-(E, II" II) be a separable Banach space and let ,U be as above (in particular U is chosen as in Remark 2.1).(i) Suo (.2), (2.), (2.) a there exists 7 >_ 0 with 27T < 1 and with c( ([0, t] a)) _< 7c(a) for any bounded subset a of E (2.7) hold.In addition suppose there is a constant M, independenl of I t, with I I Y I I 0 # M for any solution y G WI'I([o,T],E) to x'(t) It (t,x(t)) z(o) Yo (ii) Suppose (2.2), (2.4), (2.5) and (2.7) hold.In addition assume there is a constant M, independent of It, with ]] y ]] o 7 M for any solution y E WI'I([O,T],E) to (2.8)t for each It G (0, 1).Thus (2.1) has a viable solution u with I I u I I o <-M.
Proof: From Theorem 1.1 (if we are discussing (i)) or Theorem 1.2 (if we are discussing (ii)) we have immediately that (2.8)1 has a solution y (note y(t) It" for all t G [0, T] by Theorem 3.1 of [7]).Thus y is a solution of (2.1).
Remark 2.3: Suppose E is a Hilbert space and K is a closed, convex subset of E.
In addition, suppose satisfies (1.9) (with F replaced by and E replaced by K).Then (2.7) is satisfied.To see this notice r in this case is nonexpansive.Now if is a bounded subset of E then since  ~such that r(x)=l if Ilxll _<M and re(x)=0 if Ce(t,x) re(x) (t,x) and we look at the differential inclusion Ilxll >-M+e" Let x'(t) E(t,x(t)) x(O) o" a.e.t E [0, T] Theorem 2.2: Let E (E, I1" II) b a separable Banach space and assume (2.2) and (2.5) hold.In addition, suppose :[O,T]K--C(E) satisfies either (2.3) or (2.4).Assume there is a constant M with I] Y l]o < M for any,,])ossible viable solution y E WI'I([o,T],E) to (2.1).Let >0 be given and let 7, be as above.
Proof: From Corollary 1.4 or Corollary 1.6 (note (1.8) is satisfied with F replaced by Ca) we have immediately that_(2.9)has a solution y.By assumption I1Y I I 0 < M and so by definition Ca(t, y(t))= (t, y(t)).Thus y is a solution of (2.6).Now Theo- rem 3.1 of [7] implies y(t) E g for every t E [0, T] and so y is a solution of (2.1).El Finally, in this section we examine the differential inclusion x'E expconv (t,x) a.e. on [0, T] (2.11) x(0) Y0 E K. New results will be obtained for (2.11) (these extend and complement results in the literature [1-3] and these automatically lead to new existence criterion for (2.1).
For the remainder of this section we will let G(t,x) expconv(t, x).
As before, K will be a proximate retract (i.e.(2.2) holds).We also assume the follow- ing conditions hold: for each r > 0 there exists h r E LI[0, T] such that for a.e.E [0, T] and every x E K with I I I I _< (2.15) a(t,x) c_ TK(X for all x E K and a.e.t E [0, T] (2.16) and (t, K) is compact for a.e.t E [0, T]. (2.17) Recall the following results [3, pp.71-72].
Theorem 2.3: (Krein-Milman) Let X be a Banach space and 5 M C X be com- pact.Then exp convM is the smallest closed subset of M such that conv exp cony M cony M.
Theorem 2.4: Let X be a Banach space, D C_ X be closed and suppose F'D2 X have closed values with F(D) compact and conv F continuous.Then expconvF is l.S.C.
Thus if we assume (1.7) (with F replaced by ) then (2.15) is automatically satisfied.
In addition if we assume (2.5) then (2.16) holds.Let U,A and r be as in the beginning of Section 2 and define G" [0, T] x E+C(E) by (t,x) A(x)G(t,r(x)) if x C U {0} ifxV.
Suppose there is a constant M with I I I I 0 < M for any possible viable solution to (2.11 Let e>0 be given and let re:E[0,1 be as before.Let Ge(t,x)- r e(x)G (t,x) and we now look at the differential inclusion x'(t) eG(t,x(t)) (0)-t e [0, r] (2.21) Now Theorem 2.2 (together with the ideas in Theorem 2.5) immediately yields the following result.
(2.29) Theorem 2.7: Let E (E, I1" II) b a separable Banach space and assume (2.24)- (2.29) hold.In addition, suppose there is a constant Mo, independent of ., with I I Y I I o # Mo for any solution y e WI'I([0,T],E) to x' G e---pconv Remark 2.7: We could also obtain an analog of Theorem 1.5 and Corollary 1.6 for the differential inclusion (2.23).R.emark 2.8: Notice (2.25) and (2.29) could be removed if we assume (2.31).
K a.e.t G [0, T] for each It E (0, 1).Thus (2.1) has a viable solution u with I I u[I o <-M.
It is also possible to useCorollary 1.4 or Corollary 1.6 to establish an existence principle for (2.1).Suppose there is a constant M with I I Y I I 0 < M for any possible viable solution to (2.1).Let c > 0 be given and let -:E--[0, 1] be the Urysohn func- tion for (B (0, M), E\B(0, M + c)) In addition, suppose there is a constant Mo, independent of A, with I I Y [[ o 5 Mo for any solution y WI'I([o,T],E) to (1.10).for each ,kG(0,1).Then(1.2) has a solution u G W 1' 1([0, T], E) with I[ t I I o Mo" F(t, x) a.e.t E [0, T] (2.3o) for each (0,1).Thus (2.23) (and so (1.2)) has a solution u WI'I([O,T],E) with I I I I o < M.Proof: As in Theorem 2.5 it is easy to see that xH(t,x) is 1.s.c. for a.e.t G [0, T].(2.31)Now apply Theorem 1.2 to H. [:l