EXISTENCE OF SOLUTIONS OF FUNCTIONAL DIFFEINTIAL INCLUSIONS

We prove the existence of solutions of a functional differential 
inclusion. By using the variation of parameters formula we convert the 
functional differential inclusion into an integral inclusion and prove the 
existence of a fixed point of the set-valued mapping with the help of the 
Kakutani-Bohnenblust-Karlin fixed point theorem.


INTRODUCTION
Fixed point theorems are widely used as a tool to prove the existence of solutions of differential inclusions.Schauder's fixed point theorem is used to prove the existence of solutions of differential inclusions [2] and functional differential inclusions with nonconvex right-hand side [8] in Banach spaces.Ky Fan's fixed point theorem is used in [7,10].In [5], Marino used a fixed point theorem due to Martelli [6] for establishing the existence of solutions of a nonlinear differential inclusion in Banach spaces.Angell [1] obtained an existence theorem for integral inclusions of Urysohn type by using Kakutani-Bohnenblust-Karlin fixed point theorem.Papageorgiou [9] proved the same for the nonconvex case by using Schauder's fixed point theorem.
In this paper we prove the existence of solutions of functional differential inclusions via integral inclusions.First we convert the functional differential inclusion into an integral inclusion by using the variation of parameters formula.Then we use the Bohnenblust-Karlin extension of Kakutani's fixed point theorem [3] to prove the existence of solutions of the integral inclusion which is the solution of differential inclusion.iReceived: March, 1992..Revised" May, 1992.  . BASIC IIYPOTIIESES Consider the differential inclusion h(t) E L(t,zt) + F(t, zt), a.e. on [O,b   x(t) = (t) on [-r,O]   (1) where :t'[-r, Ol--..R n is continuous such that In order to ensure the existence of solutions of the differential inclusion (1), we shall make the following assumptions: = N clU{F,); il -g II < ,} 6>0 that is, F is upper semicontinuous in the sense of Kuratowski with respect to the variable .Note that, as the intersection of closed sets each F(7,) is closed, there exists a measurable set-valued function P: [0, b]---.E1, a constant M > 0, and for each >0, a function , e L' ([O,b];Rn), t(t)>0, such that for given x e Lcc ([-r,b];Rn) and selection v(t) q.F(t, xt) there exists a selection rl(t) P(t), with b <_ v(t) < ,,(t) + co(t).
where the operator T(t, tr): C--.C is given by T(t,a) zt(tr, ), tr _< _< b such that T(a,a) = I, T(t, tr)T(tr, s)= T(t,s) is a solution of the homogeneous equation 5:(t) = L(t, zt) and X 0 is defined by Xo(O) ={ O, -r<_O<O I, 0=0 where I is the identity matrix.
So in order to prove the existence of solutions of the differential inclusion (1), we have to prove the existence theorem for the integral inclusion (2).We prove this existence theorem by using Bohnenblust-Karlin extension of Kakutani's fixed point theorem.that Theorem I: (Bohnenblust-Karlin) [3] Let E be a nonempty, closed convex subset of a Banach space .If F:E---,2 is such (a) F(a) is nonempty and conve: for each o" E, (b) the graph of F, ((F)C EE is closed, (c) U{ r(a); a fi E} is contained in a sequentially compact set C , then the set-valued map F has a fi:ed point, that is, there exists a o" o E such that o" 0 F(ao).

EXISTENCE
Since our interest is to study the existence of the solutions of the differential inclusion (1), we will need to give a precise definition of the term solution.
Theorem 2: [1]: Under lhe hypothesis (it), for each z S, O(z) is not empty and the set O(S) defined by the relalion (4) is an equi-absolulely integrable set and is weakly compact in L' ([0, bl; Now we prove the relative compactness of the set q(S) and the convexity of @(:).
Theorem g: Under hypotheses (iii) and (iv), for each z S, *(z) is not empty and the set (S) defined by the relation ( 4) is a relatively sequentially compact subset of L([O,b];Rn).
To prove the theorem, it remains to show that (S) is equibounded.
Theorem 4: For each z S, the set @(x) defined by the relation (4) is convez.
Therefore Hence q(r) is convex.
Next we will prove that the graph of q,{]() is closed.For that we use the following closure theorem.
Theorem 5:[1] Let I = [O,b], consider the set-valued mappin9 F: I x Loo---,2 En and assume that F satisfies the hypothesis (ii) with respect to .Let v, vk, : and z k be functions measurable on I, z, :t ounded, and let v, vk L(I;R").Then if vk(t F(t, ztt) a.e. in I and vk---,v weakly in L'(I;Rn) while ztcz uniformly on I, then v(t) F(t, xt) a.e. in I.
Theorem 6: Under the hypotheses (ii), (iii) and (iv) the map q'S--2 S has a closed graph, that is, {(z,V) e S x S: V e P(x)} is closed.
0 Without loss of generality, we may assume that vtc--v weakly in LI ([O,b];Rn) and from Theorem 5, v(s) F(s, zs).
To prove y E P(z) we wish to show that y satisfies the equation y(t) = T(t,O) (0) + / T(t,s)Xov(s)ds 0 [0, b] x L[ r, b] be continuous and C = C([ r, b]; Rn).parameters formula for the initial value problem The variation of