REGULAR AND SINGULAR PERTURBATIONS OF UPPER SEMICONTINUOUS DIFFERENTIAL INCLUSION

. In the paper we study the continuity properties of the solution set of upper semicontinuous differential inclusions in both regularly and singularly perturbed case. Using a kind of dissipative type of conditions introduced in [1] we obtain lower semicontinuous dependence of the solution sets. Moreover new existence result for lower semicontinuous differential inclusions is proved.


INTRODUCTION
In the paper we consider the following regularly perturbed multivalued differential equation: (t) 6_ F(x(t),a), x(0) x0; 6 [0,1] ( Where z 6 H (Hilbert space), a 6 D (metric space), F is a multi from, H x D into H and has closed convex bounded images.Moreover F(.,a)is upper semicontinuous, F(z,.)is continuous in the sense of graph.Let H H1 x H2, H, is Hilbert 1,2.The following Cauchy problem: i() e (,, ( called singularly perturbed is also considered.For 0 one h 0 e F(z,y), z(0) x0 (1.3) The lt system is called reduced inclusion.The pair of AC x(.) and L-y(.) is a solution of (1.3), when (1.3) holds for a.e.t.Suppose F is one side Lipschitz on x we prove that the solution set Z(a) of (1.1) depends continuously on a in C(I,H).In the literature the continuous properties of Z(.) are studied when F(., a) is Lipschitz (in that ce f(.)is continuous).So our results are new also in ce of finite dimensional spies.For F(x, .) with convex graph the upper semicontinuous properties of the solution set of (1.2) are studied in [2].The lower semicontinuous properties of the lt set are studied in [3] under different type of hypotheses then thse of [2].The existnce T. DONCHEV AND V. ANGELOV of Lipschitz solution of (1.3) is proved in [4].Using refined version of the lemma of Plis, Veliov shows in [3] that the solution set of (1.2) is LSC at 0 + with respect to C(I,Rn) L2(I,R") topology.In both papers F is assumed to be Lipschitz.In our paper the Lipschitz continuity requirement of F is dispenced with.The LSC of the solution set for more general systems than (1.1) is investigated in [1] for one side Lipschitz F. However F is assumed to be continuous.When   F is only USC it is difficult to show the existence of solutions when F does not satisfy additional compactness hypotheses.Such a problem is considered in [5] when H" is uniformly convex Banach space.Here we use the techniques developed there (we generalise theorem of [5]).In section 2 we extend the well known lemma of Plis [6].In paragraph 3 as a trivial consequence of the refined version of the last lemma we show the continuous dependence of Z(.) on o for (1.1).We also obtain existence result for lower semicontinuous diffrential inclusions which do not satisfy any compactness conditions.In the last section using similar ideas as in [3] we prove the LSC dependence on e of the solution set of (1.2) at 0+.We note that the main results in the paper can be proved also for Banach H with uniformly convex dual H'. 2. PRELIMINARIES.
In the paper I := [0, T] (commonly T 1), H (for system (1.2) H H1 x H2) is a Hilbert space with scalar product < >, while a(x,A) is the support function supaeA < x,a >.The graph of the multi F H Pf(H) (Nonempty closed convex bounded subsets of H) is the set raphF := {(x,y) E H x H y E F(x)}.When this set is closed in g x H we say that F has a closed graph.We denote by d(z,A) inf{[z-y[" y A}.The Hausdorff distance is DH(A,B) := max{supeAd(a,B),supbesd(b,A)}.The multi F is called USC (LSC) at z, when to > 0 there exists > 0 such that F(x)+eU D F(y));(F(x) C F(y)+eV) whenever Ix-y[ <_ .
Here V {x "Ix _< 1}.The multi F from I x E into Pf(E)is said to be almost upper (lower)   semicontinuous (AUSC) if to e > 0 there exists Ic with meas(I\Ic) > such that F is USC (LSC) on/ x E. The Lipschitz function x with constant _< N will be callcd N-Lipschitz.For the system (1.2) we will use the following hypotheses: A1. F(.,.) is USC, closed convex valued bounded on bounded sets.A2. (One side Lipschitz condition) There exist positive constants L1, L2, L3, #.
If (xl,y),(x2, y2) _ H x H2 and f F(x,y), then there exists g _ F(x2, y2) such that: Here f" and f' are the projections of f on H1 and H2 respectively.
< Y Y2, f(Yx)-f(Y2) >< -,ulYx Y2I and V(.) is Lipschitz.If f(x) Az (f is linear) and H is finite dimensional then A2 is fulfilled, when the eigenvalues of the matrix A have negative real parts.Various prototypes of A2 are common in the singular perturbation literature.
REMARK.In view of proposition 2.1 we suppose [F(x,y)[ <_ M, since we consider only AC functions (x, y), satisfying the conditions of proposition 2.1.
The following lemma extend the well known lemma of Plis [6].Using similar arguments as in [5] we relax the continuity and Lipschitz assumptions of [6] and refine the estimation as well.
Obviously their cluster points x(.) and y(.) are solutions satisfying the conclusion of the lemma. QED.
The fashion however is the same and the proof is omitted.QED Fix c and consider the system (1.1) under the assumptions: C1.F(.)is USC closed convex valued bounded on the bounded sets.
Using lemma 2.1 and corollary 2.1 we will prove our main results, which are similar in the regularly and singularly perturbed case.
Let M be metric space and let the parameter c M. Suppose that C1, C2 hold uniformly on c.Let A C H be compact.Denote the restriction of F on A by FA and the solution set of (1.1)   by Z(c).The following theorem is valid.THEOREM :.1.If lim,t..GraphFA(.,aGraphFa(.,l)for every compact A C H in the sense of the Hausdorff distance, then Z(.)is LSC.I.e. to every solution z/ (.) of (1.1B) there exits a net x=(.)of solutions of (1.1a) such that x(.) converges uniformly to x(.) as o .M oreor if lim,,.,GraphF(., o) GraphF(.,), then Z(.) is continuous.
Consider also the discretized version of (3.1).
Using this result one can obtain interesting existence result for LSC differential inclusions.Corollary :.1.Let G be closed valued almost LSC multi satisfying the inequality of theorem 3.3.Denote F(t,x):= fq,>oclco{u: u G(t,y): lY x[ < e}.If F satisfies H1 then the following differential inclusion admits a solution (t) G(t,x), PROOF.Let N be as in the proof of theorem 3.2.From theorem 2 of [8] we know that there exists a F g+l continuous selection g(t,z) G(t,z).Recall that f(.,.) is called F g+l continuous at (t,x) when f(t,,x,) f(t,x) whenever Iz,-zl < (N + 1)(t,-t) and t, t.An obvios modification of the proof of theorem 6.1 of [9] shows the existence of solution of 5: g(t,x).QED.REMARK.The question of the approximation of the solution set of (1.1) is studied in [10] for general nonauthonomous systems.We note only that to the author's knowledge all the existence refults in the litherature use compactness conditions on G or the nonemptiness of the interior of clcoG(.,.).(see e.g. 9, 10 of [9]) 4. SINGULARLY PERTURBED CASE.
In this section we consider the differential inclusion (1.2).The next theorem shows the LSC dependence of Z(e) at 0 + with respect to C L2 topology.THEOREM 4.1.Suppose A1, A2 hold.Let (x,y) be solution of (1.3) and let y(.) be continuous.If r (0.1) and if is fixed then there exists e() such that to every < e() we have Ix(t)-x,(t)[ < and ly(t) y,(t)[L < 6 for some solution (x,,y,) of (1.2).
Obviously the solution set of this system is not LSC at 0, because the first inclusion is not Lipschitz.Consider however (t) -'/ + I + [0, ] (0) 0.
The solution set of last system is LSC since theorem 4.1 holds, however the right-hand side is not Lipschitz.This is true also for the first inclusion (without y(.) and without singular perturbation).In that case theorem 3.2 is valid.
As we have seen the LSC dependence on parameters in regulary and singulary perturbed ces can be investigated under the same approach.The USC dependence however can not.We give an example for system which is not USC at 0+. EXAMPLE 4.2.Consider the system (t) -+ (t) (0) 0, h e I-i, ].
We note that using the properties of the duality map j(.) (s theorem 3.3 for definition and [9] for the properties) one can prove similar results as theorem 3.1 and theorem 4.1 in case of uniformly convex Banach space H'.Using technique as in the proof of theorem 3.2 and by more carefull estimations one can obtain similar results also in case of nonautonomous system.