Nonconvex second-order differential inclusions with ... - Springer Link

Set-Valued Analysis 3: 7t-86, 1995. @ 1995 Kluwer Academic Publishers. Printed in the Netherlands.

71

N o n c o n v e x Second-Order Differential Inclusions with M e m o r y TRUONG XUAN DUC HA

Institute of Mathematics, PO Box 631 - Boho, 10000 Hanoi, Vietnam and M A N U E L D. P. M O N T E I R O M A R Q U E S *

Centro de Materndtica e AplicafSes Fundamentais (CMAF), Universidade de Lisboa Av. Prof. Gama Pinto, 2, 1699 Lisboa, Portugal (Received: 26 April 1994; revised: 26 September 1994) Abstract. We prove several existence theorems for the second-order differential inclusion of the form ~(t) E G(x(t)), ~c(t) E --NG(x(t))~c(t) +F(t,T(t)x) in the case when F or both G and F are maps with noncouvex values in an Euclidean or Hilbert space and F(t, T(t)x) is a memory term ([T(t)x](O) = x(t + 0)). Mathematics Subject Classifications (1991). 34A60, 35K22. Key words: differential inclusion, second-order, nonconvex sets, normal cones, memory.

0. Introduction The second-order differential inclusion

it(t) e

(0.1)

has been studied b y m a n y authors (e.g., [4, 5, 9, 12, 15, 16]). In particular, Gautier [12] considered the differential inclusion (0.i) with the constraint x ( t ) E X , where X is a closed subset o f a Banach space. Recently, Castaing proved some existence theorems for a differential inclusion o f the form 5~(t) E G(x(t)), it(t) E --NG(x(t))~(t ) with c o n v e x or n o n c o n v e x valued map G, where -Na(x(t))~c(t) denotes the (inward) Clarke's normal cone to G(x(t)) at 5:(t) E G(x(t)). The aim o f this paper is to establish several existence theorems for the secondorder differential inclusion o f the form

• a(x(t)),

(0.2)

it(t) • -Na(~(t))ic(t) + F ( t , T ( t ) x ) , in the Euclidean space I~d (or in a Hilbert space H ) when the map F or both maps F and G are nonconvex-valued, F ( t , T ( t ) x ) being a m e m o r y term * Corresponding author.

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T.X. DUC HA AND M. D. R MONTEIRO MARQUES

[T(t)x](O) = x(t + 0), for 0 E [--%0] where 7- > 0 is fixed. In other words, for a function x of time, T(t)x is, up to a translation, the restriction Xl[t_r,t]; precisely, it is the function on [-7-, 0] obtained by translation of x[[t-r,t]. To overcome the difficulties posed by nonconvexity, we use the original technique developed by Gamal in [10, 11] and the new approach proposed by Valadier [20] for the study of a differential inclusion of the form x(t) E C(t), ~(t) E -Nc(t)x(t) with nonconvex-valued map C. For a first order problem similar to (0.2), we may refer to [18, Chapter II]. In Section 1, we introduce the notation and formulate the problems. In Section 2, we solve the case of a convex-valued G with nonconvex-valued F. In Section 3, we solve the case when F and G are both nonconvex-valued (however the normal cone is then replaced by a regularization of its intersection with a ball). 1. Notation

Let I~d be the d-dimensional Euclidean space with the unit closed ball B = B(0, 1). Let A be a subset of ~d. Denote by IIAI[ the sup of llall with a E A, by 5"(., A) the support function of A, by d(x, A) the distance from x E ~d tO A, by NAX the Clarke's (outward) normal cone to A at x E A [8] and by ProjAx the set of proximal points {a E A: [la- xll = d ( x , A)}. For any two subsets A1, A2 of ]~a h(A1, A2) stands for the Hausdorff distance between A1 and A2. Denote as usual by ~([a, b],~ d) or simply by ~(a, b) the Banach space of continuous functions from [a, b] to ~a. Let 7- > 0 be given. By ~0 we mean the Banach space ~([-7-,0],]t~ a) with the norm given by I1¢110= max{ll¢(0)ll:

0

[-7-, 0]).

Let to /> 0, a > 0 be given scalars. For any t E [to, to + a] we define the map T(t) from ¢(to - 7., to + a) into ¢-o as follows: [T(t)x](O) = x(t + 0),

0 e [-T,o]. Let f~ be an open subset of I~d and G be a map from ~ into nonempty closed subsets of ~a. We associate to the map G the following maps F0, Fp and

¢~p(p >

0):

Fo(x, y) = -NC(x)y, if y E G(x),

and

Fo(x, y) = ~, otherwise;

r p ( x , y ) = ro(x,y)npB(O, 1), and ~)p is the smallest map containing Fp with nonempty convex closed values and a closed graph. Let (t0,(P0) E [O, oo[×~o and F be a map from [to, o0[×¢.o into nonempty subsets of ~ d Assume that xo = (po(O), Yo E G(xo). We shall be concerned with the following second-order differential inclusion and initial conditions:

T(to)x = ~o, ~(t) e a ( ~ ( t ) ) ;

~(to) = Yo;

(1.1) (1.2)

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NONCONVEX SECOND-ORDER DIFFERENTIAL INCLUSIONS

!i(t) E -Na(x(t))2(t) + F ( t , T ( t ) x ) .

(1.3)

In the case when G is nonconvex-valued, then, by technical masons (see [20]), instead of (1.3) we are forced to consider !i(t) E q~n(x(t),x(t)) + F ( t , T ( t ) x ) .

(1.3;)

Given a > 0 and a function x from [to - -r, to + a] into t~a, we say that x is a solution to the differential inclusion (1.1)-(1.3) (or to (1.1), (1.2) and (1.3~)) on [to, to + a] if there exists an absolutely continuous function y E ¢(to, to + a) such that: x ( t ) ~ zo + ft,lY(s) ds

( t E [to, to+a]);

y(to) = Yo,

(1.4)

(1.5) T(to)x = ~Po, (1.6) y(t) E G(x(t)), Vt E [to, to + a], for almost all t E [to, to + a], (1.7) ~1(t) 6 --NG(x(t))y(t) + F ( t , T ( t ) x ) , or:

~)(t) E O p ( x ( t ) , y ( t ) ) + F ( t , T ( t ) x ) ,

for almost all t E [to, to+a]. (1.7')

2. Existence Theorems for the Second-Order Differential Inclusion with Nonconvex-Valued Perturbation

In this section we study the differential inclusion (1.1)-(1.3). We have the following: THEOREM 2.1. Let f~ C Nd be an open subset and (to, ~o) E [0, c~[x¢o with xo := g)o(0) E ft. Let G: ~ ~ Nd and F: [to, c~[X¢o ~ 1~ be set-valued maps and Yo E G(xo). Assume that the following conditions are satisfied: (i) The function ~o is l-Lipschitz and satisfies, for some w > O,

~o(t) e ¢o(O) + ~ B c a,

vt e [-7, o];

(ii) G is an £,-Lipschitz map with nonempty convex compact values in ~B; (iii) For any a > O, b > O, F is a continuous map from [to, to + a] x Bb(~O) into nonempty compact subsets of Nd such that

sup {llF(t,¢)ll: (t,¢) E [to, to + aJ x Bb@o)} < ~ , where Bb(qoo) = {~b E ~o: I1¢ - ~ollo < b}. Then, if a E]0, w/~], there exist absolutely continuous functions x: [to-% to+ a] --* -0 and y: [to, to + a] --* 1~d such that (1.4)-(1.7) hold.

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T.X. DUC HA AND M, D, R MONTEIRO MARQUES

Proof. We adapt the technique of Gamal [10, 11] to the present problem. (1) Construction of the approximants. After a has been chosen in ]0, w/~], we can choose b/> a max {l, ~} and then take ff > 0 large enough to ensure that (by assumption (iii)) F([to, to + a] x Bb(~O)) C ~B.

(2.1)

Let J~ C Bb(CP0) be the compact set of functions whose Lipschitz constant is ~< max {t, ~}. It is clear that the restriction of F to [to, to + a] x J~ is uniformly continuous. Let e m = 2 -,'~ (m = 1,2,...). Since the restriction of F is uniformly continuous, then by [10, Lemma 1], there is a strictly decreasing sequence of positive numbers (era) converging to 0 as m ~ c~ such that a/em-i and e m - l / e m are integers/> 2 and

II(t~,C~)-(t2,C2)llR×~

1} is contained in K (by (2.6) and (2.12)), hence it is relatively compact in the norm of H. Thus (Ym) is relatively compact for uniform convergence. It follows from (2.20) that (Szm) is also relatively compact for uniform convergence. By integrating, the same can be said of (xm), since x,~(to) = xo for a][1 m. Finally, (zm) is again 'equioscillating' and we need to show that {Zm(t): m >/ 1} is a relatively strongly compact subset of H, for all t. Notice that z,~(t) belongs to era(t) := F(6m(t), T(Sm(t))Xm). If we define ¢(t) := F(t, T(t)x), which is a strongly compact set, then d(zm (t), ¢(t)) ~< h(¢,~ (t), ¢(t)) 0, as m --~ oo, by continuity of F. This implies the relative compactness of (zm(~)). Remark 2.4. Another improvement on the existence theorems, suggested by a referee, is the following. Let ~0 = ~ ( [ - % 0 ] , H ) and let us assume that F: [t0, oo[x¢~0 ~ H is a lower semicontinuous multifunction, whose values are nonempty closed subsets of a fixed strongly compact set K C H. As shown above, we need only consider the restriction of F to a compact subset [t0, t0 + a] x J~ of the Banach space N x ~0. Thus, by an obvious extension of Theorem 3 in [3] (which is an application of the technique of directionally continuous selections), there is an upper semicontinuous, compact convex-valued

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~. x. DUC HA AND M. D. P. MONTEIRO MARQUES

multifunction/~: [to, to + a] x J~ ~ ~-6 K such that every solution of the differential inclusion ii(t) • -Na(x(t))ic(t ) + F ( t , T ( t ) x ) ,

a.e.,

(2.30)

is also a solution of problem (0.2). The point is that (2.30) can be solved in a simpler way: since the perturbation F is convex-valued, the relatively weak compactness of (Zm) is sufficient and it does not require such a careful and technical construction.

3. Existence Theorem for a Second-Order Nonconvex-Valued Differential Inclusion with Nonconvex Constraint

In this section, the map G is no longer assumed to be convex-valued. In order to overcome this difficulty, we shall use the approach proposed by Valadier in .[20]. Let us recall the notation: F0(x, y) = - N a ( x ) y , if y • G(x), and P0(x, y) = 0, otherwise; Fp(x, y) = F0(x, y)M pB(O, 1); and q~p is the smallest map containing Fp with nonempty convex closed values and a closed graph. The main result on the existence of a local solution reads as follows: THEOREM 3.1. Let (to,~o) • [0, oc[×¢([-~-,0],I~ d) be given. Let G: -~ ~ Nd and F: [to, OO[X¢([--T,O],I~d) ~ I~d be set-valued maps and Yo • G(xo), where xo := cflo(0). Assume that." (i) (Po is l-Lipschitz and for some w > 0, ¢flo(t) • qOo(0) + w B C ft, Vt • [-% 0]; (ii) G is an £-Lipschitz map with nonempty compact values in ( B ; (iii) for any a, b > O, F is a continuous map from [to,to + a] × Bb(~O) into nonempty compact subsets of l~d such that sup{[IF(t,¢)lf: ( t , ¢ ) • [to, to+a] x Bb(990)} < oo. Then, for a •]0, w/~] and for large enough p > 0 (see (3.5), (3.7) below), there exists an absolutely continuous function x: [to - % to + a] --+ f~ which is a solution to (1.1), (1.2), (1.3~). That means the existence of an absolutely continuous function y: [to, to + a] --+ I~d such that:

x(t)=xo+

y(s)ds, Vt•[to, to+a]; y(to)=Yo,

T(to)x = ~o,

y(t) e a(x(t)),

(3.1) (3.2)

vt • [to, to + a],

(3.3)

~l(t) C q~p(x(t),y(t)) + F ( t , T ( t ) x ) , f o r almost all t e [to, to + a].(3.4) The set of such solutions x is closed in the uniform convergence on [to, to + a].

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NONCONVEX SECOND-ORDER DIFFERENTIAL INCLUSIONS

Proof. By assumption (iii), let a, b, ff be positive real numbers such that: a m a x {/,~}