One-sided continuous dependence of maximal solutions

JOURNAL

0:

DIFFERENTIAL

EQUATIONS

One-Sided

39, 413-425

(198 1)

Continuous Dependence Maximal Solutions*

of

ERIC SCHECHTER’ Mathematics

Research Center, University of Wisconsin, Madison, Wisconsin 53706

Received December 13, 1979; revised May 13, 1980

We consider the maximal solution of u’(s) =f(s, u(s)), where f satisfies a onesided variant of Caratheodory’s conditions. A best-possible condition is proved for the dependence of u on J Also we show that a function v satisfies v(t) - v(r) < j:f(s, v(s)) ds if and only if v is dominated by the maximal solution IL

1. INTRODUCTION In this paper we consider solutions ZJof the integral relations

u(t) - u(r) = j-‘f(s, u(s)) ds I

(a < r < t < T),

(l-2)

where T is strictly greater than a and may be 00, and the indicated integrals are finite, Lebesgue integrals. We assume that u is nonnegative and that J lRxR++lR+ is a function such that f(t, y) is locally integrable in t for each fixed y, and nondecreasing and rightcontinuous in y for almost every fixed t.

(1.3)

Here R, = [0, +a~). A solution of (1.2) is a solution of the differential equation u’(s) = f(s, U(S)) (a ,< s < 7) in the sense of Caratheodory; i.e., u is absolutely continuous on compact subsets of [a, r) and satisfies the differential * Sponsored by the United States Army under Contract DAAG29-75-C-0024. This material is based upon work supported by the National Science Foundation under Grant MCS78-09525. .’ Current address: Mathematics Department, Vanderbilt University, Nashville, Tennessee 37235.

413 0022.0396/8 l/O303 1 l-34$02.00/0 Copyright c‘ 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.

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ERIC SCHECHTER

equation almost everywhere on [a, ZJ. We shall consider (1.2) mainly as an initial value problem, i.e., with u(a) given. These conditions do not necessarily determine u uniquely. (Consider for instance u’(s) = (U(S))‘/*, u(a) = 0.) But the initial value problem does have a uniquely determined maximal, i.e., largest, solution. Most of the literature on the existence of Caratheodory solutions requires f(t, y) to be continuous in y, but not necessarily nondecreasing; see for instance [ 21 or 181. Wu [ 91 proves existence of solutions, and of maximal solutions, under assumptions slightly weaker than (1.3). But existence follows easily from (1.3) via some lemmas which we shall need for other purposes, and so for completeness we shall prove existence in Section 2. Note that the monotonicity condition in (1.3) runs the “wrong” way: solutions are not damped, and can blow up in finite time. (Consider for instance u’(s) = u(s)‘, u(a) = 1.) Nevertheless the monotonicity is a rather strong hypothesis. It makes comparison results easy (as in [S ]), and so makes possible the simple results presented below. The author suspects that the results of this paper can be extended to the more general setting of Wu [ 9 ], but at a cost of increased complexity. In Section 3 we shall show that a function satisfies (1.1) if and only if it grows no faster than the maximal solutions of u’(s) =f(s, U(S)). One reason for interest in such a characterization is the following: Let (X, (( ]I) be a Banach space, and let F be a time-dependent discontinuous nonlinear operator in X. Thus x’(s) = F(s, x(s)) may be an abstract representation of a nonlinear partial differential equation. For many purposes it is important to have at least a rough estimate of ]]x(s)]]. In some applications U(S) = ]Ix(s)l] satisfies (1.1) for some f. Replacing f by a larger function, we can often assume (1.3). Hence ]ix(s)i] is dominated by the maximal solutions of (1.2). This estimate is extremely crude, perhaps the worst possible. But it is very simple and general, and it is adequate for some applications, particularly in the theory of local existence of solutions. Moreover, it is amenable to some analysis by approximation, as indicated below and as shown in detail in Section 4. This type of estimate is applied to some nonlinear partial differential equations in [ 61. Our main result, in Section 4, is a special case of a more general principle, which we now describe in general terms. Consider functions f; not necessarily satisfying (1.3), not even necessarily scalar-valued, but satisfying some conditions which enable us to make sense of the initial value problem u’(s) = f(s, u(s))

(a,
u(a) = x. Write U(S) = u(s; a, x) to display the dependence on a and x. How does u vary when f is varied?

MAXIMAL

415

SOLUTIONS

In several different contexts (discussed below), the answer is roughly as follows: Let f,, fi, f, ,... and f, be such functions, and let u,, u2, u3 ,... and U, be the corresponding solutions. Then lim ~~(t; a, x) = u,(t; a, x)

for all initial

k *cc

times a, initial values X, and t > a,

(1.4)

if and only if

;h jdfk(h y>ds? jdfm& Y>ds

c with d > c.

for all d, c, y

-c

(1.5)

(We emphasize the universal quantifiers in (1.4) and (1.5). It does not seem to be possible to fix the initial time a in (1.4) and replace (1.5) with a simple corresponding condition.) In practical applications we are mainly interested in the fact that (1.5) implies (1.4); the converse is of interest because it tells us that the practical part cannot be improved. In applications the functions f are more accessible than the solutions U, and (1.5) often can be verified directly. Condition (1.5) is particularly interesting when the functions f, oscillate rapidly. For instance, let fk(s, x) = sin(ks), f,(s, x) = 0. Then (1.5) holds, but fk does not converge pointwise to f,. The equivalence of (1.4) and (1.5) says that u is affected by the average values off, not the instantaneous values. The equivalence of (1.4) and (1.5) was proved by Artstein [ 11, under an assumption that the functions f are uniformly integrable in their first argument and Lipschitz in their second argument. Piccinini [4] weakened the Lipschitz condition to a uniform continuity condition. Piccinini refers to the above convergences as G-convergence and relates this to the notion of rconvergence developed for partial differential equations by DeGiorgi [3]. Stassinopoulos and Vinter [ 71 proved a version of the equivalence principle applicable to differential inclusions u’(s) E f (s, U(S)), where f is multivalued and u is not uniquely determined. Again, uniform integrability and Lipschitz conditions (in a sense) are assumed. All these results are in finitedimensional settings and so compactness is readily available, but it is not essential. In [S] the author shows that (1.5) implies (1.4), with f, x, and u taking values in an arbitrary Banach space. The functions f are assumed to be continuous and dissipative in their second argument, but not necessarily uniformly continuous. (The dissipativeness condition says, roughly, that f(s, y) is a decreasing function of y.) All the above results have complicated hypotheses which may obscure the underlying equivalence principle. In Section 4 we present a very simple

416

ERIC

SCHECHTER

version of the principle. The functions f must satisfy (1.3), but they need not be uniformly integrable in their first argument, nor continuous in their second argument. We pay for this weakness of hypotheses, however: we must replace the lim’s and equalities in (1.4) and (1.5) with lim sup’s and inequalities. We obtain a one-sided version of the equivalence principle. We note that in some of their multi-valued results, Stassinopoulos and Vinter [7] replace (1.5) with a condition of roughly the following sort:

lim ,/A fk@,Y>ds= 1 f&, Y>ds

k-m

for all y and

A

all measurable sets A.

(1.6)

In other words, fk + f, in the weak topology of the Banach space L’. Obviously (1.6) implies (1.5); the two conditions are equivalent when the functionsf, are uniformly integrable. In the following example (suggested by P. Billingsley), (1.5) holds but (1.6) does not. For k = 1,2,3 ,..., let A, = *k-l, 2-kj]:j= 1, 2)..., 2k}, let I, be the characteristic function lJ{[2-kj-2of A,, and letfk(s) = 2 k+‘lk(.s). Let A be the union of the A,‘s, and let f, be the characteristic function of the interval [0, 11. Then (, fk(s) ds = 1 for all k, but s, f,(s) ds < j, so (1.6) does not hold. The reader can verify that (1.5) does hold, and that such functions fk, f, do satisfy the hypotheses of Section 4 of the present paper.

2. NOTATIONS AND TECHNICAL LEMMAS LEMMA 1. Suppose that [ p, Q) -+ R + satisfies

e: R x R, -+ R,

u(t) - v(p) < (’ e(s, u(s)) ds < 03 P

satisfies

(1.3),

and

U:

(p
Assume that at least one of Q, lim suptTp v(t) is co. Also assume that h > 0, q > p are numbers satisfying fi e(s, v(p) + h) ds < h. Then p < q < Q, and v(t) < v(p) + h for all t in [p, q]. Proof. The proof is standard but brief so we include it for completeness. The Lebesgue integral ji e(s, u(s)) ds is a continuous function of t which vanishes at t = p, so u(t) - v(p) < Iie(s, U(S)) ds < h for all t in some interval [p, z) of positive length. By choosing z as large as possible we may assume that either (i) (ii)

z = Q, and u(t) - v(p) < h for all t in [ p, Q), or z < Q, and J‘; e(s, U(S)) ds = h.

MAXIMAL

417

SOLUTIONS

In case (i) we have lim sup,,c v(t) < u(p) + h < co, so Q = co. Then the conclusions of the lemma follow. In case (ii), it suffices to show q < z. If q > z then Jf e(s, u(s)) ds < J”; e(s, V(P) + h) ds < h, a contradiction. This completes the proof of the lemma.

Notation 1. Let f satisfying (1.3), a E R, x E R + , and E > 0 be given. We shall specify an approximate solution w of

u’(s) = f(s, u(s))

(a
(2.1)

u(a) =x, having certain convenient properties which will be used later. Let i = i(x, E) be the positive integer determined by

*+C’x
(2.2)

Let ti = a, and define (2.3) recursively by

‘j [c+f(s,jc)]ds=~(j=i+ I

l,i+2,...);

U=syptj.

(2.4)

lj-I

This determines (2.3) uniquely integrable. Let w(t)=

(j-

l)& + ’ I

since f(., js) is nonnegative

[E +f(s,je)l

ds

(tj-

1< t <

and locally

tj)*

(2.5 1

tj-l

We leave it to the reader to verify that w is strictly increasing, w(tj) = js (j = i, i + 1, i + 2,...), w(t) + co as t increases to U, and w is the unique solution of the initial value problem w’(s) = E +f(s, E + &[&-‘w(s)l) (a < s < U), P.6)

w(a) = 3~ + E([E-~x~,

where [rn is the greatest integer less than or equal to r. Later we shall show that for E = 2-“, the corresponding sequence w, decreases to the maximal solution of (2.1). The main advantage of this approximation over those in [2, 8, 91 is that in (2.6) the second argument of f is a step-function of s. This is crucial to the proof of the theorem in Section 4.

418

ERIC SCHECHTER

LEMMA 2. Let f satisfying (1.3), a E I?, x E I?,, and E > 0 be given. DeJine i, {tj}, U, and w as in (2.2~(2.6). Let Z be some integer greater than i. Let e: RXR+--+R+ be another function satisfying (1.3), and let v: [a, Q) -+ R + be a solution of

v(t) - v(r) < 1’ e(s, v(s)) ds r

(a
Suppose that at least one of Q, hm sUpITov(t) is co. Also assume that v(a) + E < w(a), and that

*itIe(s, je)

Ilj

ds < E

(i < j < I).

Then tt < Q, and v < w everywhere in [a, t,].

tj

< Q and v(tj) < Proof: As an induction hypothesis, assume that (j - 1)s for some j < I. (This is clear for j = i, since = a and w(tj) = ie.) Then

ti

fj+l I

Ij

e(s, V(tj) + E) ds <

tjtl I

e(s, je) ds < E.

Ii

By Lemma 1, therefore, tit, < Q, and v(t) < v(tj) + E < je = w(tj) < w(t) for all t in [tj, tj+,]. This completes the induction, and the proof of the lemma. Notation 2. Let f satisfying (1.3), a E I?, and x E R + be given. Suppose, for the moment, that at least one solution of (2.1) exists for some T > a. A solution ur: [a, T,) + R + is a continuation (to the right) of another solution u2: [a, T2) + R + if T, < T, and u, = u2 on [a, Tz). We shall omit the words “to the right” since this is the only type of continuation which will concern us. A solution u: [a, T) + R + is noncontinuable if it has no proper continuation; then we shall call T the jinal time for u. By Zorn’s Lemma, every solution of (2.1) has a noncontinuable continuation. A noncontinuable solution m: [a, T) + R, is maximal if for all other noncontinuable solutions v: [a,Q)-I?+ wehaveavon [a,T).Obviously(2.1)has at most one maximal solution. Note that since f is nonnegative, any solution u(t) of (2.1) increases to a limit (possibly 00) when t increases to T. LEMMA 3. Let f satisfying (1.3), a E R, x E R, be given. Then (2.1) has at least one solution for some T > a. A solution v: [a, Q) + R + of (2.1) is noncontinuable tf and only if at least one of Q, lim,tp v(t) is 00. Among the

MAXIMAL

419

SOLUTIONS

noncontinuable solutions there exists a maximal solution m: [a, T)-+ R + . This maximal solution can be approximated as follows: For n = 1, 2, 3,..., let w,: [a, U,)+ R, be the approximate solution defined in (2.2~(2.6), using E= 2-“. Then the numbers U, increase to T, and w,,(t) decreasesto m(t) uniformly on compact subsetsof [a, T). Proof. Much of this proof is standard, so we shall only sketch it. Lemma 2 is applicable with w = w,, u = w,+ i. Hence U, + i > U,, and < w, on [a, U,). Therefore the numbers U,, increase to a limit T o
w,(t) - w,(r) = [’ { 2-” + f(s, 2-” + 22”[2”w,(s)])

(a < r < t < U,). Fix r and t, and take limits as n -+ co. By the Dominated Convergence Theorem,

m(t) - m(r) = ~‘J(s, m(s)) ds I

(a < r < t < T);

i.e., m is a solution of (2.1). Suppose that T and ME lim,lT m(t) are both finite. Fix some 6 > 0 small enough so that

I=

f&M+

2)ds < 1.

T-S

Let e(s, y) = 2-” +f(s, 2-” + 2-“[2”y]). U,>T--&w,(T-6)
I

T

e(s, w,(T-6)+

For n sufficiently

large we have

T

l)ds< I T-6

T-6

e(s, M + 2) ds < 1.

Apply Lemma 1 with v = w,, [ p, Q) = [T - 6, U,), h = 1, q = T. It follows that T < U,, a contradiction. So at least one of T, limtTT m(t) is co. Now let u: [a, Q)- R, be any solution of (2.1), and let L = lim,?c v(t). First suppose Q and L are both finite. By the preceding construction, there exists at least one solution of

z’(t) = f(t, z(t)) z(Q) = L

(Q < t < Q + ~1,

420

ERIC SCHECHTER

for some E > 0. Combine z and u to obtain a proper continuation of u. Conversely, suppose at least one of Q, L is co. Then obviously u is noncontinuable. To show that m is maximal, let U: [a, Q) + R, be any other noncontinuable solution of (2.1). Apply Lemma 2 with e = f and E = 2-“; then Q > U,, and u < w, on [a, 17,). Now let n * co. It follows that Q > T, and c’ < m on [a, 7’). This completes the proof of Lemma 3.

3. CHARACTERIZATION

OF SOLUTIONS OF THE INTEGRAL INEQUALITY

A function satisfies (1.1) if and only if, after every initial dominated by the maximal solution of (1.2). More precisely:

time, it is

THEOREM. Let f be a function satisfying (1.3), and let v: [a,,, a~) -+ IR, be a function which is bounded on compact subsets of [a,, 00). For each initial time a in [a,, co), let m(t) = m(t, a) be the maximal solution of (2.1) with initial value x = v(a), and let T(a) be its final time. Then:

v is measurable, and v(t) - u(r) <

fora,
co

(3.1)

if and only if

for every initial time a in [a,, a~), v(a) < m(., a) everywhere on [a, T(a)). Moreover, if (3.1)-(3.2) hold, then v has bounded variation subintervals of [a,, 00).

(3.2) on compact

Remark. The theorem is also applicable for functions v defined on a bounded interval [a,, b,) or [a,, b,], since we can define v = 0 everywhere to the right of that interval. Proof of theorem. The relation (3.1)+ (3.2) is standard; for completeness we briefly sketch a proof. Fix any a > a, and E > 0. Define w: [a, U) + R, as in (2.2)-(2.6), with x = v(a). Apply Lemma 2 with e =S; thus v < w. Now Lemma 3 yields u < m and thus (3.2). For the converse, assume (3.2). Fix any r and t with a, < r Q t < co. We shall use (3.2) to show v has bounded variation on [r, t]; then we shall use this fact together with (3.2) to prove u(t) - v(r) < s: f(s, v(s)) ds.

MAXIMAL

421

SOLUTIONS

By hypothesis, ME sup{u(s): r < s < t) is finite. Since f(., integrable on [r, t], there is some 6 > 0 such that

Iclbf(s,M+

1)ds < 1

A4 + 1) is

if r
It follows by Lemma 1 (applied to m(., a) rather than U) that if r < a < b < t and b - a < 6, then and

a < b < T(a)

m(b, a) < v(a) + 1 GM + 1,

(3.3)

hence v(b) - v(a) < m(b, a) - u(a) = jb f(s, m(s, a)) ds a

(3.4)

< bf(~,M+‘l)d~. Ia For real numbers w let w+ = max { W,0). Let any partition r = so < s, < s2 < ... < si = t

(35)

of [r, t ] be given. Choose a refinement r=sl,
\‘ Iu(si) - u(siI)1
-i. sif(s,M+

+ u(sb) - u(sb)

l)ds+M+O

'i-1

=2

‘f(s,M+ Ir

I)ds+M

using (3.4). The last expression is independent of the choice of partition (3.5), so 21has bounded variation on [r, t]. Therefore u has at most countably many discontinuities in [r, t], each discontinuity is a jump, and the magnitudes of the jumps are summable. Hence for each positive integer n, the set A, = {s E (r, t): Iu(s+) - u(s)1 > 2-” or ]v(s-) - u(s)1 > 2-“}

422

ERIC SCHECHTER

is finite. The sets A,, form an increasing sequence, and v is continuous at every point in (r, t)\lJ,“, A,,. The sets B, = {r + 2-“k(t - r): k = 0, 1, 2,..., 2”) also form an increasing sequence of finite sets; hence so do the sets C,=A.uB,. Temporarily fix any positive integer it > log,((t - r)/6). Suppose C, consists of the points C,:r=s,

(3.6)

Then si-siPi, < 2-“(t - r) < 6, so T(si- i) > si by (3.3). Therefore we can define an approximate solution w, by taking w,(.) = m(., siPi) on each interval [sip,, si), and w,,(t) = u(t). It follows from (3.2) and (3.3) that u(si) = w,(si) < w,(si-) = m(si, sip ,) for i = 1, 2,..., j; and also that u < w, ,< M + 1 everywhere on [r, t]. By (3.2~(3.4), also, W”(S) - U(Si- I> = JS f(s, si-I

w,(q))

&

Q I” f(q, si-l

M + 1) &

(si-l U(t)

-

u(r)

=

C

I”Csi)

-

u(si-

111 GC

lwn(si-)

< s < si), -

wn(si-

(3.7)

111 (3.8)

= ’ f(s, w,(s)) ds. Ir Now suppose that n > 1 + log,((t - r)/6). Then the preceding analysis is applicable to both w, and w,-i, and C,-, is a subset of C,. Let si-,, si be two consecutive points in C,. Then w,(si- i) = v(si- i) < w,- i(si- i). On the interval [si-i, sJ, both w, and w,,- i may be viewed as maximal solutions of w’(s) = f(s, w(s)), with initial values w,(si- ,) and w, _ ,(si- ,), respectively. Hence w, < w,-, on [si-i, sJ. This holds for all i, so w, < wnml on [r, t]. Thus

w, > w,+1> w,+z> ***> u

on [r, t],

W)

and w,(s) = V(S) for all s in C,. We wish to show that {w,(s)} decreases to u(s) for all s in [r, t]. This is clear for every s in U,“=, C,. Fix any s in [r, t]\U.“=l C,. Then u(e) is continuous at s. Temporarily fix some large n, and let C, be as in (3.6). Chooseisothatsi-1
423

MAXIMAL SOLUTIONS

Now take limits in (3.8). By the Dominated Convergence Theorem, u(r) - u(r) < j:f(s, U(S))ds. This completes the proof of the theorem. 4. ONE-SIDED LIMITS OF MAXIMAL

SOLUTIONS

For any f satisfying (1.3) any initial time a E R, and any initial value XE R+, let m(t) = m(t;f, a, x) be the maximal solution of (2.1), and let T(f, a, x) be its final time. THEOREM. Let (fk} be a sequence (or more generally, a net) of functions satisfying (1.3), and let f, be another such function. Then

foreueryaEiRandxEiR+,

lim inf T(fk, a, x) > T(f,,

a, x); and

k-co

liy s,p m(t; fk, a, x) < m(t; f,,

a, x) for every

t in [a, T(f, , a, x)),

(4.1)

if and only f

liy s,upIdf&(s, y) ds
Y) dsfor all

d, c, y ECIR with d > c aid y 2 0.

(4.2)

Remark. As noted in Lemma 3, if T(f, a,~) < co then m(t;f, a, x) increasesto 00 when t increasesto T(f, a, x). Hence it is reasonable to define m(l; f, a, x) = co for all I > T(f, a, x). With this convention, (4.1) can be restated more simply as follows:

liy * zp m(t; f&, a9x) < m(t; f,, with t>a

a, x) for all t, c, x E R

and x20.

Proof of theorem. First we show that (4.2) implies (4.1). Fix any a E R, xER+, and E > 0. Define i, (fj), and w: [a, 17)+ R, as in (2.2)-(2.6), with f = f,. Fix any integer Z > i. By (4.2) and (2.4), for k sufficiently large we have

Itj““fk(s,jC)

ds < (tj+l-

tj)e + /f’“fa(S,j&) dS
424

ERIC SCHECHTER

By Lemma 2, therefore, tI < T(fk, a, x); and m(t; fk, a, x) < w(t) for all t in [a, t,]. Let k + co and then I-+ co ; this proves that U < lim inf T(‘&, a, x) and that lim sup m(t; fk, a, x) < w(t) for all t in [a, u>. Now let E = 2 -’ 1 0; then (4.1) follows by Lemma 3. The converse takes a bit longer. Suppose that (4.1) holds, but lim sup J”ffk(s, y) ds > Jf f,,(s, y) ds for some numbers y > 0 and d > c. Since sz fm(s, .) ds is right-continuous, we have in fact lim sup J”: fk(s, y) ds > Jf fm(s, y + a) ds for some E > 0. Partition the interval [c, d] into n pieces [a, b] of length (d - c)/n. For n large enough, all of the pieces must satisfy f&

y + E) ds < E.

(4.3)

On the other hand, at least one of the pieces must satisfy

lim sup Ibf& a

v) ds > jb fm(s, Y + E) ds. a

(4.4)

Fix this choice of a and 6. Taking initial time a and initial value y, we consider the maximal solutions m,(t) = m(t; fk, a, Y) and m,(t) = m(t; f,, a, y). By hypothesis (4.1) we have lim inf T(fk, a, Y) > T(f,, a, Y) and lim sup m,(t) ,< m,(t)

for all t in [a, T(f,,

By (4.3) and Lemma 1 we have T(f,, in [a, b]. Hence

a, y)).

(45)

a, y) > b and m,(s) < y + E for all s

+ E) ds.

(4.6)

Since lim inf T(fk, a, y) > T(f, , a, y) > 6, for all k sufficiently have T(fk, a, y) > b and

large we

m,(b) - Y =Ibf&, (I

m,(s)) ds < j”f-‘s, (2

m,(b) - Y = I” f,Js, q(s)) LI

ds > lb fh, a

Y

y) ds.

(4.7)

Substitute t = b in (4.5), and combine with (4.4), (4.6), and (4.7). This yields Hence (4.1) implies (4.2), m,(b) -y < m,(b) - y, a contradiction. completing the proof of the theorem.

MAXIMAL SOLUTIONS

425

ACKNOWLEDGMENT The author suggestions.

is grateful

to C. Dafermos

and the referee and others for many helpful

REFERENCES I. 2. ARTSTEIN, Continuous dependence on parameters: On the best possible results, J. Differential Equations 19 (1975), 214-225. 2. E. A. CODDING~ON AND N. LEVINSON, “Theory of Ordinary Differential Equations,” McGraw-Hill. New York, 1955. 3. E. DEGIORGI, f-convergenza e G-convergenza, Boll. Un. Mat. Ital. 5, 14-A (1977), 2 13-220. 4. L. PICCININI, G-convergence for ordinary differential equations, in “Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis” (E. DeGiorgi et al., Eds.), Pitagora Editrice, Bologna, 1979. 5. E. SCHECHTER, Existence and limits of Carathtodory-Martin evolutions, J. Nonlin. Anal., Theory, Methods, Applicns., to appear. 6. E. SCHECHTER, in preparation. 7. G. 1. STASSINOPOULOS AND R. B. VINTER, Continuous dependence of solutions of a differential inclusion on the right hand side with applications to stability of optimal control problems, SIAM J. Con. Optimization 17 (1979), 432-449. 8. W. WALTER. “Differential and Integral Inequalities,” Springer-Verlag, Berlin/New York, 1970. 9. Z. WU, The ordinary differential equations with discontinuous right members and the discontinuous solutions of the quasilinear partial differential equations, Sci. Sinica 13 (1964), 1901-1917.