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On integral solutions on nonautonomous delay equations and their propagation
Nonlinear Analysis, Theory, Methods & Applications, Vol. 24, No. 5, pp. 693-710, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/95 $9.50 + .00
Pergamon 0362-546X(94)00094-8
ON INTEGRAL SOLUTIONS ON NONAUTONOMOUS EQUATIONS AND THEIR PROPAGATION
DELAY
JANET DYSON 1 and ROSANNA VILLELLA-BRESSAN~; Mansfield College, Oxford, U.K.; and ~:Dipartimento di Matematica Pura e Applicata, Universita' di Padova, Padova, Italy (Received 1 December 1991; received in revisedform 23 September 1993; receivedfor publication 4 May 1994) Key words and phrases: Nonautonomous functional differential equations, dissipative operators, integral solution, propagation of solutions.
0. I N T R O D U C T I O N
We consider integral solutions of nonautonomous delay equations. We establish some properties of such solutions and look at their relationship to other types of solution. We then discuss their propagation. The equation under consideration is (FDE)
Jc(t) ~ f ( t , x(t)) + g(t, xt), x ( s ) = h,
0 <_ s <_ t <_ T, x,=~,
set in a Banach space X. Here xt ~ L 1 = L l ( - r , 0; X) is defined pointwise by xt(O) = x(t + 0), r, 0 < r < 00, being the delay and I $ , h ] e L l x X . There is an tx such that f - t x is m-dissipative, and f and g satisfy certain smoothness conditions. When discussing propagation, we shall confine our attention to the semilinear case, however, our discussion of integral solutions relates to the full nonlinear case. The concept of integral solution was first introduced by Benilan [1, 2] for autonomous evolution equations without delay and has since been exploited and extended by many authors; see for example [3, 4]. It provides a unification of the various different notions of solution. Thus, if we consider the evolution equation
(E)k
du dt ~ A ( t ) u + k(t),
0 <- s <- t <- T,
u(s) = ¢~,
set in a Banach space Y, under suitable conditions a strong solution of (E)k will be an integral solution, so too will a discrete scheme limit solution and, in the linear case, a mild solution. Further, if the integral solution is differentiable for some t then it will satisfy (E)k in the classical sense for that t. We start with the evolution equation without delay, (E)k, and give a partial extension of the concept of integral solution discussed by Pavel in [4]. The conditions set on A ( t ) are those first used by Crandall and Pazy, and by Evans [5-7] and they allow operators A ( t ) of the form h(t)A, where A - to is m-dissipative and also allow operators A ( t ) where D(A(t)) depends on t. We denote the unique integral solution of (E)k by Ua+k(t, s)cb. UA+k is then an evolution operator. 693
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J. DYSON and R. VILLELLA-BRESSAN
This definition of integral solution is then extended to functional differential equations. It is well known [8-11 and others] that (FDE) is closely related to the evolution equation (E)
du dt ~ A(t)u,
0 <_ s <_ t < T,
u(s) = 14~,h} ~ L 1 × X , set in the space of initial data where A(t) is defined by D(A(t)) = [{$, hi ~ L 1 × X ; $ ~ w l ' ~ ( - r , 0 ; X ) , 4~(0) = h ~ D ( f ( t , "))1, A(t)[4~, hi = [$',f(t, h) + g(t, $)}. We show that the projection of a discrete scheme limit solution of (E) onto X is an integral solution of (FDE) and thus, we are able to extend the notion of discrete scheme limit solution to delay equations, and to relate the integral solutions of the two problems. We find that (FDE) has a unique integral solution and that if the integral solution is differentiable for some t then it satisfies (FDE) in the classical sense for that t. The relationship between (E) and (FDE) also enables us, in the semilinear case, to establish a variation of parameters formula. Using this we are able to produce results relating the propagation of solutions of (FDE) to the propagation of solutions of the same equation without delay. We shall say that an evolution operator U(t, r), in a Banach space Y, propagates a family of subsets of Y, IXt]t>s, for t _> s, if U(t, r)X, C Xt for t _> r _> s. This notion can be used, for example, to express the fact that the wave equation has finite speed of propagation (see for example [12]). We start with a set of subsets [X,I of X, propagated by U / a n d define a related set of subsets IY,I o f L ~ x X w h i c h are propagated by UA and satisfy n2Yt C Xt. Here rtl and n2 are the projections of L 1 × X onto L ~ and X, respectively. Roughly speaking, rq Yt consists of segments of continuous functions q/such that ~,(t) e X t for all t _> s. Only if X t varies continuously with t in some sense do we have n2 Yt = X t . We also give a converse result which shows that, in certain circumstances, if UA propagates [Yt] then U/must propagate [Xt]. These propagation results extend the results of [12, 13] for the autonomous, semilinear and nonlinear cases. Propagation for other types of delay equations has been discussed by Grimmer and Zeman, and by Desch and Grimmer in [14, 15]. 1. I N T E G R A L S O L U T I O N S OF N O N L I N E A R E V O L U T I O N E Q U A T I O N S
Here we consider the nonlinear, nonautonomous evolution equation (E)k
du d--t ~ A(t)u + k(t),
0 <_ s <_ t <_ T,
u(s) = ¢
set in a Banach space Ywith norm II" [I- A(t): D(A(t)) C Y ~ 2 v is a time dependent (possibly multivalued) nonlinear operator with time dependent domain D(A(t)). We give a partial generalization of the definition of integral solution given by Pavel in [4] and discuss its relation to limit solutions. We shall assume throughout that A(t) satisfies (M)
there is a 20 > 0 such that R ( I - 2A(t)) = Y for 0 < ;t < 2 o.
Nonautonomous delay equations
695
We also set the condition (A.1) for all [ x l , y l ] c A ( t ) and [x2,y 2] c A ( r ) there is an co >_ 0 and an h: [0, T] --, Y which is continuous and of bounded variation, and a m o n o t o n e increasing function LA: [0, oo) ~ [0, o0) such that (Yl
-
x211 +
Y2,XI - X2)- -< collXl -
(1 +
IlY2ll)tA(llx211)llh(t)
- h(0ll.
Recall that IIx + hell - Ilxll htO h
( y , x ) = lim
The notation [x,y] c A ( t ) means x c D ( A ( t ) ) and y ~ A ( t ) x . Then, following [4, Chapter 1, proposition 2.2], it can be seen that (A. 1) is equivalent to the condition (A.2) for all [ g l , Yd c A ( t ) and [x2, Y2] c A(r) there is an co ___ 0, and functions h and L A as in (A. 1) such that for X > 0 (1 - ~co)llx~ - x211 ~ IIx, - x~ - X(y, - y2)ll + A(1 + Ilyell)LAIIx21l)llh(t)
-
h(Ol[.
Also, if we set Jx(t) = ( / - hA(t)) -1 and Ax(t) = (1/A)(Jx(t) - I ) , it will be seen that (A.2) is equivalent to condition (A.3) which for x~ = x2 is condition (C.2) in [6, 7]. (A.3) There exist ~0 > 0, co >_ 0, a function h as in (A.I) and a function LA : [0, co) ~ [0, oo) which is m o n o t o n e increasing and continuous such that for all Xl, x2 c Y and 0 < g < go 1
- x211 + AIIh(t) - h(OIIZ,~(llx~ll)(1 + IIAx(r)x211).
IIJ×(t)x~ - J×(r)x211 <- --IIx, 1 - ACO
To prove that (A.2) implies (A.3), in (A.2) set Xl = J x ( t ) u l , x2 = / x ( ~ ' ) / / 2 , Yl = A×(t)ul and Y2 = Ax(r)u2 to get
IIJx(t)ul -/x(0u211 < -
1
Ilu, - u211
1 -~.co
+ - I[h(t) - h(OllzA(l[Jx(OU2ll)(1 1 - ACO
+ IIAx(0u2[I).
Showing that there is a n / ' A such that (1/(1 - Aco))La(llJ×(Ou~ll) is bounded by LA(Ilu~II) is now straightforward. To prove the converse, given [Ul, vl] e A ( t ) a~d [u2, v2] e A(r) we let xi = ui - Avi, i = 1,2 in (A.3) and rearrange to get 1 -~(llul
-
u~ -
~(Vl
-
v~)ll
collul - u211 + Ilh(t)
-
-
Ilul -
u211)
h(0tlLA(llu2
(A.1) and hence (A.2), now follows by letting A -:, 0.
- Aw21D0 + IIv21D.
696
J. DYSON and R. VILLELLA-BRESSAN
Note also that, since IlZx(t)xll < - -
1 -
1 ),to)
(A.3) also implies condition (C.2) of [5]. Following [5-7, 16] we define a set DA(A(t))
limlla~(r)xll, ~,0
C Y by
D^(A(t))=Ix~D(A(t));limllAx(t)xl[<°° " x ~ o 1 Note that, as in [5], if A(t) satisfies (A.3) then DA(A(t)) = D ^ is independent of t. Thus, as D(A(t)) C DA(A(t)) C D(A(t)), D(A(t)) = b is also independent o f t. We shall now define an integral solution of (E) k as follows. (I.1) A continuous function u is an integralsolution of (E) k on [s, T] if u(s) = 4, and there is an to > 0 and a function hi" [0, T] --, X w h i c h is continuous and of bounded variation, and a monotone increasing function L1 : [0, oo) ~ [0, oo) such that for all s _ r _< t ___ T, r e [s, T] and [x, y] e A(r) Ilu(t) - xll ---
f'
[lu(r) - xll
( t o l l u ( a ) - xll + + + C l l h l ( a ) - hl(r)lll
do',
7
w h e r e C = (1 + Ilyll)L~(llxll).
Recall that (y, x)+ = lim h~0
We will see that if
IIx + hyll -
IIxll
h
A(t) satisfies condition (A.1) then hi can be taken as h and L 1 as L.4.
This definition generalizes that given by Pavel in [4] in that C now depends on r. Most proofs carry through with little modification. Difficulties occur when proving uniqueness, but these can be overcome by requiring 4~ e/Y~ and using a suitable D-S approximation. There is a possible inconsistency in that, instead of treating (E)k as a nonhomogeneous problem, we could have considered it as the homogeneous problem (E)o
du d---te
C(t),
u(s) = ¢b,
where C(t) = A(t) + k(t). We will see that C(t) also satisfies (M) and (A. 1) (for appropriate h), but using C(t) gives rise to a different definition of integral solution, (1.2) below. These definitions are, however, equivalent provided h and k are of bounded variation. LE~ 1. Suppose that k(t) is continuous and of bounded variation. Then if and only if C(t) satisfies this condition for appropriate h and LA.
A(t) satisfies (A.2)
Nonautonomous delay equations
697
Proof. Suppose that A ( t ) satisfies (A.2) so that for [Xl, Yl] e C(t) and [x2, Y2] E C(z), (1
Z~)ilx,
-
- x=ll - IIx, -
xz -
A(y,
+ x(1 + IIc(m=ll
- yz)ll + AIIk(t)
- k(r)ll
+ Ilk(r)ll)LA(llx2l[)llh(t)
-
h(r)[I.
Set Tk(t) = sup E Ilk(ti) - k(ti-1)ll, where the sup is taken over all possible partitions o f [s, t]. Then for t > r, Ilk(t) - k(r)[I -< TAt) - TAr). Define Th analogously. Then if we define h(t) = TAt) + Th(t), h(t) is continuous and o f b o u n d e d variation, Ilh(t) - h(r)l[ + Ilk(t) + k(OIl Ih(t) - h(r)[, and the result follows with L c = 1 + [IkllooLA + LA. Note also that A ( t ) satisfies (M) if and only if C(t) satisfies (M). This now leads to an alternative definition o f integral solution. (I.2) A continuous function u is an integral solution of (E)k in [s, T] if u(s) = 4~, and there is an o~ >_ 0 and a function h2: [0, T] --, X which is continuous and o f b o u n d e d variation, and a m o n o t o n e increasing function L2 : [0, oo) ~ [0, oo) such that for all s ___ r ___ t <_ T, r e [s, T] and Ix, y] e A(r) Ilu(t)
-
_
xll
-
Ilu(v)
f' I,ollu(o)
-
xll
- xll + ( y + k ( r ) , u ( a ) - x ) + + C l l h E ( a )
- hz(r)llJda,
7"
where C = (1 +
IlYll)Lz(llxll).
H o w e v e r , we have the following lemma. LEMMA 2. Let k be continuous and o f b o u n d e d variation. Then u satisfies (I.1) if and only if it satisfies (1.2). Proof. Let u satisfy (I.1). We have ( y + k(a), u(a) - x)+ < ( y + k(r), u(a) - x)+ + Ilk(a) - k(r)ll. The p r o o f n o w follows as in l e m m a 1. We shall n o w relate the notion of integral solution to that o f limit solution. Following m a n y other authors, we say that u,(t) is a D.S. (discrete scheme) approximate solution o f (E)k if it satisfies the following. Let s = t~ < t~' < ... < tTv, = T be a partition of [s, T]. Set u.(t) = xg
t = t~ = s t e ( t ?1k - l , ttk],
= x~
k = 1. . . . . N , ,
where x~ ~ D(A(t~)), t~
-
"
tk_ 1
k~ ~ A ( t ~ ) x ~ ,
698
J. DYSON and R. VILLELLA-BRESSAN
d. = max[t~ - t~,_ll -~ O, x~ ~ ~b as n ~ oo, and if k"(t) = k~ = k~,
t = t~ = s n t e (tk-1, t~,l,
then Ilkn - k[lLl(0,r; y) ~ 0. If u,(t) is convergent to a continuous function u(t), then u is called a D.S. limit solution or simply a limit solution. We have the following result. PROPOSITION 1. (a) Suppose that k(t) is continuous and A ( t ) satisfies (M) and (A.1). Then any approximate solution satisfying
(n)
t~
- xtnl,-1 _l kzI <-
N,
N a constant,
converges uniformly to a continuous function u and u is the unique limit solution. (b) If k is continuous and A ( t ) satisfies (M) and (A.1) for t = r, then there exists a D.S. approximate solution with k~ = k(t~) and x~ = ~b. If ~b e D ^ and k is also of bounded variation, then any D.S. approximate solution with k~, = k(t~) and x.~ = ~ satisfies (H), and so converges; hence a unique limit solution exists. Proof. This is proved in [6] for the case o9 = 0. For general o9 we might be able to apply [17, theorem 2.3] by setting v(t) = e-~tu(t), B(t) = e-OtA(t) e ~', D(B(t)) = {x; eOtx ~ D(A(t))}. Then u is a D.S. limit solution of (E) if and only if v is a D.S. limit solution of dv
d--t ~ B(t)v - o)v + e -~t k,
v(s) = e).
Since it is clear that JxS(t) -- e -~'t J~(t) e ~t (where Jxa(t) = ( / - 2A(t)) -1 and similarly for Jff(t)), A ( t ) is m-m-dissipative if and only if the same is true of B(t). The condition (A. 1), however, has to be verified separately for any particular case. The general result can anyway be verified directly by adapting the proofs o f [4, 6]. Thus, if ~b e D A and k is continuous and of bounded variation, there is a D.S. approximate solution which converges to a continuous function u. We shall now see that this u is an integral solution of (E) k . The proof is an easy generalization of [4, p. 19]. PROPOSITION 2. Suppose that k(t) is continuous and A ( t ) satisfies (M) and (A. 1). Then any D.S. limit solution of (E)k is an integral solution, where in (I.1) hi = h. The following proposition will enable us to prove that, for 4~ e D A and k continuous and of bounded variation, the notions of integral solution and limit solution coincide, and that there is a unique such solution. In addition it will enable us to prove, using a density argument, that there exists a unique integral solution for ~ e / ) and k continuous, and that this is also a limit solution.
Nonautonomous delay equations
699
PROPOSITION 3. Suppose that A f t ) satisfies (M) and (A. 1). Let u(t) be an integral solution of (E)k where 4~ 6. b and k is continuous. Let v(t) be the limit solution of dv dt 6. A ( t ) v + re(t),
(1)
0 <_ s <_ t <_ T,
v(s) = q/,
where q/6./Y~, and m ( t ) is continuous and of bounded variation. Then Ilu(t) - v(t)l[ -< e~(t-S)llu(s) - v(s)ll +
f'
e~(t-¢)lLk0)
-
m(r)ll d r .
$
Thus, if k is continuous and of bounded variation and
t = t~ = s n t 6. ( tkn_ l , tk],
= v~
such that v"(t) converges uniformly to v(t) and w~. =
v~-vL~ t~ - t kn- I
m ( t ~ ) 6. A(t~)v~,
and a constant N1 such that IIw~ll -< N1. Also the proof of [6, theorem 7.1] can be adapted to show that there is a constant N2 such that I]v~ II -< N2. So taking r = t~,, x = v~, and y = w~, in the definition of integral solution of (E) we have Ilu(t) - v~:li -
Ilu(r) - v~ll -<
I'
{~ollu(~) - v~ll ++
T
+ (1 + IIw~ll)L(llv~ll)llhl(~)-
hl(t~)lllda.
However,+
+ llm(tT,) - m(cr)H + lira(a) - k(cr)ll. The proof now proceeds as in [4, Chapter 1, theorem 3.2]. Finally, we have the following theorem. THEOREM 1. Suppose that A ( t ) satisfies (M) and (A. 1). Then equation (E)k has a unique integral solution, u, provided only that k(t) is continuous and qb 6. D. In addition, if v(t) is the integral solution of (1) for m continuous and q/6./), then Ilu(t) - v(t)ll -- e~<'-S)llu(s)
- v(s)ll +
l' $
e~('-')llk(~')
- m(z)ll d z .
(2)
700
J. DYSON and R. VILLELLA-BRESSAN
Proof. There exist k,(t) which are continuous and o f bounded variation, and 4~n e DA such that Ilk, - kilL1 --' 0 and II~, - 411 - " 0 as n --, 00. Let u, be the integral solution of dun - - ~ A(t)u. + k.(t), dt u.(s)
= 4~..
Then, by proposition 3, Ilu.(t) - v(t)ll -< e~
-
v(s)ll +
f'
e~<'-')ll/c.(r)
- m(r)[I dr
$
and letting n ~ oo we get (2), and hence the uniqueness of the integral solution. We may use a density argument to show that, in fact, (E)k has an integral solution for k • L 1 and that (2) is satisfied if u and v are these integral solutions for k, m e L ~. For k continuous and 4> • / ) we may define UA+k(t, S)dp tO be the unique integral solution o f (E)k. If k is also o f bounded variation this is the evolution operator obtained by Crandall and Pazy [5]. We thus have UA+k(t, S): D A ~ D A and UA+k(t, S)
HUA+k(t,s)cb- UA+e(t, S)ctl]--< e ' ° < t - S ) l l ~ -
~11.
Finally, we can show the following theorem. TrmOREM 2. Suppose that A(t) satisfies (M) and (A. 1). Then for all k continuous and ¢ • / 5 there is a unique limit solution o f (E) k which is UA+k(t, S)ck.
Proof. We take the D.S. approximate solutions described in proposition l(b). For ¢ e D ^ this converges to UA+k(t, S)4~. If ~ e / 3 , then there exist 4~, e D A such that 11¢,- 4~]IL' ~ 0. SO, letting U"(S, (~)(t) be the D.S. approximate solution with initial data u"(s, 4~)(s) = ~b, we may approximate u"(s, 4~) by u~(s, 4~m) and hence show that u~(s, 4~)(t) converges to UA+k(t, S)4~ as required. We define the strong solution of (E) as follows. A strong solution o f (E) is a function u: [s, T] ~ Y such that: (i) u is continuous on [s, T] and u(s) = 4~; (ii) u is absolutely continuous on compact subsets of (s, T); (iii) u is differentiable a.e. and du
d--~~ A(t) + k(t)
a.e.
PROPOSITION4. Let A(t) satisfy (M) and (A. 1), k(t) be continuous and of bounded variation and ~b e D ^. Then, if dUA+k(t, s)
UA+k(t, S)
and
dUA+k(t, s)
Nonautonomous delay equations
701
Proof. This is theorem 3.3 of [5]. COROLLARY 1. If Y is reflexive and ~ e D A, then U,+k(t, s)~, is a strong solution of (E) k. Proof. UA+k(t, S)Cb is Lipschitz continuous, hence absolutely continuous and differentiable a.e. We now suppose that A is also linear and show that, in this case, the mild solution and integral solution coincide. Suppose that A(t) is linear and satisfies (M) and (A.1) and that D(A(t)) = Y. We define the mild solution of (E)k as the function u such that u(t) = Ua(t, s)x o +
I'
gA(t, 0k(r) dr.
$
We have the following proposition. PROPOSITION 5. Let A(t) be linear, such that D(A(t)) = Y and satisfies (M) and (A. 1). Let k be continuous. Then the mild solution of (E)k is an integral solution. Proof. We use the Yosida approximations Ax(t). Recall that, for A small and positive, Ax(t ) satisfies (A.I) with 209 in place of 09. Also Ax(t)~b is Lipschitz continuous in ~ and continuous in t [5]. Consider (E)×
du× = Ax(t)Ux + k(t) dt ux (s) = ~.
The integral solution of (E) × is continuously differentiable and thus is also the strong solution. However, the strong solution is also the mild solution [18, Chapter VI, proposition 5.2] and thus, since each type of solution is unique, the three notions of solution coincide. Denote this solution by ux(t). Now, if Ux(t, s)~ is the integral solution of the homogeneous version of (E) ×, U×(t, s)~ converges to UA(t, S)Cb, uniformly with respect to t, t e [s, T], [5]. So that, letting A ~ 0 in Ux(t) = Ux(t, s)dp +
f'
Ux(t, r)k(z) dr
$
we see that Ux(t) converges to a function u(t) which is the mild solution of (E)k. However, u× is also the integral solution of (E) ×, so that, for all w e Y Ilux(t) - wll - Ilux(r) - wll -<
l'
12~ollux(cr) - wll +
7"
+ (1 + IlZx(r)wll)t(llwll)" IIh(a) - h(r)lll d o .
(3)
Now, let [ x , y ] ~ A(r) and set w = x × = x - ).y so that Xx ~ x as ~. ~ 0, and Ax(r)w = y. Hence, letting A ~ 0 in (3) shows that u is also an integral solution of (E)k as required.
702
J. DYSON and R. VILLELLA-BRESSAN 2. F U N C T I O N A L
DIFFERENTIAL
EQUATIONS
We n o w turn to delay equations, set in a Banach space X, with initial data in L 1 × X, where L 1 = LI( - r, 0; X ) , r being the delay. Denote the n o r m in X by [. J, in L ~ by I1" II and let Ill~, hill = I1~,11 + Ihl for [~, hi e L ~ × X. Consider the functional differential equation
Yc(t) e f ( t , x(t)) + g(t, xt),
(FDE)
x(s) = h,
0 <_ s <_ t <_ T,
xs=~,
for IO, hi e L ~ × X. We shall suppose that f satisfies conditions (M) and (A.1) (with co = or). There will be two alternative hypotheses for g: [0, T] × L 1 --, X. (g.1) There exist y(t) e Ll(s, T), Lg: [0, oo) --, [0, oo) which i s m o n o t o n e increasing and continuous and h as in (A.I) such that
Ig(t, u) - g(r, v)J < ),(z)llu - VIIL®t-r.O;X) + tg(llull)lh(t)
- h(0[
for all u, v e L l ( - r , 0; X ) , u - v e L~°( - r, 0; X ) . (g.2) There exists fl, and Lg and h as in (g. 1) such that Jg(t, u) - g(r, v)l - #llu
-
vii +
Zg(llull)lh(t)
-
h(r)J.
Note that (g.2) implies (g. 1). Following [12, 19] we say that a continuous function x(t), with [xs, x(s)l = I¢, hi, is an integral solution of (FDE) if there exists a function h~ : [0, T] ~ X which is continuous and o f b o u n d e d variation, and a m o n o t o n e increasing function L/: [0, oo) --. [0, oo) such that for all [x,y] e f ( r , .), r e [s, T], 0 _
{(Y + g(a, x~), x(a) - x)+
Ix(t) - xl - Ix(r) - xl -< T
+ aJx(a) - xl + C J h l ( a ) - h~(r)J] d a for C = (1 + lyl)Z/(Ixl), with L and hi functions as in (I.1). Using a similar p r o o f to that given in [4] for ordinary differential equations we can see that, if (FDE) has a strong solution, then this is an integral solution. The definition o f a D.S. a p p r o x i m a t e solution o f (FDE) is not immediate as the equation contains xt as well as x(t). T h e following seems to be the natural definition. We associate with (FDE) the evolution equation in L ~ × X (E)
du d--}"e A(t)u,
0 <_ s <_ t <_ T,
u(s) = [tb, hi,
where A(t) is defined by
D(A(t)) = I[~p, hi e L ~ × X ; dp e A(t){O, hi = {O',f(t, h) + g(t, ¢)1-
wl"l(-r,
0; X ) , dp(O) = h ~ D ( f ( t , "))l
Nonautonomous delay equations
703
W e will prove that the projection of the D.S. limit solution o f (E) o n t o X is the integral solution of (FDE). H e n c e it is natural to say that u . ( t ) is a D.S. a p p r o x i m a t e solution o f (FDE) if it is a D.S. a p p r o x i m a t e solution o f the evolution equation (E) and that u(t) is a limit solution of (FDE) if it is a limit solution of (E). W e first prove the following lemma. LEMMA. 3. I f f satisfies condition (A.2) with to = a and g satisfies (g.2), then A also satisfies condition (A.2) with m = ct + fl + 1. P r o o f . T a k e [14'ihi], Wi] ~.A(ti), i = 1,2, tl = t, t 2 = r, where Wi = [4'1, wi} and w i ~_ f ( t i , hi) + g(ti, 4'3. Set [q/i, ki] = 14'i, hi] - 2W/, i = 1, 2. So, suppressing i and t, 4' - 24'' = q/
and
h - 2w = k.
(4)
Thus, 4'(0) = h e °/x +
o q/(s) ~-(~-o)/× no T ~ ~'
f
0
and
114', - 4'zll -<
l°
Ihl -
hzl e °/x dO
+
--r
I°f°l~l(S)-~z(s)le-(~-°)/XdsdO ~--r
<- 2lht - h2l +
0
(5)
Ilq/, - q/zl[.
H o w e v e r , f satisfies (A.2) and so, using (4) and (g.2), (1 - 2a)lhl
-
hE] <- [hi -
h2 -
2((wl
-
g(t,
4'1)) -
(w2
-
g(r, 4'2))1
4'z)l)tAIhzl) 4'zll + 2z~(ll4'zll)lh(t) - h(r)l IIw211 + [g(r, %)l)LAIh21).
+ X[h(t) - h(r)l(l + ]w2 - g(z,
Ik, - kzl + &/~l14', + 2[h(t)-
h(r)l(1 +
(6)
So, combining (5) and (6), (1 -
2#)114,1
-< Ilq/1 -
-
4'211 + (1 -
~zll
2a
-
2)[h,
-
+ Ikl - k2l + 2lh(t) -
hz[
h(r)llL~(ll4,zll
+ (1 +
IIw~[I +
Ig(r,
4,2)[)Lf(Ih21)l.
However,
Ig(r, 4'2)[ -< [g(v, '2) - g(0, 0)l + Ig(0, 0)l 8114,211 + Ih(0-
h(0)ltg(ll011)+
Ig(0, 0)l
-< L([14,ztl), w h e r e / , is a function o f the same type as L, and the result follows. Suppose that Ua(t, S)[4,, hi is the integral solution o f (E). Define x(t) = rt2 UA(t, s)14,, h} =4,(t-s)
t >- s, s - r <- t <- s.
(7)
704
J. DYSON and R. VILLELLA-BRESSAN
Then, by [20], if {~, hi ~ b = D(A(t)), UA(t, S)l~b, hi is a translation. That is, if x(t) is defined as above, then xt = rtl UA(t, S)[~, hi, t - s. The proof of the following proposition is an easy generalisation of [12, proposition 1]. PROPOSITION 6. Let f a n d g satisfy (M), (A.1), with to = a and (g. 1), and suppose that (E) has a D.S. limit solution u(t). Then x(t) = (~(t - s) = rt2u(t)
s - r <_ t < s s<_ t < T
is an integral solution of (FDE). Thus we see that if, in addition, g satisfies (g.2), then x(t) given in (7) is an integral solution of (FDE) for 1~, hi e / 3 . LEMMA4. Let f s a t i s f y (M) and (A.1), with 09 = a and g satisfy (g.2). Then, for [~b, hi e / ) the integral solution of (FDE) is unique. Proof. If x(t) is an integral solution of (FDE) then k(t) = g(', xt) is continuous. The result now follows from repeated applications of (2). PROPOSITION 7. Suppose that f satisfies (M) and (A. 1), with 09 = or, and that g satisfies (g.2). Let [~, h} e D ^. Then, if x(t) is the integral solution of (FDE) and d x / d t exists at t, dx d-~ ~ f ( t , x(t)) + g(t, xt). Proof. Set k(t) = g(t, xt). If [4~, hi e D ^ then as xt = nl UA(t, S)[~, hi it is Lipschitz continuous, and thus g(t, xt) is continuous and of bounded variation. Clearly x(t) = Uy+k(t, s)h; so, applying proposition 4, as h e / Y ' ( f ) , [11, theorem 141, dUy+k(t, s)h e f ( t , Uy+k(t, s)h) + k(t), dt and the result follows. COROLLARY 2. If X is reflexive, then for 1~, hi ~ D ^, n2 UA (t, S){¢, hi is a strong solution of (FDE) f o r 0 _ < s _ < t _ < T. Proof. UA(t,s)lc~,h] is Lipschitz continuous and, hence, n2UA(t,s)[th, hl is Lipschitz continuous and so differentiable a.e. Now suppose that, in addition, f is linear and DA(f(t, ") = X . We prove a variation of parameters formula. We denote by B(t) the operator D(B(t)) = 1{4~,h} e L l x X; 4~ e w l ' l ( - r , 0; X ) , (o(O) = h e D ( f ( t , .))} B(t)[4~, hi = {4a',f(t, h)], that is the operator A ( t ) with g = 0.
Nonautonomous delay equations
705
PROPOSITION 8. L e t f be linear with Df(t, -) = X and satisfy (M) and (A.1) and g satisfy (g.2). Then, for [4~,hi e L ~ × X, U~(t, s)14~, hi = UB(t, s){~, hi +
f' l'
Us(t, OlO, g(r, rq U~(t, 014~, hl)l dr.
$
From proposition 6 we have that n2UB(t,s)[~, h} = Us(t,s)h and so, projecting onto X, we have x(t) = Us(t, s)h +
Uy(t, r)g(r, x,) dr,
s
where x(t) is the integral solution of (FDE). Proof. Set G(t)[4, h} = [0, g(t, 4~)1 so that A f t ) = B(t) + G(t). Then IIG(t){~,, hi - G(O{~,
kill - B[I~ - ~l[ + Lg(ll~ll)lh(t)
- h(01.
Now, if v(t) is the unique integral solution of (E), it is also the unique integral solution of du d t ~ B(t)u + k(t),
0 <_ s <_ t <_ T,
u(s)
[4, hi,
(8)
where k(t) = G(t)v(t) and is continuous. For, if [x, y] e B(r), then [x, y + G(r)x} e A(r) so [IV(t) - xll - Hv(r) - x[I <-
I
[(y + G(r)x,
v(a)
- x)+
t
+ (~o +/~)llv(a) - xll + L(llxll)(1 + Ily +
G(r)xll)lh(~)
-
h(r)l} da.
However,+ _< + + [IG(~)v(a) - G(r)xl[
_<
+ G(~)v(~,),
v(~,) - x>+ +/~llv(~)
- xll + lib(a) - h(r)[lLg(llxll),
and IIGtr)xll
so that
<- IIG(r)x -
G(0)011 + IIG(0)011 --/~llxll + [h(r) - h(0)[Lg(0) + IIG(0)01,
v(t) is indeed the integral solution of (8). Hence v(t) is also the mild solution of (8) v(t) = UB(t, s)[4~, h} +
f'
UB(t, r)G(r)v(r) dr.
$
However, v(t) = UA(t, s){4~, h} and the result follows. 3. P R O P A G A T I O N
OF S O L U T I O N S
We shall now use proposition 8 to relate the propagation of solutions of (FDE) and the related ordinary equation (DE)
2(0 e f ( t , x(t)), x(s) = h.
0 <_ s <_ t <_ T,
706
J. DYSON and R. VILLELLA-BRESSAN
We say that the evolution operator U(t, s)propagates the subsets {Xt]t > o if U(t, s)X s C X t for
t>_s>_tr. From proposition 6 we have that n2Un(t, s)[O, hi = U:(t,s)h. It follows that if UB(t,s) propagates {Xt]t~. o, then Uy(t, s) propagates [n2Xt]t>o. Now, as in the autonomous case [12], given a family [Xt} propagated by U:(t, s) we want to find families [Y,} propagated by UB(t, s) and UA(t, s) such that rt2 Yt C Xt. Let P be a subset o f L 1 and define Yo(a) = P x Xo. Set Q(a) = [q/: [a - r, oo) --, x ; q/o ~ P, ~'lto,®) is continuous, ~u(t) ~ Xt for all t ___ a] and Qt(o-) = [i/tt : q / e Q(a)], so that Qo(a) = P, and let Yt(o') = [{t~, hi E Ot(o') x xt; h = t#(0)] = [[q~, q~(0)]; $ E Qt(tT)]. Note that we have rt2 Yt C Xt, but for equality we need some sort of smoothness on the t-dependence of Xt. In lemma 5 o f [12] it was proved that if Xt is an increasing family o f closed subspaces o f X, then n2 ~ = Xt if and only if U X , = Xt. r
We first prove that LEMMA 5. If Uf(t, s) propagates [Xtlt>~, a family of subsets of X, then Us(t, s) propagates
I~(a)]. Proof. Let [~, hi e YAa) so that there is a ~ a Q(a) such that 4~ = ¢ts and h = ~u(s). Then n2 Us(t, s){¢, hi = U:(t, s)h e Xt and
n~ Us(t, s){¢ hi = U:(t + O, s)h = ~u(t + O)
t+O>_s s-r<_t+O<_s.
So if we define ~ e Q(a) by ~ ( r ) = ~u(r) = U:(r, s)h
a - r _< r ___ s r > s,
t h e n nl Us(t, s)[th, h} = ~ t a n d ~t(0) = Uf(t, s)h = rtz Us(t, s)14~, hi, so t h a t
Us(t, s)16, hi ~ Yt(a). We now consider the propagation of the solutions of the delay equation. We restrict f to be linear. THEOREM 3. Let f(t) be linear and satisfy Df(t, .) = X, and (A. 1), and (M) and g satisfy (g. 1). Let x(t) be the integral solution of (FDE). Suppose that: (a) Uf(t, s) propagates the family of closed subspaces [Xt}t> o; (b) g(t, Qt(a)) c X, t >_ a. Then {q~,hi ~ Y,(a) implies that x(t) ~ Xt, t _> s _ a _> O.
Nonautonomous delay equations
707
P r o o f . Let 1~, hi e Y~(a), so that there is ~, e Q(a) such that ~ = ~u~. F r o m proposition 8 Uf(t, Og(r, x.) dr.
x(t) = Uf(t, s)h + $
We set up an iteration as follows a-r<_t<.s
x°(t) = ~(t)
t>_s,
= Uy(t, s)h
so x ° ~ Q(o) and Ix °, x ° ( t ) l ~ Yt(a), and x"(t) = ~u(t)
a - r <_ t <_ s t
I
= Uf(t,s)h +
t>_s.
Uf(t,r)g(r, x T - 1 ) d r ,
$
Then Ix'+'(t) - x(t)l <
i
t
IUy(t, r)g(r, xT) - Uf(t, r)g(r, xDI dr
$
and so IIXn+l -- xllL=(s,,) -<
I te~('-')~'(r)llx~
- x, llL®(-r.X)dr --%
$
.l
e~ry(r)llx ~ - xl[L®(s.,)dr.
$
So
y.(r)ilx ° - xllL~(~,o dr,
IIx "+' - xllL=(.,,) ~ s
where Yn(r) = y(r) e ~r
~,_l(a) da,
y0(r) = e a r y(z).
s
Hence, xn(t) converges uniformly to x(t). Now suppose x ' - l ( t ) ~ Q(a), so that, if r > a, [x~ -1, x"-l(z)] ~ Y,(a). Thus g(r, x~ -1) ~ X~, so Uy(t, r)g(r, x~ -1) ~ X t and x ' ( t ) ~ X t, t > a. Hence x ~ ~ Q(a) as it is also continuous by induction. So [x~, x"(r)} ~ YT(a) and the induction is complete. Finally, x(t) = lim x"(t) ~ X t as X t is closed. n~co
COROLLARY3. Suppose further that g satisfies (g.2). Then UA(t,s) propagates
Yt(a) for
r~S>_G.
P r o o f . Let [~, hi e Y~(a), so there exists gJ e Q(a) such that ~ = ~s and h = ~(s). If x(t) is the integral solution of (FDE), UA(t, s)[(h, h} = [xt, x(t)]. So, if we define
y ( r ) = ~(r)
=x(r)
a-r_
r>_s,
then y ~ Q(a) and UA(t, S)14), h l { y t , y ( t ) l E Yda).
708
J. DYSON and R. VILLELLA-BRESSAN
We now prove a partial converse of this corollary. TrmOREM 4. Suppose that f i s linear, Df(t, .) = X , and satisfies (A.I), with to = o~, and (M); suppose that g is linear and satisfies (g.2). For some tr _> 0, let [Xt}t, o be an increasing family of closed subpaces of X. Suppose that: (a) n 2 ~ ( t r ) = X t , t >_ a;
(b) UA(t, S) propagates [Yt(tr)}t~.o; (c) g(t, Qt(tr)) c X t t >_ ix. Then Uy(t, s) propagates [Xt}t~ o. Proof. Since B(t) = A ( t ) - G(t), G(t) = 10, g(t, -)} we can use proposition 8 to obtain
UB(t, S)[4', hi = gA(t, S)I¢, h} + f t UA(t,0{0, g(r, rq UB(r, S))}l~, h} dr. s
Define u°(t) = UA(t, S)[¢, hi, un(t) = UA(t, S)I~, h} +
t ___ a
f'
UA(t, s)G(Ou~-'(O dr.
$
Then, using a similar proof to that in theorem 3, it can be proved that u~(¢) converges uniformly
to UB(t, S)[~, hi. Since UA(t,s) propagates IYt(a)lt>~, UA(t,s) also propagates I ~ ( a ) b , ~ . Also, by [12, lemma 6], if {qL h} e Yt(a), G(t)[~, h} = {0, g(t, 4~)} e ~(a). Thus G(t, "): ~ ( a ) ~ ~(a). Suppose now that h ~ Xs, s >_ a. Then there exists q~ such that I~, h} e ~ ( a ) and, therefore, u°(t) = UA(t, s)l~b, hi e ~(tr). Suppose that u"-l(r) e g ( a ) , t _> a; then G(r)u"-l(r) e g(tr) so that jts UA(t, s)G(r)u"-l(r)dr e ~(a) and, therefore, u~(t) e Yt(a). Thus, Un(t, s){¢, hi e ~(a), so that Uf(t, s)[$, hi e n2 ~(a) = X t . 4. A N E X A M P L E
We consider the first order partial differential delay equation au(x t) Ot
~.. au(x t) [o 2(t+O--~x +j G(t, u(x, t + O)) dO, -r
u(x, s) = h(x),
h ~ C o = Co(-Oo, co),
-oo < x < oo , t >_ s >_ O,
us(x, 0) = $(x, 0),
~b: R x I - r , 01 ~ R,
where G: [0, o0) × R ---, R satisfies IG(t, u) - G(r, v)l---fllu - vl + Ih(t) - h(r)lt(lul) for some fl > 0, where h: [0, co) ~ R is continuous and of bounded variation, and L is a monotone increasing continuous function which satisfies the condition L(x) <_ ax + b, a and b e R (this is equivalent to saying that there is a monotone increasing function L1 such that for every ~b e L l ( - r , 0; Co),
f--ro L(I~(o)I)
dO _ L,
(f --ro I~(o)1 d 0)
.
Nonautonomous delay equations
709
We m a y r e f o r m u l a t e our p r o b l e m as the abstract functional differential equation in the Banach space Co
2(0 = f ( t , x(t)) + g(t, xt), x(s) = h,
0 <_ s <. t < T,
xs = 6
for (~, h) ~ L l ( - r , 0; Co) × Co, where
Ou f(t,u)=2(t+
Df(t,.)=[ueCo;u'eCo]
1)~xx,
and g: [s, T] × L l ( - r , 0; Co) --, C o is defined by
g(t, 4,) =
f ° a ( t , 4~(0)) dO. --r
Set J× (t)u = (I - i f ( t ,
Ilf(t, u) - f(-c,
-))-lu,
u)ll
fx (t, u) = f(t, J× (t)u), then
= 2It -
lliu'll
= Ilf(r, u)ll It - rl ~ Ill(r, u)lllt - r[. r+l
Hence, by [4, C h a p t e r 1, proposition 6.1], [IA(t)u - Jx(r)ul[ -< Z[[fx(t, u)lllt - r[. It is routine to show that f ( t , .) is m-dissipative and Df(t, .) = Co. Hence condition (A.1) holds, with aJ = 0. It can be seen that, if k e C 1 ( - o o , oo), the strong solution o f du
d---t = f ( t , u),
u(s) = k
is u(x, t) = k(x + (t + 1) 2 - (s + 1) 2) and, hence,
Uf(t,s)k = k(x + (t + 1) 2 - (s + 1) 2)
(9)
for k e C 1. H o w e v e r , as C 1 0 C O is dense in Co, (9) also holds for k e C 0. So, if we s e t B ( a , t ) = l x ; - a < _ x + ( t + 1) 2 - 1 < a l a n d Zt = {k e C0(R); supp k C B(a, t)l the family {Ztl is p r o p a g a t e d by Uy(t, s). H o w e v e r , the Zt are not increasing and this will be required later. So instead we define C(a, t) = Ix; x + (t + 1) 2 - 1 _> - a , x __. a} and
S t = {k ~ Co(R); supp k C C(a, t)},
X o = Z o.
These sets are closed, increasing and are also p r o p a g a t e d by Uy(t, s). Suppose now that: (i) P C [q~ ~ L l ( - r , 0; Col; ~b(0) ~ Xo, 0 ~ [ - r , 0]}; and
(ii)
G(t, Xt) C X t .
Then g(t, Qt(~r)) c X t . Thus, we m a y apply t h e o r e m 2 to find that if [0, hi e Y~(a), then
x(t) e X , .
710
J. DYSON and R. VILLELLA-BRESSAN REFERENCES
1. BENILAN PH., Equations d'evolution dans un espace de Banach quelconque et applications, Thesis. Universit6 de Paris XI, Orsay (1972). 2. BENILAN PH., Solutions integrales d'equations d'evolution dans un espace de Banach, C.r. Acad. Sc. Paris 274,47-50 (1972). 3. CRANDALL M. G. & EVANS L. C., On the relation of the operator a/Os + O/Or to evolution governed by accretive operators, Israel J. Math. 21,261-278 (1975). 4. PAVEL N. H., Nonlinear Evolution Operators and Semigroups, Lecture Notes in Mathematics. Springer, Berlin (1987). 5. CRANDALL M. G. & PAZY A., Nonlinear evolution equations in Banach spaces, Israel J. Math. 11, 57-94 (1972). 6. EVANS L. C., Nonlinear evolution equations in an arbitrary Banach space, Tech. Summary Report no 1568. Math. Res. Center Madison, Wisconsin (1975). 7. EVANS L. C., Nonlinear evolution equations in Banach spaces, Israel J. Math. 26, 1-42 (1977). 8. WEBB G. F., Asymptotic stability for abstract nonlinear functional differential equations, Proc. Am. math. Soc. 54, 225-395 (1976). 9. DYSON J. & VILLELLA-BRESSAN R., Functional differential equations and nonlinear evolution operators, Proc. R. Soc. Edinb. 75A, 223-234 (1976). 10. DYSON J. & VILLELLA-BRESSAN R., Nonlinear functional differential equations in L ~ spaces, Nonlinear Analysis 1, 383-395 (1977). 11. DYSON J. & VILLELLA-BRESSAN R., Semigroups of translations associated with functional and functional differential equations, Proc. R. Soc. Edinb. 82A, 171-188 (1979). 12. DYSON J. & VILLELLA-BRESSAN R., Propagation of solutions of a non linear functional differential equation, Diff. Integral Eqs. 4, 293-303 (1991). 13. DYSON J. & VILLELLA-BRESSAN R., On the propagation of solutions of a delay equation, Ren. Sem. Mat. Univ. Padova 80, 55-63 (1988). 14. GRIMMER R. & ZEMAN M., Wave propagation for linear integrodifferential equations in a Banach spaces, J. diff. Eqns 54, 274-282 (1984). 15. DESCH W. & GRIMMER R., Invariance and Wave propagation for nonlinear integrodifferential equations in Banach spaces, J. Integral Eqns 8, 137-164 (1985). 16. CRANDALL M. G., A generalised domain for semigroup generators, Proc. Am. math. Soc. 37, 434-440 (1973). 17. DORROH J. R. & RUIZ RIEDER G., A singular quasilinear parabolic problem in one space dimension, J. diff. Eqns 91, 1-23 (1991). 18. MARTIN, R., Nonlinear Operators and Differential Equations in Banach Spaces. Wiley, New York (1976). 19. CRANDALL M. G. & NOHEL J. A., An abstract functional differential equation and related nonlinear Volterra equation, Israel J. Math. 29, 313-328 (1978). 20. PLANT A., Nonlinear semigroups of translations in Banach spaces generated by functional differential equations, J. math. Analysis. Applic. 60, 67-74 (1977).
Pergamon 0362-546X(94)00094-8
ON INTEGRAL SOLUTIONS ON NONAUTONOMOUS EQUATIONS AND THEIR PROPAGATION
DELAY
JANET DYSON 1 and ROSANNA VILLELLA-BRESSAN~; Mansfield College, Oxford, U.K.; and ~:Dipartimento di Matematica Pura e Applicata, Universita' di Padova, Padova, Italy (Received 1 December 1991; received in revisedform 23 September 1993; receivedfor publication 4 May 1994) Key words and phrases: Nonautonomous functional differential equations, dissipative operators, integral solution, propagation of solutions.
0. I N T R O D U C T I O N
We consider integral solutions of nonautonomous delay equations. We establish some properties of such solutions and look at their relationship to other types of solution. We then discuss their propagation. The equation under consideration is (FDE)
Jc(t) ~ f ( t , x(t)) + g(t, xt), x ( s ) = h,
0 <_ s <_ t <_ T, x,=~,
set in a Banach space X. Here xt ~ L 1 = L l ( - r , 0; X) is defined pointwise by xt(O) = x(t + 0), r, 0 < r < 00, being the delay and I $ , h ] e L l x X . There is an tx such that f - t x is m-dissipative, and f and g satisfy certain smoothness conditions. When discussing propagation, we shall confine our attention to the semilinear case, however, our discussion of integral solutions relates to the full nonlinear case. The concept of integral solution was first introduced by Benilan [1, 2] for autonomous evolution equations without delay and has since been exploited and extended by many authors; see for example [3, 4]. It provides a unification of the various different notions of solution. Thus, if we consider the evolution equation
(E)k
du dt ~ A ( t ) u + k(t),
0 <- s <- t <- T,
u(s) = ¢~,
set in a Banach space Y, under suitable conditions a strong solution of (E)k will be an integral solution, so too will a discrete scheme limit solution and, in the linear case, a mild solution. Further, if the integral solution is differentiable for some t then it will satisfy (E)k in the classical sense for that t. We start with the evolution equation without delay, (E)k, and give a partial extension of the concept of integral solution discussed by Pavel in [4]. The conditions set on A ( t ) are those first used by Crandall and Pazy, and by Evans [5-7] and they allow operators A ( t ) of the form h(t)A, where A - to is m-dissipative and also allow operators A ( t ) where D(A(t)) depends on t. We denote the unique integral solution of (E)k by Ua+k(t, s)cb. UA+k is then an evolution operator. 693
694
J. DYSON and R. VILLELLA-BRESSAN
This definition of integral solution is then extended to functional differential equations. It is well known [8-11 and others] that (FDE) is closely related to the evolution equation (E)
du dt ~ A(t)u,
0 <_ s <_ t < T,
u(s) = 14~,h} ~ L 1 × X , set in the space of initial data where A(t) is defined by D(A(t)) = [{$, hi ~ L 1 × X ; $ ~ w l ' ~ ( - r , 0 ; X ) , 4~(0) = h ~ D ( f ( t , "))1, A(t)[4~, hi = [$',f(t, h) + g(t, $)}. We show that the projection of a discrete scheme limit solution of (E) onto X is an integral solution of (FDE) and thus, we are able to extend the notion of discrete scheme limit solution to delay equations, and to relate the integral solutions of the two problems. We find that (FDE) has a unique integral solution and that if the integral solution is differentiable for some t then it satisfies (FDE) in the classical sense for that t. The relationship between (E) and (FDE) also enables us, in the semilinear case, to establish a variation of parameters formula. Using this we are able to produce results relating the propagation of solutions of (FDE) to the propagation of solutions of the same equation without delay. We shall say that an evolution operator U(t, r), in a Banach space Y, propagates a family of subsets of Y, IXt]t>s, for t _> s, if U(t, r)X, C Xt for t _> r _> s. This notion can be used, for example, to express the fact that the wave equation has finite speed of propagation (see for example [12]). We start with a set of subsets [X,I of X, propagated by U / a n d define a related set of subsets IY,I o f L ~ x X w h i c h are propagated by UA and satisfy n2Yt C Xt. Here rtl and n2 are the projections of L 1 × X onto L ~ and X, respectively. Roughly speaking, rq Yt consists of segments of continuous functions q/such that ~,(t) e X t for all t _> s. Only if X t varies continuously with t in some sense do we have n2 Yt = X t . We also give a converse result which shows that, in certain circumstances, if UA propagates [Yt] then U/must propagate [Xt]. These propagation results extend the results of [12, 13] for the autonomous, semilinear and nonlinear cases. Propagation for other types of delay equations has been discussed by Grimmer and Zeman, and by Desch and Grimmer in [14, 15]. 1. I N T E G R A L S O L U T I O N S OF N O N L I N E A R E V O L U T I O N E Q U A T I O N S
Here we consider the nonlinear, nonautonomous evolution equation (E)k
du d--t ~ A(t)u + k(t),
0 <_ s <_ t <_ T,
u(s) = ¢
set in a Banach space Ywith norm II" [I- A(t): D(A(t)) C Y ~ 2 v is a time dependent (possibly multivalued) nonlinear operator with time dependent domain D(A(t)). We give a partial generalization of the definition of integral solution given by Pavel in [4] and discuss its relation to limit solutions. We shall assume throughout that A(t) satisfies (M)
there is a 20 > 0 such that R ( I - 2A(t)) = Y for 0 < ;t < 2 o.
Nonautonomous delay equations
695
We also set the condition (A.1) for all [ x l , y l ] c A ( t ) and [x2,y 2] c A ( r ) there is an co >_ 0 and an h: [0, T] --, Y which is continuous and of bounded variation, and a m o n o t o n e increasing function LA: [0, oo) ~ [0, o0) such that (Yl
-
x211 +
Y2,XI - X2)- -< collXl -
(1 +
IlY2ll)tA(llx211)llh(t)
- h(0ll.
Recall that IIx + hell - Ilxll htO h
( y , x ) = lim
The notation [x,y] c A ( t ) means x c D ( A ( t ) ) and y ~ A ( t ) x . Then, following [4, Chapter 1, proposition 2.2], it can be seen that (A. 1) is equivalent to the condition (A.2) for all [ g l , Yd c A ( t ) and [x2, Y2] c A(r) there is an co ___ 0, and functions h and L A as in (A. 1) such that for X > 0 (1 - ~co)llx~ - x211 ~ IIx, - x~ - X(y, - y2)ll + A(1 + Ilyell)LAIIx21l)llh(t)
-
h(Ol[.
Also, if we set Jx(t) = ( / - hA(t)) -1 and Ax(t) = (1/A)(Jx(t) - I ) , it will be seen that (A.2) is equivalent to condition (A.3) which for x~ = x2 is condition (C.2) in [6, 7]. (A.3) There exist ~0 > 0, co >_ 0, a function h as in (A.I) and a function LA : [0, co) ~ [0, oo) which is m o n o t o n e increasing and continuous such that for all Xl, x2 c Y and 0 < g < go 1
- x211 + AIIh(t) - h(OIIZ,~(llx~ll)(1 + IIAx(r)x211).
IIJ×(t)x~ - J×(r)x211 <- --IIx, 1 - ACO
To prove that (A.2) implies (A.3), in (A.2) set Xl = J x ( t ) u l , x2 = / x ( ~ ' ) / / 2 , Yl = A×(t)ul and Y2 = Ax(r)u2 to get
IIJx(t)ul -/x(0u211 < -
1
Ilu, - u211
1 -~.co
+ - I[h(t) - h(OllzA(l[Jx(OU2ll)(1 1 - ACO
+ IIAx(0u2[I).
Showing that there is a n / ' A such that (1/(1 - Aco))La(llJ×(Ou~ll) is bounded by LA(Ilu~II) is now straightforward. To prove the converse, given [Ul, vl] e A ( t ) a~d [u2, v2] e A(r) we let xi = ui - Avi, i = 1,2 in (A.3) and rearrange to get 1 -~(llul
-
u~ -
~(Vl
-
v~)ll
collul - u211 + Ilh(t)
-
-
Ilul -
u211)
h(0tlLA(llu2
(A.1) and hence (A.2), now follows by letting A -:, 0.
- Aw21D0 + IIv21D.
696
J. DYSON and R. VILLELLA-BRESSAN
Note also that, since IlZx(t)xll < - -
1 -
1 ),to)
(A.3) also implies condition (C.2) of [5]. Following [5-7, 16] we define a set DA(A(t))
limlla~(r)xll, ~,0
C Y by
D^(A(t))=Ix~D(A(t));limllAx(t)xl[<°° " x ~ o 1 Note that, as in [5], if A(t) satisfies (A.3) then DA(A(t)) = D ^ is independent of t. Thus, as D(A(t)) C DA(A(t)) C D(A(t)), D(A(t)) = b is also independent o f t. We shall now define an integral solution of (E) k as follows. (I.1) A continuous function u is an integralsolution of (E) k on [s, T] if u(s) = 4, and there is an to > 0 and a function hi" [0, T] --, X w h i c h is continuous and of bounded variation, and a monotone increasing function L1 : [0, oo) ~ [0, oo) such that for all s _ r _< t ___ T, r e [s, T] and [x, y] e A(r) Ilu(t) - xll ---
f'
[lu(r) - xll
( t o l l u ( a ) - xll +
do',
7
w h e r e C = (1 + Ilyll)L~(llxll).
Recall that (y, x)+ = lim h~0
We will see that if
IIx + hyll -
IIxll
h
A(t) satisfies condition (A.1) then hi can be taken as h and L 1 as L.4.
This definition generalizes that given by Pavel in [4] in that C now depends on r. Most proofs carry through with little modification. Difficulties occur when proving uniqueness, but these can be overcome by requiring 4~ e/Y~ and using a suitable D-S approximation. There is a possible inconsistency in that, instead of treating (E)k as a nonhomogeneous problem, we could have considered it as the homogeneous problem (E)o
du d---te
C(t),
u(s) = ¢b,
where C(t) = A(t) + k(t). We will see that C(t) also satisfies (M) and (A. 1) (for appropriate h), but using C(t) gives rise to a different definition of integral solution, (1.2) below. These definitions are, however, equivalent provided h and k are of bounded variation. LE~ 1. Suppose that k(t) is continuous and of bounded variation. Then if and only if C(t) satisfies this condition for appropriate h and LA.
A(t) satisfies (A.2)
Nonautonomous delay equations
697
Proof. Suppose that A ( t ) satisfies (A.2) so that for [Xl, Yl] e C(t) and [x2, Y2] E C(z), (1
Z~)ilx,
-
- x=ll - IIx, -
xz -
A(y,
+ x(1 + IIc(m=ll
- yz)ll + AIIk(t)
- k(r)ll
+ Ilk(r)ll)LA(llx2l[)llh(t)
-
h(r)[I.
Set Tk(t) = sup E Ilk(ti) - k(ti-1)ll, where the sup is taken over all possible partitions o f [s, t]. Then for t > r, Ilk(t) - k(r)[I -< TAt) - TAr). Define Th analogously. Then if we define h(t) = TAt) + Th(t), h(t) is continuous and o f b o u n d e d variation, Ilh(t) - h(r)l[ + Ilk(t) + k(OIl Ih(t) - h(r)[, and the result follows with L c = 1 + [IkllooLA + LA. Note also that A ( t ) satisfies (M) if and only if C(t) satisfies (M). This now leads to an alternative definition o f integral solution. (I.2) A continuous function u is an integral solution of (E)k in [s, T] if u(s) = 4~, and there is an o~ >_ 0 and a function h2: [0, T] --, X which is continuous and o f b o u n d e d variation, and a m o n o t o n e increasing function L2 : [0, oo) ~ [0, oo) such that for all s ___ r ___ t <_ T, r e [s, T] and Ix, y] e A(r) Ilu(t)
-
_
xll
-
Ilu(v)
f' I,ollu(o)
-
xll
- xll + ( y + k ( r ) , u ( a ) - x ) + + C l l h E ( a )
- hz(r)llJda,
7"
where C = (1 +
IlYll)Lz(llxll).
H o w e v e r , we have the following lemma. LEMMA 2. Let k be continuous and o f b o u n d e d variation. Then u satisfies (I.1) if and only if it satisfies (1.2). Proof. Let u satisfy (I.1). We have ( y + k(a), u(a) - x)+ < ( y + k(r), u(a) - x)+ + Ilk(a) - k(r)ll. The p r o o f n o w follows as in l e m m a 1. We shall n o w relate the notion of integral solution to that o f limit solution. Following m a n y other authors, we say that u,(t) is a D.S. (discrete scheme) approximate solution o f (E)k if it satisfies the following. Let s = t~ < t~' < ... < tTv, = T be a partition of [s, T]. Set u.(t) = xg
t = t~ = s t e ( t ?1k - l , ttk],
= x~
k = 1. . . . . N , ,
where x~ ~ D(A(t~)), t~
-
"
tk_ 1
k~ ~ A ( t ~ ) x ~ ,
698
J. DYSON and R. VILLELLA-BRESSAN
d. = max[t~ - t~,_ll -~ O, x~ ~ ~b as n ~ oo, and if k"(t) = k~ = k~,
t = t~ = s n t e (tk-1, t~,l,
then Ilkn - k[lLl(0,r; y) ~ 0. If u,(t) is convergent to a continuous function u(t), then u is called a D.S. limit solution or simply a limit solution. We have the following result. PROPOSITION 1. (a) Suppose that k(t) is continuous and A ( t ) satisfies (M) and (A.1). Then any approximate solution satisfying
(n)
t~
- xtnl,-1 _l kzI <-
N,
N a constant,
converges uniformly to a continuous function u and u is the unique limit solution. (b) If k is continuous and A ( t ) satisfies (M) and (A.1) for t = r, then there exists a D.S. approximate solution with k~ = k(t~) and x~ = ~b. If ~b e D ^ and k is also of bounded variation, then any D.S. approximate solution with k~, = k(t~) and x.~ = ~ satisfies (H), and so converges; hence a unique limit solution exists. Proof. This is proved in [6] for the case o9 = 0. For general o9 we might be able to apply [17, theorem 2.3] by setting v(t) = e-~tu(t), B(t) = e-OtA(t) e ~', D(B(t)) = {x; eOtx ~ D(A(t))}. Then u is a D.S. limit solution of (E) if and only if v is a D.S. limit solution of dv
d--t ~ B(t)v - o)v + e -~t k,
v(s) = e).
Since it is clear that JxS(t) -- e -~'t J~(t) e ~t (where Jxa(t) = ( / - 2A(t)) -1 and similarly for Jff(t)), A ( t ) is m-m-dissipative if and only if the same is true of B(t). The condition (A. 1), however, has to be verified separately for any particular case. The general result can anyway be verified directly by adapting the proofs o f [4, 6]. Thus, if ~b e D A and k is continuous and of bounded variation, there is a D.S. approximate solution which converges to a continuous function u. We shall now see that this u is an integral solution of (E) k . The proof is an easy generalization of [4, p. 19]. PROPOSITION 2. Suppose that k(t) is continuous and A ( t ) satisfies (M) and (A. 1). Then any D.S. limit solution of (E)k is an integral solution, where in (I.1) hi = h. The following proposition will enable us to prove that, for 4~ e D A and k continuous and of bounded variation, the notions of integral solution and limit solution coincide, and that there is a unique such solution. In addition it will enable us to prove, using a density argument, that there exists a unique integral solution for ~ e / ) and k continuous, and that this is also a limit solution.
Nonautonomous delay equations
699
PROPOSITION 3. Suppose that A f t ) satisfies (M) and (A. 1). Let u(t) be an integral solution of (E)k where 4~ 6. b and k is continuous. Let v(t) be the limit solution of dv dt 6. A ( t ) v + re(t),
(1)
0 <_ s <_ t <_ T,
v(s) = q/,
where q/6./Y~, and m ( t ) is continuous and of bounded variation. Then Ilu(t) - v(t)l[ -< e~(t-S)llu(s) - v(s)ll +
f'
e~(t-¢)lLk0)
-
m(r)ll d r .
$
Thus, if k is continuous and of bounded variation and
t = t~ = s n t 6. ( tkn_ l , tk],
= v~
such that v"(t) converges uniformly to v(t) and w~. =
v~-vL~ t~ - t kn- I
m ( t ~ ) 6. A(t~)v~,
and a constant N1 such that IIw~ll -< N1. Also the proof of [6, theorem 7.1] can be adapted to show that there is a constant N2 such that I]v~ II -< N2. So taking r = t~,, x = v~, and y = w~, in the definition of integral solution of (E) we have Ilu(t) - v~:li -
Ilu(r) - v~ll -<
I'
{~ollu(~) - v~ll +
T
+ (1 + IIw~ll)L(llv~ll)llhl(~)-
hl(t~)lllda.
However,
+ llm(tT,) - m(cr)H + lira(a) - k(cr)ll. The proof now proceeds as in [4, Chapter 1, theorem 3.2]. Finally, we have the following theorem. THEOREM 1. Suppose that A ( t ) satisfies (M) and (A. 1). Then equation (E)k has a unique integral solution, u, provided only that k(t) is continuous and qb 6. D. In addition, if v(t) is the integral solution of (1) for m continuous and q/6./), then Ilu(t) - v(t)ll -- e~<'-S)llu(s)
- v(s)ll +
l' $
e~('-')llk(~')
- m(z)ll d z .
(2)
700
J. DYSON and R. VILLELLA-BRESSAN
Proof. There exist k,(t) which are continuous and o f bounded variation, and 4~n e DA such that Ilk, - kilL1 --' 0 and II~, - 411 - " 0 as n --, 00. Let u, be the integral solution of dun - - ~ A(t)u. + k.(t), dt u.(s)
= 4~..
Then, by proposition 3, Ilu.(t) - v(t)ll -< e~
-
v(s)ll +
f'
e~<'-')ll/c.(r)
- m(r)[I dr
$
and letting n ~ oo we get (2), and hence the uniqueness of the integral solution. We may use a density argument to show that, in fact, (E)k has an integral solution for k • L 1 and that (2) is satisfied if u and v are these integral solutions for k, m e L ~. For k continuous and 4> • / ) we may define UA+k(t, S)dp tO be the unique integral solution o f (E)k. If k is also o f bounded variation this is the evolution operator obtained by Crandall and Pazy [5]. We thus have UA+k(t, S): D A ~ D A and UA+k(t, S)
HUA+k(t,s)cb- UA+e(t, S)ctl]--< e ' ° < t - S ) l l ~ -
~11.
Finally, we can show the following theorem. TrmOREM 2. Suppose that A(t) satisfies (M) and (A. 1). Then for all k continuous and ¢ • / 5 there is a unique limit solution o f (E) k which is UA+k(t, S)ck.
Proof. We take the D.S. approximate solutions described in proposition l(b). For ¢ e D ^ this converges to UA+k(t, S)4~. If ~ e / 3 , then there exist 4~, e D A such that 11¢,- 4~]IL' ~ 0. SO, letting U"(S, (~)(t) be the D.S. approximate solution with initial data u"(s, 4~)(s) = ~b, we may approximate u"(s, 4~) by u~(s, 4~m) and hence show that u~(s, 4~)(t) converges to UA+k(t, S)4~ as required. We define the strong solution of (E) as follows. A strong solution o f (E) is a function u: [s, T] ~ Y such that: (i) u is continuous on [s, T] and u(s) = 4~; (ii) u is absolutely continuous on compact subsets of (s, T); (iii) u is differentiable a.e. and du
d--~~ A(t) + k(t)
a.e.
PROPOSITION4. Let A(t) satisfy (M) and (A. 1), k(t) be continuous and of bounded variation and ~b e D ^. Then, if dUA+k(t, s)
UA+k(t, S)
and
dUA+k(t, s)
Nonautonomous delay equations
701
Proof. This is theorem 3.3 of [5]. COROLLARY 1. If Y is reflexive and ~ e D A, then U,+k(t, s)~, is a strong solution of (E) k. Proof. UA+k(t, S)Cb is Lipschitz continuous, hence absolutely continuous and differentiable a.e. We now suppose that A is also linear and show that, in this case, the mild solution and integral solution coincide. Suppose that A(t) is linear and satisfies (M) and (A.1) and that D(A(t)) = Y. We define the mild solution of (E)k as the function u such that u(t) = Ua(t, s)x o +
I'
gA(t, 0k(r) dr.
$
We have the following proposition. PROPOSITION 5. Let A(t) be linear, such that D(A(t)) = Y and satisfies (M) and (A. 1). Let k be continuous. Then the mild solution of (E)k is an integral solution. Proof. We use the Yosida approximations Ax(t). Recall that, for A small and positive, Ax(t ) satisfies (A.I) with 209 in place of 09. Also Ax(t)~b is Lipschitz continuous in ~ and continuous in t [5]. Consider (E)×
du× = Ax(t)Ux + k(t) dt ux (s) = ~.
The integral solution of (E) × is continuously differentiable and thus is also the strong solution. However, the strong solution is also the mild solution [18, Chapter VI, proposition 5.2] and thus, since each type of solution is unique, the three notions of solution coincide. Denote this solution by ux(t). Now, if Ux(t, s)~ is the integral solution of the homogeneous version of (E) ×, U×(t, s)~ converges to UA(t, S)Cb, uniformly with respect to t, t e [s, T], [5]. So that, letting A ~ 0 in Ux(t) = Ux(t, s)dp +
f'
Ux(t, r)k(z) dr
$
we see that Ux(t) converges to a function u(t) which is the mild solution of (E)k. However, u× is also the integral solution of (E) ×, so that, for all w e Y Ilux(t) - wll - Ilux(r) - wll -<
l'
12~ollux(cr) - wll +
7"
+ (1 + IlZx(r)wll)t(llwll)" IIh(a) - h(r)lll d o .
(3)
Now, let [ x , y ] ~ A(r) and set w = x × = x - ).y so that Xx ~ x as ~. ~ 0, and Ax(r)w = y. Hence, letting A ~ 0 in (3) shows that u is also an integral solution of (E)k as required.
702
J. DYSON and R. VILLELLA-BRESSAN 2. F U N C T I O N A L
DIFFERENTIAL
EQUATIONS
We n o w turn to delay equations, set in a Banach space X, with initial data in L 1 × X, where L 1 = LI( - r, 0; X ) , r being the delay. Denote the n o r m in X by [. J, in L ~ by I1" II and let Ill~, hill = I1~,11 + Ihl for [~, hi e L ~ × X. Consider the functional differential equation
Yc(t) e f ( t , x(t)) + g(t, xt),
(FDE)
x(s) = h,
0 <_ s <_ t <_ T,
xs=~,
for IO, hi e L ~ × X. We shall suppose that f satisfies conditions (M) and (A.1) (with co = or). There will be two alternative hypotheses for g: [0, T] × L 1 --, X. (g.1) There exist y(t) e Ll(s, T), Lg: [0, oo) --, [0, oo) which i s m o n o t o n e increasing and continuous and h as in (A.I) such that
Ig(t, u) - g(r, v)J < ),(z)llu - VIIL®t-r.O;X) + tg(llull)lh(t)
- h(0[
for all u, v e L l ( - r , 0; X ) , u - v e L~°( - r, 0; X ) . (g.2) There exists fl, and Lg and h as in (g. 1) such that Jg(t, u) - g(r, v)l - #llu
-
vii +
Zg(llull)lh(t)
-
h(r)J.
Note that (g.2) implies (g. 1). Following [12, 19] we say that a continuous function x(t), with [xs, x(s)l = I¢, hi, is an integral solution of (FDE) if there exists a function h~ : [0, T] ~ X which is continuous and o f b o u n d e d variation, and a m o n o t o n e increasing function L/: [0, oo) --. [0, oo) such that for all [x,y] e f ( r , .), r e [s, T], 0 _
{(Y + g(a, x~), x(a) - x)+
Ix(t) - xl - Ix(r) - xl -< T
+ aJx(a) - xl + C J h l ( a ) - h~(r)J] d a for C = (1 + lyl)Z/(Ixl), with L and hi functions as in (I.1). Using a similar p r o o f to that given in [4] for ordinary differential equations we can see that, if (FDE) has a strong solution, then this is an integral solution. The definition o f a D.S. a p p r o x i m a t e solution o f (FDE) is not immediate as the equation contains xt as well as x(t). T h e following seems to be the natural definition. We associate with (FDE) the evolution equation in L ~ × X (E)
du d--}"e A(t)u,
0 <_ s <_ t <_ T,
u(s) = [tb, hi,
where A(t) is defined by
D(A(t)) = I[~p, hi e L ~ × X ; dp e A(t){O, hi = {O',f(t, h) + g(t, ¢)1-
wl"l(-r,
0; X ) , dp(O) = h ~ D ( f ( t , "))l
Nonautonomous delay equations
703
W e will prove that the projection of the D.S. limit solution o f (E) o n t o X is the integral solution of (FDE). H e n c e it is natural to say that u . ( t ) is a D.S. a p p r o x i m a t e solution o f (FDE) if it is a D.S. a p p r o x i m a t e solution o f the evolution equation (E) and that u(t) is a limit solution of (FDE) if it is a limit solution of (E). W e first prove the following lemma. LEMMA. 3. I f f satisfies condition (A.2) with to = a and g satisfies (g.2), then A also satisfies condition (A.2) with m = ct + fl + 1. P r o o f . T a k e [14'ihi], Wi] ~.A(ti), i = 1,2, tl = t, t 2 = r, where Wi = [4'1, wi} and w i ~_ f ( t i , hi) + g(ti, 4'3. Set [q/i, ki] = 14'i, hi] - 2W/, i = 1, 2. So, suppressing i and t, 4' - 24'' = q/
and
h - 2w = k.
(4)
Thus, 4'(0) = h e °/x +
o q/(s) ~-(~-o)/× no T ~ ~'
f
0
and
114', - 4'zll -<
l°
Ihl -
hzl e °/x dO
+
--r
I°f°l~l(S)-~z(s)le-(~-°)/XdsdO ~--r
<- 2lht - h2l +
0
(5)
Ilq/, - q/zl[.
H o w e v e r , f satisfies (A.2) and so, using (4) and (g.2), (1 - 2a)lhl
-
hE] <- [hi -
h2 -
2((wl
-
g(t,
4'1)) -
(w2
-
g(r, 4'2))1
4'z)l)tAIhzl) 4'zll + 2z~(ll4'zll)lh(t) - h(r)l IIw211 + [g(r, %)l)LAIh21).
+ X[h(t) - h(r)l(l + ]w2 - g(z,
Ik, - kzl + &/~l14', + 2[h(t)-
h(r)l(1 +
(6)
So, combining (5) and (6), (1 -
2#)114,1
-< Ilq/1 -
-
4'211 + (1 -
~zll
2a
-
2)[h,
-
+ Ikl - k2l + 2lh(t) -
hz[
h(r)llL~(ll4,zll
+ (1 +
IIw~[I +
Ig(r,
4,2)[)Lf(Ih21)l.
However,
Ig(r, 4'2)[ -< [g(v, '2) - g(0, 0)l + Ig(0, 0)l 8114,211 + Ih(0-
h(0)ltg(ll011)+
Ig(0, 0)l
-< L([14,ztl), w h e r e / , is a function o f the same type as L, and the result follows. Suppose that Ua(t, S)[4,, hi is the integral solution o f (E). Define x(t) = rt2 UA(t, s)14,, h} =4,(t-s)
t >- s, s - r <- t <- s.
(7)
704
J. DYSON and R. VILLELLA-BRESSAN
Then, by [20], if {~, hi ~ b = D(A(t)), UA(t, S)l~b, hi is a translation. That is, if x(t) is defined as above, then xt = rtl UA(t, S)[~, hi, t - s. The proof of the following proposition is an easy generalisation of [12, proposition 1]. PROPOSITION 6. Let f a n d g satisfy (M), (A.1), with to = a and (g. 1), and suppose that (E) has a D.S. limit solution u(t). Then x(t) = (~(t - s) = rt2u(t)
s - r <_ t < s s<_ t < T
is an integral solution of (FDE). Thus we see that if, in addition, g satisfies (g.2), then x(t) given in (7) is an integral solution of (FDE) for 1~, hi e / 3 . LEMMA4. Let f s a t i s f y (M) and (A.1), with 09 = a and g satisfy (g.2). Then, for [~b, hi e / ) the integral solution of (FDE) is unique. Proof. If x(t) is an integral solution of (FDE) then k(t) = g(', xt) is continuous. The result now follows from repeated applications of (2). PROPOSITION 7. Suppose that f satisfies (M) and (A. 1), with 09 = or, and that g satisfies (g.2). Let [~, h} e D ^. Then, if x(t) is the integral solution of (FDE) and d x / d t exists at t, dx d-~ ~ f ( t , x(t)) + g(t, xt). Proof. Set k(t) = g(t, xt). If [4~, hi e D ^ then as xt = nl UA(t, S)[~, hi it is Lipschitz continuous, and thus g(t, xt) is continuous and of bounded variation. Clearly x(t) = Uy+k(t, s)h; so, applying proposition 4, as h e / Y ' ( f ) , [11, theorem 141, dUy+k(t, s)h e f ( t , Uy+k(t, s)h) + k(t), dt and the result follows. COROLLARY 2. If X is reflexive, then for 1~, hi ~ D ^, n2 UA (t, S){¢, hi is a strong solution of (FDE) f o r 0 _ < s _ < t _ < T. Proof. UA(t,s)lc~,h] is Lipschitz continuous and, hence, n2UA(t,s)[th, hl is Lipschitz continuous and so differentiable a.e. Now suppose that, in addition, f is linear and DA(f(t, ") = X . We prove a variation of parameters formula. We denote by B(t) the operator D(B(t)) = 1{4~,h} e L l x X; 4~ e w l ' l ( - r , 0; X ) , (o(O) = h e D ( f ( t , .))} B(t)[4~, hi = {4a',f(t, h)], that is the operator A ( t ) with g = 0.
Nonautonomous delay equations
705
PROPOSITION 8. L e t f be linear with Df(t, -) = X and satisfy (M) and (A.1) and g satisfy (g.2). Then, for [4~,hi e L ~ × X, U~(t, s)14~, hi = UB(t, s){~, hi +
f' l'
Us(t, OlO, g(r, rq U~(t, 014~, hl)l dr.
$
From proposition 6 we have that n2UB(t,s)[~, h} = Us(t,s)h and so, projecting onto X, we have x(t) = Us(t, s)h +
Uy(t, r)g(r, x,) dr,
s
where x(t) is the integral solution of (FDE). Proof. Set G(t)[4, h} = [0, g(t, 4~)1 so that A f t ) = B(t) + G(t). Then IIG(t){~,, hi - G(O{~,
kill - B[I~ - ~l[ + Lg(ll~ll)lh(t)
- h(01.
Now, if v(t) is the unique integral solution of (E), it is also the unique integral solution of du d t ~ B(t)u + k(t),
0 <_ s <_ t <_ T,
u(s)
[4, hi,
(8)
where k(t) = G(t)v(t) and is continuous. For, if [x, y] e B(r), then [x, y + G(r)x} e A(r) so [IV(t) - xll - Hv(r) - x[I <-
I
[(y + G(r)x,
v(a)
- x)+
t
+ (~o +/~)llv(a) - xll + L(llxll)(1 + Ily +
G(r)xll)lh(~)
-
h(r)l} da.
However,
_<
+ G(~)v(~,),
v(~,) - x>+ +/~llv(~)
- xll + lib(a) - h(r)[lLg(llxll),
and IIGtr)xll
so that
<- IIG(r)x -
G(0)011 + IIG(0)011 --/~llxll + [h(r) - h(0)[Lg(0) + IIG(0)01,
v(t) is indeed the integral solution of (8). Hence v(t) is also the mild solution of (8) v(t) = UB(t, s)[4~, h} +
f'
UB(t, r)G(r)v(r) dr.
$
However, v(t) = UA(t, s){4~, h} and the result follows. 3. P R O P A G A T I O N
OF S O L U T I O N S
We shall now use proposition 8 to relate the propagation of solutions of (FDE) and the related ordinary equation (DE)
2(0 e f ( t , x(t)), x(s) = h.
0 <_ s <_ t <_ T,
706
J. DYSON and R. VILLELLA-BRESSAN
We say that the evolution operator U(t, s)propagates the subsets {Xt]t > o if U(t, s)X s C X t for
t>_s>_tr. From proposition 6 we have that n2Un(t, s)[O, hi = U:(t,s)h. It follows that if UB(t,s) propagates {Xt]t~. o, then Uy(t, s) propagates [n2Xt]t>o. Now, as in the autonomous case [12], given a family [Xt} propagated by U:(t, s) we want to find families [Y,} propagated by UB(t, s) and UA(t, s) such that rt2 Yt C Xt. Let P be a subset o f L 1 and define Yo(a) = P x Xo. Set Q(a) = [q/: [a - r, oo) --, x ; q/o ~ P, ~'lto,®) is continuous, ~u(t) ~ Xt for all t ___ a] and Qt(o-) = [i/tt : q / e Q(a)], so that Qo(a) = P, and let Yt(o') = [{t~, hi E Ot(o') x xt; h = t#(0)] = [[q~, q~(0)]; $ E Qt(tT)]. Note that we have rt2 Yt C Xt, but for equality we need some sort of smoothness on the t-dependence of Xt. In lemma 5 o f [12] it was proved that if Xt is an increasing family o f closed subspaces o f X, then n2 ~ = Xt if and only if U X , = Xt. r
We first prove that LEMMA 5. If Uf(t, s) propagates [Xtlt>~, a family of subsets of X, then Us(t, s) propagates
I~(a)]. Proof. Let [~, hi e YAa) so that there is a ~ a Q(a) such that 4~ = ¢ts and h = ~u(s). Then n2 Us(t, s){¢, hi = U:(t, s)h e Xt and
n~ Us(t, s){¢ hi = U:(t + O, s)h = ~u(t + O)
t+O>_s s-r<_t+O<_s.
So if we define ~ e Q(a) by ~ ( r ) = ~u(r) = U:(r, s)h
a - r _< r ___ s r > s,
t h e n nl Us(t, s)[th, h} = ~ t a n d ~t(0) = Uf(t, s)h = rtz Us(t, s)14~, hi, so t h a t
Us(t, s)16, hi ~ Yt(a). We now consider the propagation of the solutions of the delay equation. We restrict f to be linear. THEOREM 3. Let f(t) be linear and satisfy Df(t, .) = X, and (A. 1), and (M) and g satisfy (g. 1). Let x(t) be the integral solution of (FDE). Suppose that: (a) Uf(t, s) propagates the family of closed subspaces [Xt}t> o; (b) g(t, Qt(a)) c X, t >_ a. Then {q~,hi ~ Y,(a) implies that x(t) ~ Xt, t _> s _ a _> O.
Nonautonomous delay equations
707
P r o o f . Let 1~, hi e Y~(a), so that there is ~, e Q(a) such that ~ = ~u~. F r o m proposition 8 Uf(t, Og(r, x.) dr.
x(t) = Uf(t, s)h + $
We set up an iteration as follows a-r<_t<.s
x°(t) = ~(t)
t>_s,
= Uy(t, s)h
so x ° ~ Q(o) and Ix °, x ° ( t ) l ~ Yt(a), and x"(t) = ~u(t)
a - r <_ t <_ s t
I
= Uf(t,s)h +
t>_s.
Uf(t,r)g(r, x T - 1 ) d r ,
$
Then Ix'+'(t) - x(t)l <
i
t
IUy(t, r)g(r, xT) - Uf(t, r)g(r, xDI dr
$
and so IIXn+l -- xllL=(s,,) -<
I te~('-')~'(r)llx~
- x, llL®(-r.X)dr --%
$
.l
e~ry(r)llx ~ - xl[L®(s.,)dr.
$
So
y.(r)ilx ° - xllL~(~,o dr,
IIx "+' - xllL=(.,,) ~ s
where Yn(r) = y(r) e ~r
~,_l(a) da,
y0(r) = e a r y(z).
s
Hence, xn(t) converges uniformly to x(t). Now suppose x ' - l ( t ) ~ Q(a), so that, if r > a, [x~ -1, x"-l(z)] ~ Y,(a). Thus g(r, x~ -1) ~ X~, so Uy(t, r)g(r, x~ -1) ~ X t and x ' ( t ) ~ X t, t > a. Hence x ~ ~ Q(a) as it is also continuous by induction. So [x~, x"(r)} ~ YT(a) and the induction is complete. Finally, x(t) = lim x"(t) ~ X t as X t is closed. n~co
COROLLARY3. Suppose further that g satisfies (g.2). Then UA(t,s) propagates
Yt(a) for
r~S>_G.
P r o o f . Let [~, hi e Y~(a), so there exists gJ e Q(a) such that ~ = ~s and h = ~(s). If x(t) is the integral solution of (FDE), UA(t, s)[(h, h} = [xt, x(t)]. So, if we define
y ( r ) = ~(r)
=x(r)
a-r_
r>_s,
then y ~ Q(a) and UA(t, S)14), h l { y t , y ( t ) l E Yda).
708
J. DYSON and R. VILLELLA-BRESSAN
We now prove a partial converse of this corollary. TrmOREM 4. Suppose that f i s linear, Df(t, .) = X , and satisfies (A.I), with to = o~, and (M); suppose that g is linear and satisfies (g.2). For some tr _> 0, let [Xt}t, o be an increasing family of closed subpaces of X. Suppose that: (a) n 2 ~ ( t r ) = X t , t >_ a;
(b) UA(t, S) propagates [Yt(tr)}t~.o; (c) g(t, Qt(tr)) c X t t >_ ix. Then Uy(t, s) propagates [Xt}t~ o. Proof. Since B(t) = A ( t ) - G(t), G(t) = 10, g(t, -)} we can use proposition 8 to obtain
UB(t, S)[4', hi = gA(t, S)I¢, h} + f t UA(t,0{0, g(r, rq UB(r, S))}l~, h} dr. s
Define u°(t) = UA(t, S)[¢, hi, un(t) = UA(t, S)I~, h} +
t ___ a
f'
UA(t, s)G(Ou~-'(O dr.
$
Then, using a similar proof to that in theorem 3, it can be proved that u~(¢) converges uniformly
to UB(t, S)[~, hi. Since UA(t,s) propagates IYt(a)lt>~, UA(t,s) also propagates I ~ ( a ) b , ~ . Also, by [12, lemma 6], if {qL h} e Yt(a), G(t)[~, h} = {0, g(t, 4~)} e ~(a). Thus G(t, "): ~ ( a ) ~ ~(a). Suppose now that h ~ Xs, s >_ a. Then there exists q~ such that I~, h} e ~ ( a ) and, therefore, u°(t) = UA(t, s)l~b, hi e ~(tr). Suppose that u"-l(r) e g ( a ) , t _> a; then G(r)u"-l(r) e g(tr) so that jts UA(t, s)G(r)u"-l(r)dr e ~(a) and, therefore, u~(t) e Yt(a). Thus, Un(t, s){¢, hi e ~(a), so that Uf(t, s)[$, hi e n2 ~(a) = X t . 4. A N E X A M P L E
We consider the first order partial differential delay equation au(x t) Ot
~.. au(x t) [o 2(t+O--~x +j G(t, u(x, t + O)) dO, -r
u(x, s) = h(x),
h ~ C o = Co(-Oo, co),
-oo < x < oo , t >_ s >_ O,
us(x, 0) = $(x, 0),
~b: R x I - r , 01 ~ R,
where G: [0, o0) × R ---, R satisfies IG(t, u) - G(r, v)l---fllu - vl + Ih(t) - h(r)lt(lul) for some fl > 0, where h: [0, co) ~ R is continuous and of bounded variation, and L is a monotone increasing continuous function which satisfies the condition L(x) <_ ax + b, a and b e R (this is equivalent to saying that there is a monotone increasing function L1 such that for every ~b e L l ( - r , 0; Co),
f--ro L(I~(o)I)
dO _ L,
(f --ro I~(o)1 d 0)
.
Nonautonomous delay equations
709
We m a y r e f o r m u l a t e our p r o b l e m as the abstract functional differential equation in the Banach space Co
2(0 = f ( t , x(t)) + g(t, xt), x(s) = h,
0 <_ s <. t < T,
xs = 6
for (~, h) ~ L l ( - r , 0; Co) × Co, where
Ou f(t,u)=2(t+
Df(t,.)=[ueCo;u'eCo]
1)~xx,
and g: [s, T] × L l ( - r , 0; Co) --, C o is defined by
g(t, 4,) =
f ° a ( t , 4~(0)) dO. --r
Set J× (t)u = (I - i f ( t ,
Ilf(t, u) - f(-c,
-))-lu,
u)ll
fx (t, u) = f(t, J× (t)u), then
= 2It -
lliu'll
= Ilf(r, u)ll It - rl ~ Ill(r, u)lllt - r[. r+l
Hence, by [4, C h a p t e r 1, proposition 6.1], [IA(t)u - Jx(r)ul[ -< Z[[fx(t, u)lllt - r[. It is routine to show that f ( t , .) is m-dissipative and Df(t, .) = Co. Hence condition (A.1) holds, with aJ = 0. It can be seen that, if k e C 1 ( - o o , oo), the strong solution o f du
d---t = f ( t , u),
u(s) = k
is u(x, t) = k(x + (t + 1) 2 - (s + 1) 2) and, hence,
Uf(t,s)k = k(x + (t + 1) 2 - (s + 1) 2)
(9)
for k e C 1. H o w e v e r , as C 1 0 C O is dense in Co, (9) also holds for k e C 0. So, if we s e t B ( a , t ) = l x ; - a < _ x + ( t + 1) 2 - 1 < a l a n d Zt = {k e C0(R); supp k C B(a, t)l the family {Ztl is p r o p a g a t e d by Uy(t, s). H o w e v e r , the Zt are not increasing and this will be required later. So instead we define C(a, t) = Ix; x + (t + 1) 2 - 1 _> - a , x __. a} and
S t = {k ~ Co(R); supp k C C(a, t)},
X o = Z o.
These sets are closed, increasing and are also p r o p a g a t e d by Uy(t, s). Suppose now that: (i) P C [q~ ~ L l ( - r , 0; Col; ~b(0) ~ Xo, 0 ~ [ - r , 0]}; and
(ii)
G(t, Xt) C X t .
Then g(t, Qt(~r)) c X t . Thus, we m a y apply t h e o r e m 2 to find that if [0, hi e Y~(a), then
x(t) e X , .
710
J. DYSON and R. VILLELLA-BRESSAN REFERENCES
1. BENILAN PH., Equations d'evolution dans un espace de Banach quelconque et applications, Thesis. Universit6 de Paris XI, Orsay (1972). 2. BENILAN PH., Solutions integrales d'equations d'evolution dans un espace de Banach, C.r. Acad. Sc. Paris 274,47-50 (1972). 3. CRANDALL M. G. & EVANS L. C., On the relation of the operator a/Os + O/Or to evolution governed by accretive operators, Israel J. Math. 21,261-278 (1975). 4. PAVEL N. H., Nonlinear Evolution Operators and Semigroups, Lecture Notes in Mathematics. Springer, Berlin (1987). 5. CRANDALL M. G. & PAZY A., Nonlinear evolution equations in Banach spaces, Israel J. Math. 11, 57-94 (1972). 6. EVANS L. C., Nonlinear evolution equations in an arbitrary Banach space, Tech. Summary Report no 1568. Math. Res. Center Madison, Wisconsin (1975). 7. EVANS L. C., Nonlinear evolution equations in Banach spaces, Israel J. Math. 26, 1-42 (1977). 8. WEBB G. F., Asymptotic stability for abstract nonlinear functional differential equations, Proc. Am. math. Soc. 54, 225-395 (1976). 9. DYSON J. & VILLELLA-BRESSAN R., Functional differential equations and nonlinear evolution operators, Proc. R. Soc. Edinb. 75A, 223-234 (1976). 10. DYSON J. & VILLELLA-BRESSAN R., Nonlinear functional differential equations in L ~ spaces, Nonlinear Analysis 1, 383-395 (1977). 11. DYSON J. & VILLELLA-BRESSAN R., Semigroups of translations associated with functional and functional differential equations, Proc. R. Soc. Edinb. 82A, 171-188 (1979). 12. DYSON J. & VILLELLA-BRESSAN R., Propagation of solutions of a non linear functional differential equation, Diff. Integral Eqs. 4, 293-303 (1991). 13. DYSON J. & VILLELLA-BRESSAN R., On the propagation of solutions of a delay equation, Ren. Sem. Mat. Univ. Padova 80, 55-63 (1988). 14. GRIMMER R. & ZEMAN M., Wave propagation for linear integrodifferential equations in a Banach spaces, J. diff. Eqns 54, 274-282 (1984). 15. DESCH W. & GRIMMER R., Invariance and Wave propagation for nonlinear integrodifferential equations in Banach spaces, J. Integral Eqns 8, 137-164 (1985). 16. CRANDALL M. G., A generalised domain for semigroup generators, Proc. Am. math. Soc. 37, 434-440 (1973). 17. DORROH J. R. & RUIZ RIEDER G., A singular quasilinear parabolic problem in one space dimension, J. diff. Eqns 91, 1-23 (1991). 18. MARTIN, R., Nonlinear Operators and Differential Equations in Banach Spaces. Wiley, New York (1976). 19. CRANDALL M. G. & NOHEL J. A., An abstract functional differential equation and related nonlinear Volterra equation, Israel J. Math. 29, 313-328 (1978). 20. PLANT A., Nonlinear semigroups of translations in Banach spaces generated by functional differential equations, J. math. Analysis. Applic. 60, 67-74 (1977).