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Estimates regarding the decay of solutions of functional differential equations
,Von!incnr Analysu. Theor). Printed m Great Bntain.
Methods
& Applic~~n~.
Vol. 8. NO. 12. pp. 1395-1108.
Oj62-546X8-1 S 00 - .OO @ 19s Pcrgamon Press Ltd.
1984.
ESTIMATES REGARDING THE DECAY OF SOLUTIONS FUNCTIONAL DIFFERENTIAL EQUATIONS
OF
J. R. HADDOCK* Department
of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152, U.S.A
and T. KRISZTIN~ University of Szeged, Bolyai Institute, 6720 Szeged, Hungary (Received.12
August 1983; received for publication 23 January 1984)
Key words and phrases: Asymptotic equation, Razumikhin methods.
behavior,
exponential
and nonexponential
decay, comparison
1. INTRODUCTION
of this paper is to study the rapidity of convergence (to zero) of solutions of functional differential equations (FDE) with finite delay. Our main techniques involve differential inequalities and comparison theorems, and the principal intent is to provide results that can be applied in a direct manner. Investigations that deal with convergence properties of solutions of differential equations (ordinary, partial and functional) are quite often concerned with exponential decay. Likewise, the results presented here frequently yield exponential convergence. On the other hand, we are sometimes able to estimate asymptotic behavior that is not exponential (example 4.2), while in other instances convergence that is faster than exponential can be deduced (example 4.5). In Section 2, we provide the general setting and underlying assumptions. Section 3 contains a statement of the main result, with applications and comments given in Section 4. In order to facilitate the exposition of the paper, the proofs have been delayed until Section 5.
THE PURPOSE
2. PRELIMINARIES
Let r 3 0 be given and let C = C([-r, 01, R”) denote the space of continuous functions that map the interval [--I, 0] into R”. For $ E C, define the norm of # by 1)@/I= _z;+, 1q(s) 1, . .
where ( . 1 denotes any convenient norm in R”: If x: [to - r, to + A) + R” is continuous on O
(2.1)
where F:Rf X CF+ R”. (R+ and R- denote the set of nonnegative and nonpositive real numbers, respectively, CF C C and x’(t) denotes the right derivative of x with respect to t.) This paper was supported in part by the National Science Foundation under grant 8301304. t This paper was written while the second author was a visiting faculty member at Memphis State University. l
1395
J. R. HADDOCK and T. KRISZTIN
1396
Let to E R’ and (p, E CF be given, A function ,r( .) is said to be a solution of (2.1) through (to, ho) if there exists A, 0 < A 6 3:) such that: x( . ) is defined and continuous on [rO- r, to + A), absolutely continuous on bounded subintervals of [to, to + A), xc E CF for t E [to, to + A), xb = 6 and (2.1) holds almost everywhere on [to, r. + A). We assume: for each (to, q&o)E R’ X CF, there exists a solution x(a) of (2.1) on an interval [to, to + A), where either A = = or lim,_,-,IX(f0, &o>(f)I = =.
r
2.1. If CF = C and F: R’ x C+ R” is continuous and maps closed and bounded sets into bounded sets, then (HO) holds. Moreover, x( .) satisfies (2.1) on [to, to + A) and x’( .) is continuous on (to, to + A). For general results along these lines, we refer to [8, Chapter 21. Remark
Remark
2.2. Throughout
this paper, we rely on the relationship
between the limit
~~~_~[i~(O)+6F(t,~)l-I~(O)II (WFR+xG) and the upper-right
(2.2)
Dini derivative def D+lx(t)[=
b5*;[ix(l+
S)i
-
ix(t)/].
We note that the limit (2.2) always exists (cf. [13, p. 37]), and it is straightforward that, for any solution x( .) of (2.1) on [to, to + A),
(2.3)
to show
(2.4)
a.e. on [to, to + A). We consider the following assumptions:
,“y_ i [I Q(O) + SF(r, 4)
I - I444 II 0-J
where the function h : R’ -+ R’ is continuous, strictly increasing on R’, continuously differentiable on (0, ~0)and h(0) = 0, >
0-W
Estimates regarding the decay of solutions of functional differential equations
1397
and the functions LY:R+ + R-, /3:R’ + RC, y: [-r, a) -+ (- a, 0) are continuous on their respective domains with a(f) + P(r) G 0 for all t E R’.
(H3)
Remark 2.3. We now illustrate two cases for which (Hl) holds. (i) Suppose h( .) satisfies (H2), y: [--r, 00) --, (- 00, 0) is continuous and
(2.5) Then h( II$ll) = e-n~r+T)drh(_~axo S
l@(s)I) efltiffrjdT
e-lO-rfir+r)dr _yo
{h( 1Q(s) I) efiy(r+r)dr}.
By choosing any /I( .) such that P(t) > b(t) e-Prtif+s)dr, we see that (Hl) holds. (ii) Now, suppose h satisfies (H2) and
~~*~[l~(O)+dF(r,0)l-l~(0)11 s 4VGN9I) where ,u: [--r, 0] + RC is nondecreasing
I”-7 h( / e(s)
(2.6)
+ b(t) j”-, h(I@Wl) d&), on [-r, 01. Then
I) dp(S) = J‘o h( I G(s) I) e-~~r’r)dr’~~rtr)dTd~(~)
0 4 For any continuous
--I
exp
- ’ y(t + t) dt dp(S) . mrIo (1 I 1
{h( I@(s) I) eflfif-r)dr}.
y: [-r, m) --, (--00 , 0) and /3(. ) such that P(r) 3 b(t) I’% (exp(-
6’ tit + r)) dr) d&),
we again have an inequality of the form (Hl). For given functions h, CY,p and y such that (Hl), function g: R+ x R -+ R by Y0)
cr(t>+ B(I),h’(u)
g(t’u,= [a(r)
(H2) and (H3) are satisfied, define the
h(u); u > 0, h’(u) > 0
+ p(r)]h(u); u > 0, h’(u) = 0
(2.7)
1398
J. R.
Obviously, g is continuous
on
HADDOCK and T. KRISZTIN
R' x (0, ~0). Consider the initial value problem
u’(r) = g(r, u(0); u(t0) = uo(t 2 to), where (ro, uo) E R' x R. Denote by u(&, ug)( *) the maximal solution of (2.8). result, (2.8) serves as a comparison model for solutions of equation (2.1). 3. THE
MAIN
(2.8) In our main
RESULT
We are now ready to state the main theorem. Examples are provided in the next section, while a proof of the theorem is developed in Section 5. For results that are related to those presented here see [16]. THEOREM 3.1. Suppose (HO), (Hl), (H2) and (H3) are satisfied. Then, for every (to, $00)E RC x CF,each solution x(ro, +I)( .) of (2.1) is defined on [to, a) and I&
&l)(t) I 6 4t0, uo>(t>
(t 2 to>,
(3.1)
where u(to, uo)( .) is the maximal solution of (2.8) with u. =
max h -‘[h(l~(~)i)exp(l’Y(to+ --14520 s
r)dr)].
(3.2)
The reader should be alerted that theorem 3.1 is a Razumikhin-type comparison result. General results along these lines are thoroughly developed in [12, Vol. II, pp. 81-1101. In fact, except for fairly straightforward modifications, theorem 3.1 can be considered as a special case of theorems 8.1.4 and 8.2.5 in the above reference. We emphasize, though, that the main contribution here does not reside in the proof of theorem 3.1. Rather, it lies in the applications and, more specifically, in the stipulation of the comparison ordinary differential equation (2.8) and the initial value ug given by (3.2) once the auxiliary functions a; p and Y have been chosen. The following two corollaries provide obvious, yet quite useful, cases for which (2.8) can be explicitly solved. COROLLARY3.1. In addition to the hypotheses
of theorem
3.1, suppose there exist to a
0,
IQ> 0 such that a(r) + /J(f) < 0
(t 3 to)
and Y(f) inf h’(u) 2 sup O d t)ls lx(to, h)(r) 1G h-'[I
~1,
exp(i: Y(S) ds)]
COROLLARY3.2. In addition to the hypotheses u1 > 0 such that &) + B(t) < 0
of theorem (t 3 to)
(3.3)
3.1, suppose there exist to B 0,
Estimates regarding the decay of solutions of functional differential equations
1399
and r(t) s;gU, h’(u) C inf rrro a(t) + P(r)’ Then, for 0 < ~0 = -2~~
h-1[h(
1&J(S) 1) exp@
tit0 + t) d $1
s
~1,
(3.4) where
H(z) =
&$j
4. EXAMPLES
(0 < z s Ul). AND
COMMENTS
The key idea in applying the results of the previous section Lies in the choice of u( a). Choices for a( .) and h( .f are usually obvious, while /3( *) wili often be determined after y(e) is chosen. The first example illustrates that the estimate in theorem 3.1 is, in some sense, the best possible. Example 4.1. Consider the nonautonomous, x’(t) = -(l
linear delay equation (r = 1)
-+ t-’ + (1 + t>-‘)X(t) + x(t - 1)
and set H.s) = 2/(s + 2)(-l
(x(f) H?,tQ,
(4.1)
s s c 0).
Natural choices for 4.) and h(a) are o(t) = -(l f t-’ + (1 + t)-‘) and h(u) = U. Then (2.5) holds for b(t) = 1. Let y( *) be defined by y(f) = -(t + 1)-t. From (i) of remark 2.3, we wish to choose p( - ) such that
while a(t) + p(t) < Oft L 1). By choosing h(f) = 1 + r-l, we see that the conditions of theorem 3.1 and, in particular, corollary 3.1 are satisfied. This follows since
h’(u) = 1 =
u(t) &ft)
+
@)’
Thus, uo= and, as a consequence
2 max -v-1zSasrios +2
I
s-I-2 =I. 2 I
of (3.3), Ml7 ~)~f) I 6 exp (Hyde)
To see that this estimate cannot be improved, the solution of (4.1).
=2(r+l)-?
we need only to note that x(t) = 2(t + 1)-t is
1400
J. R. HADDOCK and T. KRISTIN
For the next two examples, we consider the scalar equation X’(f) = a(f)xA(t) + b(t)xTt - r(t)) where A is the quotient OG8<1
of odd integers,
(x(t) E R,ta:),
t: R' ---, [0,t-1 is continuous
a(r) S a
(3.2)
and for some a < 0,
(f20),
(4.3)
and /b(t)] G -&z(r)
(f 2 0).
(4.4)
Let o< 0 be chosen such that 8e-” < 1,
(4.5)
and set a(r) = a i- (u(t) - u)Oe-m, h(u) = u’, y(t) f a, P(t) = -u(t)8 Example
4.2. Let L > 1. For notational
convenience,
p = a(1 - Oe-w),
e-O”.
set
q = (A - 1)-t
(4.7)
From (2.7), we have g(t, u) =
=
L
[max{u + (u(r) - c)f3e-” o. u ~ o
puA;
o
%; h
u>
I
$E C([-r,
ixq AP) 4
us0
0;
Now,let
u
- a(t)f?e-m, Au~‘@-~)}]u*;u > 0
(> AP
01, R), u. =
_zm,
1$(s) 1eto/*js
If u. c ~1, then corollary 3.2 implies MO, @)(t)l 6
(t 2 0). (1 - (A -“;)pua-+)q
If uo > ut, then 1x(0, G)(r) I 6 where tt is defined by u. e(d’)tl = ut ;
u0 e
(d4r
(t E [O, t,])
i.e.
and
I-m @,>(O I6
Ul
(1 - (A - l)p&‘(t
(4.6)
(r.2
- t#
t,).
Estimates regarding the decay of solutions of functional differential equations
Hence, for all @E C([-r,
1101
01, RI,
jx(0,@)(f)I = O(tF)
= O(t-';'"-'J)
(c -
=).
(4.8)
To see that this particular estimate cannot be improved. we need only to consider the case b(t) = 0, a(t) = CI.Then (4.2) reduces to an ordinary differential equation that can be explicitly solved with (4.8) resulting. Example 4.3. Let h < 1 and once again consider (4.2)-(4.7).
puA;
For this case
()
u >
-
(7
Q
AP
Let @E C([--r, 01,R), uo = n-
and
If uo s ui, then corollary 3.1 imphes /x(0, $~)(t)! G ug e(“l’)r
(f c 0).
If cl0> ul, then
where tl is defined by (ub-” -(A. - l)pt~)-’
= ul; i.e.
1-A
- 6” tl = u”c* _ l)p and 1x(0, p)(f) / 6 u1 e(“‘d)(t-:l) So, for any $J E C([-r,
O], R) and all o< 0 such that OeeQ< 1
(i.e. o> (l/r) In e), we have 1x(0, G)(t) 1= O(eco/A)‘)
(4.9)
1102
J. R. HADDOCK and T. KRISZTIN
For b(t) = 0 and A < 1, it is easy to solve (4.2) to see that convergence to 0 is faster than exponential. The next example indicates that the estimate (4.9) is sharp for b(r) f 0 in the sense that we cannot obtain convergence that is faster than exponential. More importantly, it illustrates that (4.9) does not hold in general if o< (l/r)ln 6’. Example
4.4. Consider an initial value problem x’(t) = -e1&*~3(t) + x’/3(f - 1) x0
=
$0,
@O(S)
2
(x(r) E R, r 2 0)
-1Gs60.
for
e-’
Suppose there exists r1 Z- 0 such that x(t) = x(0, &o)(t) 2 e-’ for t E [0, ti] and x(ti) = e-l’. Then x’(r,) =
_e’/3Xl/3(r,)
+ X’/3(t, -
1) 2
_e1/3 e-u:3)‘l
+ e’“OL- 1) = 0,
which implies x(0, &o>(r) 3 e-’ In particular,
(t 3 0).
(4.9) does not hold for a< (l/r) In 8= -l/3,
but it does hold for -l/3
< (;r< 0.
The next example illustrates that, in some circumstances, exponential. Example
convergence
can be faster than
4.5. Consider the equation
x’(t) = -2eh(x(t)) where r: R- --* [0, l] is continuous
and h: R-,
+ h(x(t - t(t))), R is defined by
c
eu; u > l/e
h(u) =
G;O
10; u s 0.
It is not difficult to verify that the conditions of corollary a(t) = -2e, y(t) = -1, P(r) = e and ui = l/e (t 2 0). Then
3.1 hold with r = 1, to = 0,
implies 1x(0,h)(t) Remark
/6
~$6 e+
(t 2 0).
4.1. It is noteworthy that the techniques in examples 4.2 and 4.3 do not directly include the linear case, 1. = 1, for (4.2)-(4.5). However, this case is not difficult to handle.
Estimcltss
regarding
the decay
of solutions
of functional
differential
equations
l-W3
For example. suppose (4.3), (4.4), (4.5) hold for A = 1 and o< 0. Then, for r(r) = (3, corollary 3.1 yields ix(t)i = O(ewr) as I* r for w = max{a(l - Oewm), a}. In particular, (4.9) holds if u is the solution of the equation (4.10)
u= a(1 - @e-“) which yields the best value for w. Note that (4.10) is precisely the characteristic
equation for
x’(f) = ax(t) - Bax(r - r);
whereas, if A < 1, we obtain, as before, a “characteristic
inequality”
l-8e-“>O, which is independent
of a and yields a smaller value for o than (4.10).
In the previous examples we relied on the estimate (2.5) of remark 2.3 to obtain (Hl). However, as the next example illustrates, it is sometimes better to use (2.6) even if (2.5) is available. Example
4.6. Consider x’(t) = ax’(t) + bxA(t - 1) + cx$
- 2),
(4.11)
where a + lb1 + jcl
I+
(4.12)
x
whenever >
(4.13)
On the other hand, (2.6) holds for cr(t) = a, b(r) = 1, h(u) = d, r = 2 and
*P(S)=
lbl + 1~1, -l
s = -2.
For tit) = u < 0 and /3(t) = J!$ em du(s) = /b 1e-O + Icl e-2b, we apply corollary 3.1 to obtain (4.12) whenever
u=ui
Ib(e-'+
Icle-*',I.=
a+(ble-‘+lcje-?“
1
(4.14)
We emphasize that (4.14) provides a better estimate than (4.13). The next logical step in examining equation (4.2) would be to try to weaken the conditions in (4.3) and/or (4.4), and still obtain estimates regarding convergence properties of solutions.
1404
J. R. HADDOCK and T. KRISZTIN
If we were to drop (4.4) then we can no longer guarantee exponential 1 (cf. example 4.1). Likewise, if we retain (4.4) and change (4.3) to IF a(t)dt= -=, i0
decay. even for J. =
(4.15)
then we cannot be assured of exponential convergence since x(f) = (t + 1))’ is a solution of (4.2) for a(t) = -(t + 1)-i, b(t) = 0, J. = 1. However, all solutions tend to zero as I--, = (cf. [7]). Even without (4.3), we can sometimes obtain asymptotic estimates of solutions of (4.2). For example, consider (4.2) with a(0 G - -
1
(t + l)P
, 1b(t)
1 s - es(t),
O
O
Then we can argue (again using corollary 3.1 with
>’ p(t) =
-+)eexp(/O YQ +9 ds), ~0) =
-(t
-r
for suitable constant c > 0) that, for each (to, &) E R’ x C([-r,
+
,.‘+ c)p
01, R), O
x(to, $%)>-(0= A=1 as c-+ m, where p, S are arbitrary numbers, provided p > 0 and 0 < 6 < 1 - 0. However, for a general function a( .) such that (4.4) and (4.15) hold, it is not yet evident how we might estimate the rapidity of convergence of solutions of (4.2). This difficulty is due to the not always obvious choice of the function r( - ). In case (4.15) holds, but (4.4) does not, sufficient conditions are still available for solutions to tend to zero as t--, m (cf. [18]). But it seems that new results are needed in order to provide asymptotic estimates for this situation. Remark
4.3. Although the previous examples hold for dimension n = 1 (x(r) E R), the techniques are also useful for higher dimension. For example, consider equation (2.1), where F(t, $) =f(t, 4(O)) + g(t, @)(fi R+ x RR-_, R”, g: R- x C+ R”), and let a: R- -+ R- and b: R’ -+ R’ be continuous such that (4.16) fortER+,xER”and (4.17)
Estimates regarding the decay of solutions of functional differential equations
1405
for t E R’, $I E C. Then (2.5) holds, and the previous techniques apply. In particular, if (4.3) and (4.4) hold, then (4.9) is valid whenever u satisfies (4.5) or (4.10) according to the case 0 < k < 1 or A = 1. Note that (4.16) is true for h(u) = u if, for example, -f(t, . ) + a(t)1 is accretive for each t E R’. (I is the identity on R”.) We refer to [13] or [6, Section 4] for details. Also, several authors have considered an autonomous version of this (cf., e.g. [15]). Remark 4.4. Equations for which each constant function is a solution are becoming increasingly important in applications (see [2,3,5,9]). Unfortunately, our techniques do not seem to extend to handle these equations. (For such equations, we would often have (2.5), but with b(r) = -o(t).) Techniques have been developed for proving that solutions of such equations often tend to constant limits, but very little is known regarding the rapidity of convergence of these solutions. Exceptions to this may be found in [ll] and [l], but the results in these two papers only “scratch the surface” and more work is needed in this direction. Remark
4.5. We have been careful to define solutions and construct proofs so that theorem 3.1 still holds if R” is replaced by a general Banach space X. For results that are related to this case, see [4]. For a general discussion of functional differential equations in Banach space, we refer to the lecture notes of Webb [19]. 5. PROOF
OF THEOREM
3.1
We are now ready to develop the proof of our main result. As paragraph following the statement of theorem 3.1, our techniques comparison results. In addition to [12, Vol. II], we refer to [lo, Razumikhin methods. Let g be defined by (2.7) and h, a, /3, y be defined as in (H2), (H3). approximating differential equation for (2.8) u’(r) = g(t, u(t)) + &
was pointed out in the stem from Razumikhin 171 for a discussion of For E L 0, consider the
(1 2 ro)
u(to) = ug + E,
(5.1)
and let u,(a) = uE(fo,UO)( *) denote the maximal solution of (5.1). Clearly, uo( .) = u( . ), where u( .) is the maximal solution of (2.8). 5.1. (i) If u. = 0, then u(ro, uo)( .) = 0. (ii) If uo>O, and, for each .s>O,
LEMMA
u,(to + s) = h-‘[h(uXto))
exp( - (’ fita + r) dr)] 5
(s El-r,
Ol),
(5.2)
then u,(.) is defined and positive on [to, r~). Moreover,
hb4))
3 _~I:~
(h(udr +s)) exp([’
y(r + t) dr)}
(r 2 ro).
(5.3)
Proof. (i) Clearly, u(to, 0) = 0 since g(t, u) = 0 for u G 0 and g(r, u) s 0 for u > 0, t 2 0. (ii) From the definition of g, it follows that
$ [h&Q))]
= h’(u&))u:(f)
2 y(f)h(u&))
+ Eh’(G))
2 Y@)h(d))
1306
J. R. HADDOCK and T. KRISZTIS
for each I 2 ro such that udr) > 0. Using a standard comparison result (see, e.g. [l?. theorem 1.4.1]), we obtain (5.4)
for each t2 3 tl 2 to such that u,(t) > 0 for t E [to, t2]. From (5.4) it follows that ue(t) > 0 for t 2 to. By equation (5.1) and the definition of g, we have u;(t) s E, t Z- to. This implies u,(e) is defined on [to, cc). (5.3) can now be readily obtained from (5.2) and (5.4), thus completing the proof. LEMMA
5.2. Suppose
(HO), (Hl),
(H2),
(H3) are satisfied.
Then
for each (to, ho> E
R’ X C, x(to, $+I)(.) is defined on [to, m) and
lx(to, 4%)(W=ll~)oll
(t 2 ro).
(5.5)
Recall that x(t0, C#AJ)( -) denotes a solution of equation (2.1) through (to, 60). Proof. Let x( *) = x(t0, c&0)(-) be a solution of (2.1) on [to, to + A), A > 0. By assumption (HO), it is sufficient to show that (5.5) holds for t E [to, to + A). Define u: [to, to + A) ---$R- by
(5.6)
U(f) = ,_“32;<‘c, I@) 1.
The absolute continuity of x( .) implies that of u( .) on bounded subintervals of [to, to t A). Evidently, u’(t) = 0 whenever u(t) > Ix(t)1 and t E [to, to + A). From (2.4), (5.6). (Hl), (H2) and (H3), it follows that D’u(t)
S max{O, D-lx(t)
I} = 0 a.e. on {t E [to, to + A): u(t) = Ix(t)
}.
That is, u( .) is absolutely continuous on bounded subintervals of [to, to + A) and u’(t) s 0 a.e. on this interval. Thus, by employing (5.6), we see that the function u( .) must be constant on [to, to + A). This completes the proof. Proof of theorem 3.1. Lemma 5.1 guarantees that x( .) = x(to, &)( .) is defined on [to, 2). If ]I&]] = 0, then (3.1) follows from lemma 5.2 and part (i) of lemma 5.1. Suppose ]I&]] > 0. Then u. > 0. Now, suppose the theorem is false. Then there exists tl > to for which Ix(tl)I > u(to, uO)(tl). Consider the approximating equation (5.1). From a well-known property of such equations (cf. [12, Vol. I, lemma 1.3.1]), it follows that for E > 0 sufficiently small, we must have
IX(Q)l >
(5.7)
4t1>.
Let u,(a) be defined on [to - r, to] according to (5.2), and let T be the first time greater than to for which (5.8)
UC(T) = lx(T)]. By the definition of g and equation (2.7), we have u:(t) s [dt) +
P(OlhbXO)+ E
(t 3 ro).
(5.9)
1407
Estimates regarding the decay of solutions of functional differential equations
Assumptions
(H2), (H3) and the definition of T imply that
(Hl),
D’l4)
I c a(v440
I) i- PC0 _y$o
{h(uAr
+ s)) expc[
(5.10)
fit + t) dr)}
a.e. on [to, T]. From (5.9), (5.10), the definition of T and part (ii) of lemma 5.1, we have Wl4)l
-40
c a(W(lx(0
I> -
@4>N - E (5.11)
G
--E
-
cu(r)[h(uXt))
-
h(
lx(t) I)]
a.e. on [to, T].
By continuity of h(a), aft), u,(a) and 1x( a) ( and the fact that uE(T) = lx(T)\, 6 > 0 such that to s T - 6 and - a(r)[h(u,(t))
- h&c(f) I)] s ;
there exists
(f E [T - 4 7.1).
(5.12)
Now, (5.11) and (5.12) together imply D’( Ix(r) 1 - uE(r)) s - _s’a.e. on [T - 6, T]. Hence, we deduce that Ix(T) I - uXT> < jx(T - S) ( - uE(T - 6) -=c0. This contradicts
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
the choice of T in (5.8), thereby completing the proof of the theorem.
REFERENCES R. & STAFF&X 0. J., Integral inequalities and exponential convergence of solutions of differential equations with bounded delay, in Proceedings of 1982 Conference at Dundee, (Edited by W. N. EVERY and R. L. LEWIS) Lecture Nores in Marhemarics 964, 5658 Springer, New York (1983). COOKEK. L., KAPLANJ. L. & SORGM.. Stability of a functional differential equation for the motion of a radiating charged particle, Nonlinear Analysis 5. 1133-1139 (1981). COOKE K. L. gi YORKE J. A., Some equations modelling-_ growth processes and eonorrhea epidemics. Marh. Biosci. 16, 75-101 (1973). DYSONJ. & VILLELLA-BRESSAN R., Functional differential equations and nonlinear evolution operators, Proc. R. Sot. Edinb. Section A, 75, 223-234 (197511976). GY(~RI I., Connections between compa;tmental systems with pipes and integro-differential equations, Lecrure Notes in Biomathematics (to appear). HADDOCKJ. R., Asymptotic behavior of solutions of nonlinear functional differential equations in Banach space, Trans. Am. Math. Sot., 231, 83-92 (1977). WDOCK J. R., Some new results on stability and convergence of solutions of ordinary and functional differential equations, Funkcialaj Ekvacioj 19, 247-269 (1976). HOYLE J. K., Theory of Functional Differentiul Equations, Springer, New York (1977). KAPLAN J. L., SORG M. & YORKE J. A., Solutions of x’(t) =f(x(t),x(t - L)) have limits when f is an order relation, Nonlinear Analysis 3, 53-58 (1979). tiT0 J., On Liapunov-Razumikhin type theorems for functional differential equations, Funkcialaj Ekuacioj 16, 225-239 (1973). KRISZTINT., On the convergence of solutions of functional differential equations, Actu Sci. Math. 43, 4j_54 (1981). LAKSHMIK_Q~A.MV. & LEELA S., Differential and Integral Inequalities, Vols I, II, Academic Press, New York (1969). ATKINSONF. V., HADDOCK J.
1408
J. R. HADDOCK and T. KRISZTIN
13. MARTINR. H. JR., Nonlinear Operators and Differential Equations in Banach Spaces, Wiley-Interscience,
New York (1976). 14. MARTIXR. H. JR., The logarithmic derivative and equations of evolution in a Banach space, /. Murh. Sot. Japan 22, 411-429 (1970). 15. PLILLTA. T., Stability of nonlinear functional differential equations using weighted norms, Housron J. Marh. 3, 99-108 (1977). 16. POZIOM., Estimates of the solutions decay to equilibria for integro-differential equations, in Eoolurion Equarions and their Applications (Edited by F. KAP~ELand W. SCHAPPA~HER)Pitman, Boston (1982). 17. RAZ~IIKHIN B. S.. On stabilitv svstems with delav. Prikl. Marh. Me&. 20. 500-512 11956). 18. TERJGKIJ., On the asymptotic stability of solutions of functional differential equations,’ Ann. Pal. Mar. 36.
299-314 (1979). 19. WEBB G. F., Nonlinear Semigroups Superiore, Pisa (1982).
and Functional
Differential
Equations,
Lecture Notes, Scuola Normale
Methods
& Applic~~n~.
Vol. 8. NO. 12. pp. 1395-1108.
Oj62-546X8-1 S 00 - .OO @ 19s Pcrgamon Press Ltd.
1984.
ESTIMATES REGARDING THE DECAY OF SOLUTIONS FUNCTIONAL DIFFERENTIAL EQUATIONS
OF
J. R. HADDOCK* Department
of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152, U.S.A
and T. KRISZTIN~ University of Szeged, Bolyai Institute, 6720 Szeged, Hungary (Received.12
August 1983; received for publication 23 January 1984)
Key words and phrases: Asymptotic equation, Razumikhin methods.
behavior,
exponential
and nonexponential
decay, comparison
1. INTRODUCTION
of this paper is to study the rapidity of convergence (to zero) of solutions of functional differential equations (FDE) with finite delay. Our main techniques involve differential inequalities and comparison theorems, and the principal intent is to provide results that can be applied in a direct manner. Investigations that deal with convergence properties of solutions of differential equations (ordinary, partial and functional) are quite often concerned with exponential decay. Likewise, the results presented here frequently yield exponential convergence. On the other hand, we are sometimes able to estimate asymptotic behavior that is not exponential (example 4.2), while in other instances convergence that is faster than exponential can be deduced (example 4.5). In Section 2, we provide the general setting and underlying assumptions. Section 3 contains a statement of the main result, with applications and comments given in Section 4. In order to facilitate the exposition of the paper, the proofs have been delayed until Section 5.
THE PURPOSE
2. PRELIMINARIES
Let r 3 0 be given and let C = C([-r, 01, R”) denote the space of continuous functions that map the interval [--I, 0] into R”. For $ E C, define the norm of # by 1)@/I= _z;+, 1q(s) 1, . .
where ( . 1 denotes any convenient norm in R”: If x: [to - r, to + A) + R” is continuous on O
(2.1)
where F:Rf X CF+ R”. (R+ and R- denote the set of nonnegative and nonpositive real numbers, respectively, CF C C and x’(t) denotes the right derivative of x with respect to t.) This paper was supported in part by the National Science Foundation under grant 8301304. t This paper was written while the second author was a visiting faculty member at Memphis State University. l
1395
J. R. HADDOCK and T. KRISZTIN
1396
Let to E R’ and (p, E CF be given, A function ,r( .) is said to be a solution of (2.1) through (to, ho) if there exists A, 0 < A 6 3:) such that: x( . ) is defined and continuous on [rO- r, to + A), absolutely continuous on bounded subintervals of [to, to + A), xc E CF for t E [to, to + A), xb = 6 and (2.1) holds almost everywhere on [to, r. + A). We assume: for each (to, q&o)E R’ X CF, there exists a solution x(a) of (2.1) on an interval [to, to + A), where either A = = or lim,_,-,IX(f0, &o>(f)I = =.
r
2.1. If CF = C and F: R’ x C+ R” is continuous and maps closed and bounded sets into bounded sets, then (HO) holds. Moreover, x( .) satisfies (2.1) on [to, to + A) and x’( .) is continuous on (to, to + A). For general results along these lines, we refer to [8, Chapter 21. Remark
Remark
2.2. Throughout
this paper, we rely on the relationship
between the limit
~~~_~[i~(O)+6F(t,~)l-I~(O)II (WFR+xG) and the upper-right
(2.2)
Dini derivative def D+lx(t)[=
b5*;[ix(l+
S)i
-
ix(t)/].
We note that the limit (2.2) always exists (cf. [13, p. 37]), and it is straightforward that, for any solution x( .) of (2.1) on [to, to + A),
(2.3)
to show
(2.4)
a.e. on [to, to + A). We consider the following assumptions:
,“y_ i [I Q(O) + SF(r, 4)
I - I444 II 0-J
where the function h : R’ -+ R’ is continuous, strictly increasing on R’, continuously differentiable on (0, ~0)and h(0) = 0, >
0-W
Estimates regarding the decay of solutions of functional differential equations
1397
and the functions LY:R+ + R-, /3:R’ + RC, y: [-r, a) -+ (- a, 0) are continuous on their respective domains with a(f) + P(r) G 0 for all t E R’.
(H3)
Remark 2.3. We now illustrate two cases for which (Hl) holds. (i) Suppose h( .) satisfies (H2), y: [--r, 00) --, (- 00, 0) is continuous and
(2.5) Then h( II$ll) = e-n~r+T)drh(_~axo S
l@(s)I) efltiffrjdT
e-lO-rfir+r)dr _yo
{h( 1Q(s) I) efiy(r+r)dr}.
By choosing any /I( .) such that P(t) > b(t) e-Prtif+s)dr, we see that (Hl) holds. (ii) Now, suppose h satisfies (H2) and
~~*~[l~(O)+dF(r,0)l-l~(0)11 s 4VGN9I) where ,u: [--r, 0] + RC is nondecreasing
I”-7 h( / e(s)
(2.6)
+ b(t) j”-, h(I@Wl) d&), on [-r, 01. Then
I) dp(S) = J‘o h( I G(s) I) e-~~r’r)dr’~~rtr)dTd~(~)
0 4 For any continuous
--I
exp
- ’ y(t + t) dt dp(S) . mrIo (1 I 1
{h( I@(s) I) eflfif-r)dr}.
y: [-r, m) --, (--00 , 0) and /3(. ) such that P(r) 3 b(t) I’% (exp(-
6’ tit + r)) dr) d&),
we again have an inequality of the form (Hl). For given functions h, CY,p and y such that (Hl), function g: R+ x R -+ R by Y0)
cr(t>+ B(I),h’(u)
g(t’u,= [a(r)
(H2) and (H3) are satisfied, define the
h(u); u > 0, h’(u) > 0
+ p(r)]h(u); u > 0, h’(u) = 0
(2.7)
1398
J. R.
Obviously, g is continuous
on
HADDOCK and T. KRISZTIN
R' x (0, ~0). Consider the initial value problem
u’(r) = g(r, u(0); u(t0) = uo(t 2 to), where (ro, uo) E R' x R. Denote by u(&, ug)( *) the maximal solution of (2.8). result, (2.8) serves as a comparison model for solutions of equation (2.1). 3. THE
MAIN
(2.8) In our main
RESULT
We are now ready to state the main theorem. Examples are provided in the next section, while a proof of the theorem is developed in Section 5. For results that are related to those presented here see [16]. THEOREM 3.1. Suppose (HO), (Hl), (H2) and (H3) are satisfied. Then, for every (to, $00)E RC x CF,each solution x(ro, +I)( .) of (2.1) is defined on [to, a) and I&
&l)(t) I 6 4t0, uo>(t>
(t 2 to>,
(3.1)
where u(to, uo)( .) is the maximal solution of (2.8) with u. =
max h -‘[h(l~(~)i)exp(l’Y(to+ --14520 s
r)dr)].
(3.2)
The reader should be alerted that theorem 3.1 is a Razumikhin-type comparison result. General results along these lines are thoroughly developed in [12, Vol. II, pp. 81-1101. In fact, except for fairly straightforward modifications, theorem 3.1 can be considered as a special case of theorems 8.1.4 and 8.2.5 in the above reference. We emphasize, though, that the main contribution here does not reside in the proof of theorem 3.1. Rather, it lies in the applications and, more specifically, in the stipulation of the comparison ordinary differential equation (2.8) and the initial value ug given by (3.2) once the auxiliary functions a; p and Y have been chosen. The following two corollaries provide obvious, yet quite useful, cases for which (2.8) can be explicitly solved. COROLLARY3.1. In addition to the hypotheses
of theorem
3.1, suppose there exist to a
0,
IQ> 0 such that a(r) + /J(f) < 0
(t 3 to)
and Y(f) inf h’(u) 2 sup O
~1,
exp(i: Y(S) ds)]
COROLLARY3.2. In addition to the hypotheses u1 > 0 such that &) + B(t) < 0
of theorem (t 3 to)
(3.3)
3.1, suppose there exist to B 0,
Estimates regarding the decay of solutions of functional differential equations
1399
and r(t) s;gU, h’(u) C inf rrro a(t) + P(r)’ Then, for 0 < ~0 = -2~~
h-1[h(
1&J(S) 1) exp@
tit0 + t) d $1
s
~1,
(3.4) where
H(z) =
&$j
4. EXAMPLES
(0 < z s Ul). AND
COMMENTS
The key idea in applying the results of the previous section Lies in the choice of u( a). Choices for a( .) and h( .f are usually obvious, while /3( *) wili often be determined after y(e) is chosen. The first example illustrates that the estimate in theorem 3.1 is, in some sense, the best possible. Example 4.1. Consider the nonautonomous, x’(t) = -(l
linear delay equation (r = 1)
-+ t-’ + (1 + t>-‘)X(t) + x(t - 1)
and set H.s) = 2/(s + 2)(-l
(x(f) H?,tQ,
(4.1)
s s c 0).
Natural choices for 4.) and h(a) are o(t) = -(l f t-’ + (1 + t)-‘) and h(u) = U. Then (2.5) holds for b(t) = 1. Let y( *) be defined by y(f) = -(t + 1)-t. From (i) of remark 2.3, we wish to choose p( - ) such that
while a(t) + p(t) < Oft L 1). By choosing h(f) = 1 + r-l, we see that the conditions of theorem 3.1 and, in particular, corollary 3.1 are satisfied. This follows since
h’(u) = 1 =
u(t) &ft)
+
@)’
Thus, uo= and, as a consequence
2 max -v-1zSasrios +2
I
s-I-2 =I. 2 I
of (3.3), Ml7 ~)~f) I 6 exp (Hyde)
To see that this estimate cannot be improved, the solution of (4.1).
=2(r+l)-?
we need only to note that x(t) = 2(t + 1)-t is
1400
J. R. HADDOCK and T. KRISTIN
For the next two examples, we consider the scalar equation X’(f) = a(f)xA(t) + b(t)xTt - r(t)) where A is the quotient OG8<1
of odd integers,
(x(t) E R,ta:),
t: R' ---, [0,t-1 is continuous
a(r) S a
(3.2)
and for some a < 0,
(f20),
(4.3)
and /b(t)] G -&z(r)
(f 2 0).
(4.4)
Let o< 0 be chosen such that 8e-” < 1,
(4.5)
and set a(r) = a i- (u(t) - u)Oe-m, h(u) = u’, y(t) f a, P(t) = -u(t)8 Example
4.2. Let L > 1. For notational
convenience,
p = a(1 - Oe-w),
e-O”.
set
q = (A - 1)-t
(4.7)
From (2.7), we have g(t, u) =
=
L
[max{u + (u(r) - c)f3e-” o. u ~ o
puA;
o
%; h
u>
I
$E C([-r,
ixq AP) 4
us0
0;
Now,let
u
- a(t)f?e-m, Au~‘@-~)}]u*;u > 0
(> AP
01, R), u. =
_zm,
1$(s) 1eto/*js
If u. c ~1, then corollary 3.2 implies MO, @)(t)l 6
(t 2 0). (1 - (A -“;)pua-+)q
If uo > ut, then 1x(0, G)(r) I 6 where tt is defined by u. e(d’)tl = ut ;
u0 e
(d4r
(t E [O, t,])
i.e.
and
I-m @,>(O I6
Ul
(1 - (A - l)p&‘(t
(4.6)
(r.2
- t#
t,).
Estimates regarding the decay of solutions of functional differential equations
Hence, for all @E C([-r,
1101
01, RI,
jx(0,@)(f)I = O(tF)
= O(t-';'"-'J)
(c -
=).
(4.8)
To see that this particular estimate cannot be improved. we need only to consider the case b(t) = 0, a(t) = CI.Then (4.2) reduces to an ordinary differential equation that can be explicitly solved with (4.8) resulting. Example 4.3. Let h < 1 and once again consider (4.2)-(4.7).
puA;
For this case
()
u >
-
(7
Q
AP
Let @E C([--r, 01,R), uo = n-
and
If uo s ui, then corollary 3.1 imphes /x(0, $~)(t)! G ug e(“l’)r
(f c 0).
If cl0> ul, then
where tl is defined by (ub-” -(A. - l)pt~)-’
= ul; i.e.
1-A
- 6” tl = u”c* _ l)p and 1x(0, p)(f) / 6 u1 e(“‘d)(t-:l) So, for any $J E C([-r,
O], R) and all o< 0 such that OeeQ< 1
(i.e. o> (l/r) In e), we have 1x(0, G)(t) 1= O(eco/A)‘)
(4.9)
1102
J. R. HADDOCK and T. KRISZTIN
For b(t) = 0 and A < 1, it is easy to solve (4.2) to see that convergence to 0 is faster than exponential. The next example indicates that the estimate (4.9) is sharp for b(r) f 0 in the sense that we cannot obtain convergence that is faster than exponential. More importantly, it illustrates that (4.9) does not hold in general if o< (l/r)ln 6’. Example
4.4. Consider an initial value problem x’(t) = -e1&*~3(t) + x’/3(f - 1) x0
=
$0,
@O(S)
2
(x(r) E R, r 2 0)
-1Gs60.
for
e-’
Suppose there exists r1 Z- 0 such that x(t) = x(0, &o)(t) 2 e-’ for t E [0, ti] and x(ti) = e-l’. Then x’(r,) =
_e’/3Xl/3(r,)
+ X’/3(t, -
1) 2
_e1/3 e-u:3)‘l
+ e’“OL- 1) = 0,
which implies x(0, &o>(r) 3 e-’ In particular,
(t 3 0).
(4.9) does not hold for a< (l/r) In 8= -l/3,
but it does hold for -l/3
< (;r< 0.
The next example illustrates that, in some circumstances, exponential. Example
convergence
can be faster than
4.5. Consider the equation
x’(t) = -2eh(x(t)) where r: R- --* [0, l] is continuous
and h: R-,
+ h(x(t - t(t))), R is defined by
c
eu; u > l/e
h(u) =
G;O
10; u s 0.
It is not difficult to verify that the conditions of corollary a(t) = -2e, y(t) = -1, P(r) = e and ui = l/e (t 2 0). Then
3.1 hold with r = 1, to = 0,
implies 1x(0,h)(t) Remark
/6
~$6 e+
(t 2 0).
4.1. It is noteworthy that the techniques in examples 4.2 and 4.3 do not directly include the linear case, 1. = 1, for (4.2)-(4.5). However, this case is not difficult to handle.
Estimcltss
regarding
the decay
of solutions
of functional
differential
equations
l-W3
For example. suppose (4.3), (4.4), (4.5) hold for A = 1 and o< 0. Then, for r(r) = (3, corollary 3.1 yields ix(t)i = O(ewr) as I* r for w = max{a(l - Oewm), a}. In particular, (4.9) holds if u is the solution of the equation (4.10)
u= a(1 - @e-“) which yields the best value for w. Note that (4.10) is precisely the characteristic
equation for
x’(f) = ax(t) - Bax(r - r);
whereas, if A < 1, we obtain, as before, a “characteristic
inequality”
l-8e-“>O, which is independent
of a and yields a smaller value for o than (4.10).
In the previous examples we relied on the estimate (2.5) of remark 2.3 to obtain (Hl). However, as the next example illustrates, it is sometimes better to use (2.6) even if (2.5) is available. Example
4.6. Consider x’(t) = ax’(t) + bxA(t - 1) + cx$
- 2),
(4.11)
where a + lb1 + jcl
I+
(4.12)
x
whenever >
(4.13)
On the other hand, (2.6) holds for cr(t) = a, b(r) = 1, h(u) = d, r = 2 and
*P(S)=
lbl + 1~1, -l
s = -2.
For tit) = u < 0 and /3(t) = J!$ em du(s) = /b 1e-O + Icl e-2b, we apply corollary 3.1 to obtain (4.12) whenever
u=ui
Ib(e-'+
Icle-*',I.=
a+(ble-‘+lcje-?“
1
(4.14)
We emphasize that (4.14) provides a better estimate than (4.13). The next logical step in examining equation (4.2) would be to try to weaken the conditions in (4.3) and/or (4.4), and still obtain estimates regarding convergence properties of solutions.
1404
J. R. HADDOCK and T. KRISZTIN
If we were to drop (4.4) then we can no longer guarantee exponential 1 (cf. example 4.1). Likewise, if we retain (4.4) and change (4.3) to IF a(t)dt= -=, i0
decay. even for J. =
(4.15)
then we cannot be assured of exponential convergence since x(f) = (t + 1))’ is a solution of (4.2) for a(t) = -(t + 1)-i, b(t) = 0, J. = 1. However, all solutions tend to zero as I--, = (cf. [7]). Even without (4.3), we can sometimes obtain asymptotic estimates of solutions of (4.2). For example, consider (4.2) with a(0 G - -
1
(t + l)P
, 1b(t)
1 s - es(t),
O
O
Then we can argue (again using corollary 3.1 with
>’ p(t) =
-+)eexp(/O YQ +9 ds), ~0) =
-(t
-r
for suitable constant c > 0) that, for each (to, &) E R’ x C([-r,
+
,.‘+ c)p
01, R), O
x(to, $%)>-(0= A=1 as c-+ m, where p, S are arbitrary numbers, provided p > 0 and 0 < 6 < 1 - 0. However, for a general function a( .) such that (4.4) and (4.15) hold, it is not yet evident how we might estimate the rapidity of convergence of solutions of (4.2). This difficulty is due to the not always obvious choice of the function r( - ). In case (4.15) holds, but (4.4) does not, sufficient conditions are still available for solutions to tend to zero as t--, m (cf. [18]). But it seems that new results are needed in order to provide asymptotic estimates for this situation. Remark
4.3. Although the previous examples hold for dimension n = 1 (x(r) E R), the techniques are also useful for higher dimension. For example, consider equation (2.1), where F(t, $) =f(t, 4(O)) + g(t, @)(fi R+ x RR-_, R”, g: R- x C+ R”), and let a: R- -+ R- and b: R’ -+ R’ be continuous such that (4.16) fortER+,xER”and (4.17)
Estimates regarding the decay of solutions of functional differential equations
1405
for t E R’, $I E C. Then (2.5) holds, and the previous techniques apply. In particular, if (4.3) and (4.4) hold, then (4.9) is valid whenever u satisfies (4.5) or (4.10) according to the case 0 < k < 1 or A = 1. Note that (4.16) is true for h(u) = u if, for example, -f(t, . ) + a(t)1 is accretive for each t E R’. (I is the identity on R”.) We refer to [13] or [6, Section 4] for details. Also, several authors have considered an autonomous version of this (cf., e.g. [15]). Remark 4.4. Equations for which each constant function is a solution are becoming increasingly important in applications (see [2,3,5,9]). Unfortunately, our techniques do not seem to extend to handle these equations. (For such equations, we would often have (2.5), but with b(r) = -o(t).) Techniques have been developed for proving that solutions of such equations often tend to constant limits, but very little is known regarding the rapidity of convergence of these solutions. Exceptions to this may be found in [ll] and [l], but the results in these two papers only “scratch the surface” and more work is needed in this direction. Remark
4.5. We have been careful to define solutions and construct proofs so that theorem 3.1 still holds if R” is replaced by a general Banach space X. For results that are related to this case, see [4]. For a general discussion of functional differential equations in Banach space, we refer to the lecture notes of Webb [19]. 5. PROOF
OF THEOREM
3.1
We are now ready to develop the proof of our main result. As paragraph following the statement of theorem 3.1, our techniques comparison results. In addition to [12, Vol. II], we refer to [lo, Razumikhin methods. Let g be defined by (2.7) and h, a, /3, y be defined as in (H2), (H3). approximating differential equation for (2.8) u’(r) = g(t, u(t)) + &
was pointed out in the stem from Razumikhin 171 for a discussion of For E L 0, consider the
(1 2 ro)
u(to) = ug + E,
(5.1)
and let u,(a) = uE(fo,UO)( *) denote the maximal solution of (5.1). Clearly, uo( .) = u( . ), where u( .) is the maximal solution of (2.8). 5.1. (i) If u. = 0, then u(ro, uo)( .) = 0. (ii) If uo>O, and, for each .s>O,
LEMMA
u,(to + s) = h-‘[h(uXto))
exp( - (’ fita + r) dr)] 5
(s El-r,
Ol),
(5.2)
then u,(.) is defined and positive on [to, r~). Moreover,
hb4))
3 _~I:~
(h(udr +s)) exp([’
y(r + t) dr)}
(r 2 ro).
(5.3)
Proof. (i) Clearly, u(to, 0) = 0 since g(t, u) = 0 for u G 0 and g(r, u) s 0 for u > 0, t 2 0. (ii) From the definition of g, it follows that
$ [h&Q))]
= h’(u&))u:(f)
2 y(f)h(u&))
+ Eh’(G))
2 Y@)h(d))
1306
J. R. HADDOCK and T. KRISZTIS
for each I 2 ro such that udr) > 0. Using a standard comparison result (see, e.g. [l?. theorem 1.4.1]), we obtain (5.4)
for each t2 3 tl 2 to such that u,(t) > 0 for t E [to, t2]. From (5.4) it follows that ue(t) > 0 for t 2 to. By equation (5.1) and the definition of g, we have u;(t) s E, t Z- to. This implies u,(e) is defined on [to, cc). (5.3) can now be readily obtained from (5.2) and (5.4), thus completing the proof. LEMMA
5.2. Suppose
(HO), (Hl),
(H2),
(H3) are satisfied.
Then
for each (to, ho> E
R’ X C, x(to, $+I)(.) is defined on [to, m) and
lx(to, 4%)(W=ll~)oll
(t 2 ro).
(5.5)
Recall that x(t0, C#AJ)( -) denotes a solution of equation (2.1) through (to, 60). Proof. Let x( *) = x(t0, c&0)(-) be a solution of (2.1) on [to, to + A), A > 0. By assumption (HO), it is sufficient to show that (5.5) holds for t E [to, to + A). Define u: [to, to + A) ---$R- by
(5.6)
U(f) = ,_“32;<‘c, I@) 1.
The absolute continuity of x( .) implies that of u( .) on bounded subintervals of [to, to t A). Evidently, u’(t) = 0 whenever u(t) > Ix(t)1 and t E [to, to + A). From (2.4), (5.6). (Hl), (H2) and (H3), it follows that D’u(t)
S max{O, D-lx(t)
I} = 0 a.e. on {t E [to, to + A): u(t) = Ix(t)
}.
That is, u( .) is absolutely continuous on bounded subintervals of [to, to + A) and u’(t) s 0 a.e. on this interval. Thus, by employing (5.6), we see that the function u( .) must be constant on [to, to + A). This completes the proof. Proof of theorem 3.1. Lemma 5.1 guarantees that x( .) = x(to, &)( .) is defined on [to, 2). If ]I&]] = 0, then (3.1) follows from lemma 5.2 and part (i) of lemma 5.1. Suppose ]I&]] > 0. Then u. > 0. Now, suppose the theorem is false. Then there exists tl > to for which Ix(tl)I > u(to, uO)(tl). Consider the approximating equation (5.1). From a well-known property of such equations (cf. [12, Vol. I, lemma 1.3.1]), it follows that for E > 0 sufficiently small, we must have
IX(Q)l >
(5.7)
4t1>.
Let u,(a) be defined on [to - r, to] according to (5.2), and let T be the first time greater than to for which (5.8)
UC(T) = lx(T)]. By the definition of g and equation (2.7), we have u:(t) s [dt) +
P(OlhbXO)+ E
(t 3 ro).
(5.9)
1407
Estimates regarding the decay of solutions of functional differential equations
Assumptions
(H2), (H3) and the definition of T imply that
(Hl),
D’l4)
I c a(v440
I) i- PC0 _y$o
{h(uAr
+ s)) expc[
(5.10)
fit + t) dr)}
a.e. on [to, T]. From (5.9), (5.10), the definition of T and part (ii) of lemma 5.1, we have Wl4)l
-40
c a(W(lx(0
I> -
@4>N - E (5.11)
G
--E
-
cu(r)[h(uXt))
-
h(
lx(t) I)]
a.e. on [to, T].
By continuity of h(a), aft), u,(a) and 1x( a) ( and the fact that uE(T) = lx(T)\, 6 > 0 such that to s T - 6 and - a(r)[h(u,(t))
- h&c(f) I)] s ;
there exists
(f E [T - 4 7.1).
(5.12)
Now, (5.11) and (5.12) together imply D’( Ix(r) 1 - uE(r)) s - _s’a.e. on [T - 6, T]. Hence, we deduce that Ix(T) I - uXT> < jx(T - S) ( - uE(T - 6) -=c0. This contradicts
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
the choice of T in (5.8), thereby completing the proof of the theorem.
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J. R. HADDOCK and T. KRISZTIN
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and Functional
Differential
Equations,
Lecture Notes, Scuola Normale