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Attractivity in non-autonomous systems
1111.J. Non-l&ear Mechanics. Printed in Great Britain.
Vol. 20. No. S!6, pp. 519-528.
ATTRACTIVITY
198s
OOZO-7462/U S3.00 + 00 Pergamon Press Ltd.
IN NON-AUTONOMOUS SYSTEMS
TARO YOSHIZAWA Department of Applied Mathematics, Graduate School, Okayama University of Science, Okayama, Japan Abstract-We discuss some sufficient conditions for attractivity of a closed set by using Liapunov functions and show how to apply our results to the study of asymptotic behaviors of solutions in nonautonomous systems. We also discuss the convergence of solutions by the same idea. These results are extendable to functional differential equations.
For autonomous systems of ordinary differential equations, the classical Liapunov’s theorem on asymptotic stability was generalized by Barbashin and Krasovskii [l ] to the case where the derivative along the solution of a Liapunov function is not negative definite. LaSalle [5] made a simple observation based on the invariance property ofthe solutions. The general result has come to be referred to as “invariance principle”. A typical example is the Lienard’s equation xn + f(x)x’ + g(x) = 0 or an equivalent system x’ = y - F(x), where f and g are continuous g(x)F(x)
Y’ = -g(x)
in XE R = (- co, co) and F(x) =
(1) Xj(~)d~. Suppose that s0
> 0 (x # 0), xg(x) > 0 (x # 0) and G(x) =
Xg(u)du + cc as 1x1--, co, and consider s a Liapunov function V(x,y) = G(x) + y2/2. Then th\ derivative along the solution of (1) v(x,y) is equal to - g(x)F(x) $0, which is not negative definite. Hence we cannot apply the classical theorem on asymptotic stability, but by the invariance principle, we can see that the null solution of (1) is globally uniformly asymptotically stable. The main problem in the invariance principle is to find a set which attracts the solutions. LaSalle and others extended the invariance principle to non-autonomous systems and dynamical systems, see [6] for the references. In order to discuss the invariance principle in a non-autonomous system x’ = f(r, x) + g(r9x)
(2)
where f and g are continuous on I x D, I = [0, co) and D is an open set in R”, Yoshizawa [ll, 131 gave a sufficient condition for the attractivity of a closed set E in D by using a Liapunov function. A scalar function W(x) defined on D is said to be positive definite with respect to a set E, if W(x) = 0 for x E E and if corresponding to each E > 0 and each compact set K in D, there exists a 8(s, K) > 0 such that W(x) 1 8(s, K) for x E W(E, E) n K, where W(E, E) denotes the complement of the s-neighborhood N(E, E) of E. Then, if there exists a non-negative Liapunov function V(t, x) on I x D with q2)(t, x) 5 - W(x), where W(x) is positive definite with respect to a closed set E in D, and iff(t, x) is bounded for all t when x belongs to a compact set in D and
mIg(s, x(s))lh -C zc when x(t) is continuous and x(t) E K I for all t 2 f0 and some compact set ‘i in D, then every bounded solution of (2) approaches E as t -+ cc. The above result for the attractivity has been improved by many authors, see [14,15] for the works and the references in this direction. In this article, we shall discuss the attractivity for the non-autonomous system x’ = f(t,x) 519
(3)
520
T. YOSHIZAWA
where f‘(t, x) is continuous on I x D, I = [0, 00) and D is an open set in R”. The theorems for the functional differential equation x(t) = f(t, x(s)); a s s I t)
for
t20,aSO
(4)
which correspond to the theorems in this paper, have been reported without proofs [15]. Equation (4) has discussed extensively by Driver [3] and by Burton [2]. Equation (4) contains Volterra equations
W) = &x(r))
+
t Nt,s, I 0
-w)ds
+
skx(.))
and functional differential equations with finite delays x(t) = f(t, x,), where x,(s) = x(t + s), -h 5 s I 0, and also (4) includes delay equations x(t) = f(t,x(t), x(t - r(t))) with 0 5 r(t) 5 t and continuous r(t), where given to 2 0, the initial set E,, consists of to and those values t - r(t) for which t - r(t) S to when t 2 to. For a given set Kin D, we say that a scalar function v(t, x) is a Liapunov function for (3) and the set K, if i. ii. iii. M i
l’(t,x) v(t,x) there V(t,x)
is continuous on I x D, is locally Lipschitzian in x, exists an M and a continuous function r(t) 2 0 such that 7 EL, [0,30), for all (t,x)~l x K and I&(t,x) s y(t) for all (t,x)EI x K, where
I&(t, x) = lirn;upi
{v(t + 6, x + 6f(t, x)) - v(t, x)}.
Let E c D be a non-empty closed set in the topology of D. A function W: D + [0, co) is said to be positive definite on K with respect to E, if for any E > 0 there exists a B(E) > 0 such that W(x) 2 6(c) for x E NC@,E) n K. Following Burton [2], we say that a function H(t, x) is a pseudo-Liapunov function for the sets E and K, if there exists an open neighborhood U of E and H: I x {U n K} + (- CG,cio) is continuous and if H(t, x) has the following properties: i. For any E > 0, there is a A > 0 such that H(t, x) > i for x E NC(&,E) n K. ii, For any i > 0, there is an q > 0 such that H(t, x) < i/2 for x E N(q, E) n K. To introduce a pseudo-Liapunov function is convenient for applications. Let p(t) 2 0 be continuous on I. If
s
p(t)dt = cc holds on every set J = z (u,,,, b,) such
b, - hJ2 6 for a 6 > 0 (m = 1, 2,. . .), we”ia; that p(t) is thatO~aa,,a, 0, lim inf p(s)ds > 0. I-z I , Now we are ready to state our results. The following theorem is related closely to Theorem 2 in [2] which is a generalization of LaSalle’s and Yoshizawa’s results. Theorem 1
Let I’(&x) be a Liapunov function for (3) and a given set Kin D. Suppose that there exists a 1+1
continuous function e(t) 2 0 and a T > 0 such that ~3)(4 x) i
-p(t)W(x)
If
e(s)ds + 0 as t + 00 and that
+ e(t)
for all t 2 T and x E K where p(t) is integrally positive and W(x) is positive definite on Kwith
Attractivity in non-autonomous
systems
521
respect to a closed set E c D. Suppose that there exists a pseudo-Liapunov function H(t, x) for E and K. Assume that (a) for any a > 0 and any solution x = x(t) of (3) which remains in Kin the future, there exist T(a,x) > 0 and #Ifa, x) > 0 such that if Q > T(a, x), t > 0 and a c H(a + t, x(a + t)) - H(q x(a)), then t > jl(a, x), or (b) for any a > 0 and any solution x = x(t) of (3) which remains in Kin the future, there exist T(a,x) > 0 and /?(a,~) > 0 such that if a > T(or,x), t > 0 and H(a + t, x(a + t)) - H(u, x(a)) < -a, then t > /3(a, x). Then every solution of (3) which remains in K in the future approaches E as t 4 co. Proof. Let x(t) be a solution of (3) such that x(t) E K for all t 2 to. First of all, we shall show that for any rl > 0 there is a sequence {z,,,} such that r, -P co as m-+ co and x(T,) E N(Q E) n K. If not, there is a T1 > 0 such that x(t) E N’(q, E) n K for all t 2 Tl. Since W(x) is positive definite on K with respect to E, there exists a 6 = S(q) > 0 such that W(x(t)) 2 6 for all t 2 Ti. Since p(t) is integrally positive,
s 1+1
lim inf l-+az I
p(s)ds = 2a > 0,
t+1
and hence there is a T2 > 0 such that T3 > 0 such that
s
p(s)ds 2 a for all t 2 Tt. Moreover, there is a
f1
I+1
OS
s a+1
since
e(s)d.s <
f
2
for all 2 2 T3
e(s)ds + o as t + e. Let To = max(T, T’, T,, T3). Since x(t)~ K and To Z ‘Z we
f
have
i’(T,+k,x(T,+k))-
V(To,x(To))6
-k$3~F~+;+1~(s)ds
+;~~~~~‘l+k)d~
j=O
Therefore V(To + k, x(To + k)) --B- co as k -+ 00, which contradicts M I; V(t, x(t)). Now suppose that x(t) does not approach E as t + co. Then for some E > 0, there exists a sequence (t,,J such that tm + co as M + 00 and x&J E NC(s,E) n K. We can assume that N(2c, E) c U, where U is the domain of the definition ofthe ~eud~Liapunov function H(r, x). We assume condition (a). For the above E > 0, there is a A > 0 such that H(t,x) > I for x E !V(E, E) n K, and for this I there is an q > 0 such that 0 c q < E and H(t, x) < A/2 for x E N(q, E) n K. From the above, there are sequences {tk) and {CL}such that tk + cc as k + 00, tk < f; < tk+ 1, dist(x(t;), E) = 8, dist(x(rk), E) = q and q < dist(x(t), E) < E
for tk < t < t;.
For r E [tk, t;], q LT;dist(x(t), E), that is, x(t) E N’(q, E) n K and hence there is a 6 = S(q) > 0 such that W(x(t)) 2 6 for tk 5 t 5 t;. On the other hand, H(t;, x(t;)) > 1 and x(tk)E N(& E) implies H(fk,x(tk) $ A/2. Thus we have A/2 < H&x@;)) - H(tk,x(tk)). Therefore, by condition (a), t; - tk > p(n/2,x) = B > 0 if k is sufficiently large. +P Let [p] be an integer such that fl= [B ] - r, 0 5 r < 1. Since lim inf p(s)ds = 2b > 0 r-m f t 1+1
and s
t
e(s)ds + 0 as t + 03, c+B p(s)& 2 b fr
+1
and
1. YOSHIZAWA
522
if t is sufhciently large, and hence
J
t*+ J 3
t*+3
p(s)ds 2 ‘b
and
e(s)ds 5 ;
1k
1k
if k is su~~ient~y large. Since we can assume that ti is sufficiently large and we have V(Cj+
=
1
,X(tj+ 1))-
jY(t,+
1, x(tj+
v(tj, x(rj)) l))
-
V(tj
+
X(tj
8,
+
B)) + v(tj + b x(cj+ 8))-
VtjT
'('j))
we have
< =JOD rWs -ji, J
v(tk+,,x(tk+l))
-
v(fl~x(tl))
~+~~(s)~(x(~~)d~ tj
tx
+ i
ff”‘e(s)ds
j=l
fj
cc
= <
y(s)ds -
II
Then V(tk+ r , x(&+ i)) + - co as k + 00, which contradicts the boundedness from below of F’(t,x(t)). Thus we can conclude that x(t) approaches E as t -+ 00. The same argument can be applied to the case where condition (b) is satisfied. Remark 1
If there exists a continuous function q(r) 2 0 such that [0, CJZ)and d(t,x(t))
J
’ q(s)ds is unifo~Iy
~ontinuo~
on
i q(t), then condition (a) is satisfied, where
IQ, x(r)) = l$nl&
(H(t + 6, x(r + 6)) - H(t, x(t))}.
Remark 2 The condition corresponding
to LaSalle’s condition in [7] is as follows: (c)Given any a > 0 and any solution x = x(t) of (3) which remains in Kin the future, there exist T(a, x) > 0 and /I(@,x) > 0 such that
+ks, ww
>a
for
a>T(a,x)
and
t>O
implies t > fi(a,x). This condition implies condition (a) for a pseudo-Liapunov function H(x) = dist(x,E). It is clear that H(x) is a pseudo-Liapunov function for E and R Take T(a,x) and #I(a,x) in condition (c). Then, if a > T&x), t > 0 and a c H(x(a + t)) - ~(x(u)), then t must be greater
than /&X,X). In fact, suppose t 5 /3(a, x). Then
H(x(a + t)) - H(x(a)) S Ix(a + t) - x(a)1 =
f(s,x(s))ds
Ii”D
+‘f(s,x(s))ds
I; a. But
5 a, which is a contradiction.
Therefore t > &a, x)_ Example 1
Consider the equation x” + h(t, x, X’)IX’pX’ + g(x) = q(t, x, x’)
Attractivity
in non-autonomous
systems
523
or an equivalent system
x’ = Y, Y' = where LY2 continuous i. there such
w,x9 Y)lYl”Y -
g(x) + q(t,x9 Y)
(5)
0, h(t, x, y) is continuous on I x R x R, g(x) is continuous on R and q(t,x,y) is on I x R x R. We assume that exists an integrally positive function p(t) and a continuous function k(x, y) 2 0 that h(t,x,y) 2 p(t)k(x,y), X
ii. G(x) 2 0, where G(x) =
s0
g(u)du, m
iii. [q(t, x, y)l 5 e(t) for (x,y)~ R2, where e(t) is continuous and
e(s)ds < 03. s 0 t
Now consider a function
U(t, x, y) = e- E(f) {G(x) + y2/2 + l}, where E(r) =
e(s)ds. s 0
Then we have
%k x,Y) I -
ewt,x, Y)lYl” + 2s 0,
s m
where E. =
e(s)ds. Since e -Eoy2/2 5 U(t, x, y) and tit,, 5 0, y(t) is bounded as long as a
solution (x(l),:~(t)) of (5) exists. As x’(t) = y(t), each solution of (5) can be continued for all future time. (I) Suppose that k(x,y) > 0 if y # 0 and G(x) + co as (xl + 00. Then for every solution (x(t), y(t)) of (5), G(x(t)) + a constant and y(t) + 0 as t + co. Moreover, in addition, if xg(x) > 0 for x # 0, x(t) ---,c for some constant c as t + co. In this case, it is clear that the solutions of (5) are uniformly bounded. Let (x(t),y(t)) be a solution of (5) defined for t 2 to. Then there is an m > 0 such that Ix(t)1 2 ml!(t)1 5 m for all t 2 to. Then V(x, y) = G(x) + y2/2 is a Liapunov function for (5) and the set K = {(x, y);
I4 6 m,lyl S 4,
and
J&(x,Y) 5 -
PW(X,Y)IY~"Y~
+ meW.
(6)
H(y) = y2 and E = {(x, y), y = O}. Then W(x, y) is positive Let WX,Y) = Mx,~)lyl”+~, definite on K with respect to E, and H(y) is a pseudo-Liapunov function for E and K which satisfies
Since the solution is bounded and e E L1 [0, co), condition (a) in Theorem 1 is satisfied. Thus it follows from Theorem 1 that y(t) + 0 as t + 03. On theother hand, since U(t, x(t),y(t)) is nonincreasing and 0 s U(t, x(t),y(t)), U(t, x(t),y(t)) -+ a constant and consequently G(x(t)) + a constant as I -+ cc. If xg(x) > 0 (x # 0), x(t) + c for some constant c as t + 00 since the positive limit set is connected. (II) Suppose that k(x, y) = k*(y), where k*(y) > 0 for y # 0, and that Ig(x)l is bounded for all x. Thus we have the same conclusion as in (I). In this case, we have
%dx,Y) S
-pW*Wlyl”y2
+
e(t)lyl.
For the solution (x(t),y(r)) of (5) defined for t 2 to, there is an m > 0 such that ly(t)l 5 m for all t 2 to. Then W(y) = PCy)(y( 3c2 is positive definite on K = {(x, y); 1x1c 00, IyI 5 m} with respect to the set E = {(x, y); y = O}. Clearly H(y) = y2 is a pseudo-Liapunov function for E and K, and we have (7). Since Ig(x)l is bounded for all x, we have the conclusion by the same argument as in (I).
T. YOSHUAWA
524
(III) suppose that G(x) + 00 as 1x14 00 and h(t, x, y) satisfies condition (c) for any compact set K and that R- = { (x,O);x < 01 and R+ = {(x, 0);~ > 0} are connected components in E - {(O,O)},wh ere E = {(x, y);yk(x,y) = 0). Then we have the same conclusion as in (I). In this case, we have (6), and k(x, y)(~r+~ 1s . positive definite on K = {(x, y);lxl 5 m,lyl 5 m} with respect to E = {(x, y),yk(x,y) = 0). Since we assume that h satisfies condition (c), the vector field of (5) satisfies condition (c) for the compact set K. Therefore we can see that (x(t), y(t)) + E as t + co. Let Q be the positive limit set of (x(t), y(t)). Then R c E. If (0,O) E Q for any 6 > 0 there exists a point (x(t,), y(Q) which is in the b-neighborhood of the origin, but (0,O) is eventually uniformly stable by Theorem 17.4 in [ 121, and hence (x(t), y(t)) + (0,O) as t -+ 00. If (O,O)$Q 0c E - {(O,O)}. H owever, R n # C#J, where R = {(x, y); y = 0}, because if n n R = 4, Iy(t)l >= c for some constant c > 0 and thus Ix(t)\ + oc as t + co which contradicts the,boundedness of x(t). Therefore Rc R- u R+ by out assumption since R is connected. Hence Rc R, that is, y(t) + 0 as t + 00. The remainder is the same as in (I). Under the assumptions in (I) with x&x) > 0 for x # 0, if we assume the growth condition on h, that is, for any bounded continuous function (x(t), y(t)) on (I), there exists a sequence of positive numbers (~3 and a constant d > 0 such that sk+ I - Sk2 d and that h(t, x(t), y(t)) 3 0 on [sI, Sk+ d] for all k or
-
I/@+ 1)
W,
x(Q,
Y(0W
= 00,
then we can see that (x(t), y(t)) + (0,O) as t + co by the same argument as in [8]. In a special case where q(t, x, y) s 0, the null solution of (5) is equiasymptotically stable, because uniform stability and attractivitl unply equiasymptotic stability. In the case (III), the growth condition is automatically satisfied, because of condition (c). Another growth condition is 1 i=O
T s
W, x(W(t)W
5 B
for some B > 0 and all T > 0 and (x(t), y(t)) in a compact set. This condition is somewhat easily checked in applications, but it satisfies actually the above growth condition which is weaker [8]. Theorem
2
Let V(t, x) be a Liapunov function for (3) and a given set Kin D. Suppose that there exists a continuous function e(r) 2 0 and a T > 0 such that
e(t)dt < 00 and that
Y3)k x) 5 - p(W(x) + e(t)
s 03
for all t 2 T and x E X, where p(t) >=0 is continuous,
p(t)dt = co and W(x) is positive
definite on Kwith respect to a closed set E c D. Suppose tOhatthere exists a pseudo-Liapunov function H(x) for E and K. Moreover, suppose that for any solution x(t) of (3) which remains in Kin the future, there exist c > 0 and continuous functions q(t) 2 0 and r(t) 2 0 such that
4)
5 PW,
I
oc
r(t)dt < 00
0
and
Ifcw I 4(t) + 40
for all large t
or
-r(t) - 4W S If(xW)
for all large t.
Attractivity in non-autonomous
systems
525
Then every solution of (3) which remains in K in the future approaches E as t + co. Proof. Let x(t) be a solution of (3) such that x(t) E Kfor t 2 to. By the same argument as in the proof of Theorem 1, if we assume that x(t) does not approach E as t + cc and that d(x(t)) 5 q(t) + r(t) for all large t, there exist 1 > 0 and sequences {tk}, {t;} such that tk + co as k + 03, tk < fi < fk+ 1, H(X(t;)) > 3. and H(x(&)) 5 l/2. Moreover, there iS a b > 0 such that w(x(t)) 2 6 for tk 5 t 5 CL. Since we can assume that tr is sufficiently large, we have
However, we have 1 j <
H(X(Cj))-
H(x(fj))
5
6:’q(s)ds + j.; r(s)ds
and hence
Therefore we have
w;, W) s
- w19 x(h))
-~16~:jpods+~~(e(s)+.i(s))ds
5 -
i
dcl:jq(s)ds
+ l; (e(s) + r(s))ds
j=l &d-
(2
f@ls)
+ l" (44
+ y(s))ds
j=l s
-ticA,2+6c
i ~~r(+fs j=l
+ ~~(44
+ y(s))ds.
Thus V(&, x(Q) + - co as k + co, which contradicts the boundedness from below of V(t,x(t)). Thus we can conclude that x(t) approaches E as t + co. Example 2
Consider the system x’ = - u(t)x + q(t)y + cl(t) Y'
= -4(W -
(8)
P(~)Y + e,(t)
where all functions in (8) are continuous, u(t) 2 0, p(t) 2 0 and Setting E(t) =
(lel(s)l + le2(s)l)ds and E. =
(Iel(
m lei(t))dt < co (i = 1,2). s0
+ le2(s)l)ds, consider a function
u(t, x, y) = emE(‘)(x2+ y2 + 1). Then emEo(x2 + y2 + 1) 6 U(t,x,y) 5 x2 + y2 + 1 and Ij&t,x,y) 5 0. Therefore the solutions of (8) are uniformly bounded. Let (x(t), y(t)) be a solution of (8) defined for t 2 to.
T. YOWZAWA
526
Then there exists an m > 0 such that )x(t)( S m and Iy(t)l i m for t 2 to. If we consider a function V(x, y) = $x2 + y’),
&(x9y) 5 - 4Qx2-
P(~)Y~ +
el@)x+ e2WY.
Thus V(x, y) is a Liapunov function for (8) and the set K = {(x, y); (x1 S m, ly( S m}. (I) Suppose that p(t) is integrally positive and
Theny(t)+Oast+zo. In this case, we have l&,(x,y) s -p(t)y’ + 2mlez(t)l,
where
H(y) = y2. Set
s
;
IdW s is uniformly continuous on [0, cc).
+ m(lel(t)l + le,(t)l)
and If(y(t)) 5 2mzlq(t)l
q*(t) = 2m’lq(t)l + 2mle,(t)l.
uniformly continuous on [0, co), and hence condition Thus the conclusion follows from Theorem 1. (II) Suppose that u(t) and p(r) are integrally positive and
Then
(a) in Theorem
. q*(s)& is J0 1 is satislied.
‘)q(s)lds is uniformly continuous s0
on [0, co). Then (x(t),y(t)) + (0,O) as t --) 03. In this case, we have also I& (x,y) 5 -a(t)x’ + m(le,(t)l + le,(t)l), and for H(x) = x2, fi(x(t)) I 2m21q(t)l + 2mlel(t)l. Thus x(t) + 0 as t + m by Theorem 1. (III) Suppose that there is an a > 0 such that jq(t)l 5 ap(t) for t >= 0 and assume that xl p(t)dt = co. Then y(t) + 0 as t + cm. s0 In this case, we have rfs, (x,y) $ -p(t)y’ + m(lel(t)l + lez(t)l) and, for H(y) = y’, Let q*(t) = 2m21q(t)l and r(r) = 2mlez(t)l. Then fi(y(r)) 6 2mZ14@)l f We&K q*(t)/2am2 5 p(t) and H(y(t)) 2 q*(t) + r(t). Thus the conclusion follows from Theorem 2. (IV) Suppose that
s0
cop(t)dt = co and
I0
m Iq(t)(dt < co. Then y(t) + 0 as
c +
CL).
Co
In this case, letting r(t) = 2m21q(t)l + 2m(e2(t)l, we have ti(y(t))
5 r(t) and
s0
r(t)dt < co.
Thus the conclusion follows from Theorem 2. Now we shall discuss the convergence of solutions of (3). To do this, we consider the system associated to (3) x’ = f‘(t, x) Y' = f(4Y).
1
(9)
For a given set Kin D, let V(t, x, y) be a Liapunov function for (9) and the set S = K x K. Let E in R” be a non-empty closed set. Then a scalar function W: D x D + [0,03) is said to be positive definite on K with respect to E if for any E > 0 there exists a 8(s) > 0 such that W(x,y) 2 8(c) whenever (x, y)~ S and x - YE NC (E,E). Also, we say that a scalar function H(x, y) is a pseudo-Liapunov function for E and K, if there exists an open neighborhood U of E and H(x, y) is defined and continuous for (x, y) E S such that x - y E II and if H(x, y) has the property that i. for any E > 0, there is a I > 0 such that H(x, y) > ). for (x, y)~ S such that x - YE NC 6, E), ii. for any d > 0, there is an q > 0 such that H(x, y) -E 2/2 for (x, y)E S such that x - YEWLO Then, corresponding to Theorem 1, we have the following theorem which can be proved by the same idea as in the proof of Theorem 1. Theorem
3
Let l’(t, x, y) be a Liapunov function for (9) and the set K x K, where K is a given set in D.
Attractivity in non-autonomous
systems
527 r+1
Suppose that there exists a continuous function e(r) 2 0 and a T > 0 such that sf
e(s)ds
+Oast+coandthat
for all t 1 T and (x, y) E K x K, where p(t) is integrally positive and W(x, y) is positive definite on K with respect to a closed set E c R”. Suppose that there exists a pseudo-Liapunov function H(x, y) for E and K. Assume that (d) for any CI> 0 and any pair of solutions (x = x(t), y = y(t)) of (3) which remain in Kin the future, there exist T = T(a, x, y) > 0 and /I = B(a, x, y) > 0 such that if a > T, t > 0 and a < H(x(a + t), y(a + t)) - H(x(a),y(a)), then t > j?, or (e) for any a > 0 and any pair of solutions (x = x(t), y = y(t)) of (3) which remain in Kin the future, there exist T = T(a, x, y) > 0 and /I = /I(a,x,y)>Osuchthatifa>T,t>Oand H(x(a + t), y(a + t)) - H(x(a), y(u)) < -a, then t > /3. Then every pair of solutions {x(t),y(t)} of (3) which remain in K in the future satisfies x(t) - y(t) -+ E as t + co. Remark 3
The condition corresponding to condition (c) is as follows: (c*) Given any a > 0 and any pair of solutions (x = x(t), y = y(t)) of (3) which remain in K in the future, there exist T(a, x, y) > 0 and /?(a,x, y) > 0 such that - f(s, y(s))]ds > a for a > T(a, x, y), t > 0 implies t > /l(a,x, y).
IJ”0
+* [f(s, x(s))
Condition (c*) implies condition (d) for a pseudo-Liapunov function H(x,y) = dist(x - y, E). For example, if If(t,x) - f(t,y)l is bounded for all t when x - y is bounded, any pair of solutions (x(t),y(t)) which remain in K in the future and whose difference is bounded satisfies x(t) - y(t) + E as t + 00. This is a generalization of a result obtained by Swick [lo]. Now we shall see an example considered by Murakami [9] and by Swick [lo]. Example 3
Consider an equation x” + p(t)f(x’)
+ b(t)x = e(t)
(10)
and an equivalent system x' = y, y' = -p(t)f(y)
- b(t)x + e(t)
(11)
where all functions are continuous. We assume that i. equation (10) has a solution x0(t) which exists for all t 2 0 and whose derivative x&(t) is bounded for t 2 0, ii. p(t) is integrally positive, iii. f(y) is strictly monotone increasing for all y, iv. 0 c bl 6 b(t) S b2, and either b’(t) $ 0 or b’(t) 10. Then for every pair of solutions of (lo), the difference of their derivatives tends to zero as t -+ m.
To see this, consider the associated system Y, y’ = -p(tlf(y) U’ = U, U‘ = -p(t)f(v)
X’ =
- b(t)x + e(t) - b(t)u + e(t).
(12)
We consider the case where b’(t) 5 0. Consider a function v(t, x,y,u, u) = b(t)(x - u)’ + (y - u)~.
(13)
528
T. YOSHIZAWA
Then we have q12,wYJv)
s -2P(~)Lw
-_fI~)lCv- 450.
(14)
Since x,(t) exists for all t 2 0 and xb(t) = ye(t) is bounded for all t 2 0, it follows from (13) and (14) that all solutions of (10) exist in the future and their derivatives are bounded, because the difference of any pair of solutions of (11) is bounded, that is, system (11) is distancebounded by Theorem 15.4 in [12]. Let (x(t), y(t)) and (u(t), u(t)) be a pair of solutions of (11) defined for t 2 t,,. Then there is an m > 0 such that b(t)1 5 m, Iv(t)15 m for all t 2 to. Let K= {(x,y)~R’: 1x1< co, Jy( $ m} and E= ((x,y)eR2: y=O}. Then V(t,x,y,u,o) is a Liapunov function for (11) .and K x K and W(x, y, U,u) = [f(y) - f(u)](y - II) is positive definite on K with respect to E. If we set HCy, u) = (y - u)~, H is a pseudo-Liapunov function for E and K and we have H(y(t), u(t)) 6 26,(x(t) - u(t)))y(t) - u(t)1 5 a constant, because Ix(t) - u(t)! and Iy(t) - u(t)1 are bounded. Therefore, by Theorem 3, y(t) - u(t) + 0 as t -P co. For the case where b’(t) 2 0, consider a function I/ = (x - u)~ + 0, - ~)~/b(t). REFERENCES 1. E. A. Barbashin and N. N. Krasovskii, On global stability of motion, Do/cl. AU. Nauk SSSR, 86,453-456 (1952). 2. T. A. Burton, Stability theory for functional differential equations, Dons Am. Math. Sot. 255,263-275 (1979). 3. R. D. Driver, Existence and stability of solutions of a delay-differential system, Archs rot. Mech. Anal., 10, 401426 (1962). 4. L. Hatvani, Attractivity theorems for non-autonomous systems of differential equations, Acta Sci. Math. (Szeged) 40,271-283 (1978). 5. J. P. LaSalle, The extent of asymptotic stability, Proc. Natn. Acod. Sci. USA 46, 363-365 (1960). 6. 1. P. LaSalle, Stability theory and invariance principle. In Dynamical Systems. An International Symposium, Vol. 1, pp. 211-222. Academic Press, New York (1976). 1. J. P. LaSalle, Stability of nonautonomous systems, Nonlinear Anal. I, 83-91 (1976). 8. S. Murakami, Asymptotic behavior of solutions of ordinary differential equations, Tohoku Math. .I. 34,559-574 (1982). 9. S. Murakami, Asymptotic behavior of solutions of some differential equations, J. Math. Anal. Appl. to be published. 10. K. E. Swick, Convergence of solutions of nonlinear differential equations, Ann. Mar. Pura Appl. 14,1- 126 (1977). 11. T. Yoshizawa, Asymptotic behavior of solutions of a system of differential equations, Contrib. Difirential Equations 1, 371-387 (1963). 12. T. Yoshizawa, Stability Theory by Liapunods Second Method. The Mathematical Society of Japan, Tokyo (1966). 13. T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, Vol. 14. Springer, Berlin (1975). 14. T. Yoshizawa, Asymptotic Behavior o/’ Solutions in nonautonomous Systems, Trends in Theory and Practice of nonlinear Difirenriol Equarions, pp. 553-562, Marcel Dekker, New York (1984). 15. T. Yoshizawa, Asymptotic behaviors of solutions of differential equations, in Proceedings of the Colloquium on the Qualitative Theory of Differential Equations, Szeged, August 27-31, 1984, to be published.
Vol. 20. No. S!6, pp. 519-528.
ATTRACTIVITY
198s
OOZO-7462/U S3.00 + 00 Pergamon Press Ltd.
IN NON-AUTONOMOUS SYSTEMS
TARO YOSHIZAWA Department of Applied Mathematics, Graduate School, Okayama University of Science, Okayama, Japan Abstract-We discuss some sufficient conditions for attractivity of a closed set by using Liapunov functions and show how to apply our results to the study of asymptotic behaviors of solutions in nonautonomous systems. We also discuss the convergence of solutions by the same idea. These results are extendable to functional differential equations.
For autonomous systems of ordinary differential equations, the classical Liapunov’s theorem on asymptotic stability was generalized by Barbashin and Krasovskii [l ] to the case where the derivative along the solution of a Liapunov function is not negative definite. LaSalle [5] made a simple observation based on the invariance property ofthe solutions. The general result has come to be referred to as “invariance principle”. A typical example is the Lienard’s equation xn + f(x)x’ + g(x) = 0 or an equivalent system x’ = y - F(x), where f and g are continuous g(x)F(x)
Y’ = -g(x)
in XE R = (- co, co) and F(x) =
(1) Xj(~)d~. Suppose that s0
> 0 (x # 0), xg(x) > 0 (x # 0) and G(x) =
Xg(u)du + cc as 1x1--, co, and consider s a Liapunov function V(x,y) = G(x) + y2/2. Then th\ derivative along the solution of (1) v(x,y) is equal to - g(x)F(x) $0, which is not negative definite. Hence we cannot apply the classical theorem on asymptotic stability, but by the invariance principle, we can see that the null solution of (1) is globally uniformly asymptotically stable. The main problem in the invariance principle is to find a set which attracts the solutions. LaSalle and others extended the invariance principle to non-autonomous systems and dynamical systems, see [6] for the references. In order to discuss the invariance principle in a non-autonomous system x’ = f(r, x) + g(r9x)
(2)
where f and g are continuous on I x D, I = [0, co) and D is an open set in R”, Yoshizawa [ll, 131 gave a sufficient condition for the attractivity of a closed set E in D by using a Liapunov function. A scalar function W(x) defined on D is said to be positive definite with respect to a set E, if W(x) = 0 for x E E and if corresponding to each E > 0 and each compact set K in D, there exists a 8(s, K) > 0 such that W(x) 1 8(s, K) for x E W(E, E) n K, where W(E, E) denotes the complement of the s-neighborhood N(E, E) of E. Then, if there exists a non-negative Liapunov function V(t, x) on I x D with q2)(t, x) 5 - W(x), where W(x) is positive definite with respect to a closed set E in D, and iff(t, x) is bounded for all t when x belongs to a compact set in D and
mIg(s, x(s))lh -C zc when x(t) is continuous and x(t) E K I for all t 2 f0 and some compact set ‘i in D, then every bounded solution of (2) approaches E as t -+ cc. The above result for the attractivity has been improved by many authors, see [14,15] for the works and the references in this direction. In this article, we shall discuss the attractivity for the non-autonomous system x’ = f(t,x) 519
(3)
520
T. YOSHIZAWA
where f‘(t, x) is continuous on I x D, I = [0, 00) and D is an open set in R”. The theorems for the functional differential equation x(t) = f(t, x(s)); a s s I t)
for
t20,aSO
(4)
which correspond to the theorems in this paper, have been reported without proofs [15]. Equation (4) has discussed extensively by Driver [3] and by Burton [2]. Equation (4) contains Volterra equations
W) = &x(r))
+
t Nt,s, I 0
-w)ds
+
skx(.))
and functional differential equations with finite delays x(t) = f(t, x,), where x,(s) = x(t + s), -h 5 s I 0, and also (4) includes delay equations x(t) = f(t,x(t), x(t - r(t))) with 0 5 r(t) 5 t and continuous r(t), where given to 2 0, the initial set E,, consists of to and those values t - r(t) for which t - r(t) S to when t 2 to. For a given set Kin D, we say that a scalar function v(t, x) is a Liapunov function for (3) and the set K, if i. ii. iii. M i
l’(t,x) v(t,x) there V(t,x)
is continuous on I x D, is locally Lipschitzian in x, exists an M and a continuous function r(t) 2 0 such that 7 EL, [0,30), for all (t,x)~l x K and I&(t,x) s y(t) for all (t,x)EI x K, where
I&(t, x) = lirn;upi
{v(t + 6, x + 6f(t, x)) - v(t, x)}.
Let E c D be a non-empty closed set in the topology of D. A function W: D + [0, co) is said to be positive definite on K with respect to E, if for any E > 0 there exists a B(E) > 0 such that W(x) 2 6(c) for x E NC@,E) n K. Following Burton [2], we say that a function H(t, x) is a pseudo-Liapunov function for the sets E and K, if there exists an open neighborhood U of E and H: I x {U n K} + (- CG,cio) is continuous and if H(t, x) has the following properties: i. For any E > 0, there is a A > 0 such that H(t, x) > i for x E NC(&,E) n K. ii, For any i > 0, there is an q > 0 such that H(t, x) < i/2 for x E N(q, E) n K. To introduce a pseudo-Liapunov function is convenient for applications. Let p(t) 2 0 be continuous on I. If
s
p(t)dt = cc holds on every set J = z (u,,,, b,) such
b, - hJ2 6 for a 6 > 0 (m = 1, 2,. . .), we”ia; that p(t) is thatO~aa,,a,
Let I’(&x) be a Liapunov function for (3) and a given set Kin D. Suppose that there exists a 1+1
continuous function e(t) 2 0 and a T > 0 such that ~3)(4 x) i
-p(t)W(x)
If
e(s)ds + 0 as t + 00 and that
+ e(t)
for all t 2 T and x E K where p(t) is integrally positive and W(x) is positive definite on Kwith
Attractivity in non-autonomous
systems
521
respect to a closed set E c D. Suppose that there exists a pseudo-Liapunov function H(t, x) for E and K. Assume that (a) for any a > 0 and any solution x = x(t) of (3) which remains in Kin the future, there exist T(a,x) > 0 and #Ifa, x) > 0 such that if Q > T(a, x), t > 0 and a c H(a + t, x(a + t)) - H(q x(a)), then t > jl(a, x), or (b) for any a > 0 and any solution x = x(t) of (3) which remains in Kin the future, there exist T(a,x) > 0 and /?(a,~) > 0 such that if a > T(or,x), t > 0 and H(a + t, x(a + t)) - H(u, x(a)) < -a, then t > /3(a, x). Then every solution of (3) which remains in K in the future approaches E as t 4 co. Proof. Let x(t) be a solution of (3) such that x(t) E K for all t 2 to. First of all, we shall show that for any rl > 0 there is a sequence {z,,,} such that r, -P co as m-+ co and x(T,) E N(Q E) n K. If not, there is a T1 > 0 such that x(t) E N’(q, E) n K for all t 2 Tl. Since W(x) is positive definite on K with respect to E, there exists a 6 = S(q) > 0 such that W(x(t)) 2 6 for all t 2 Ti. Since p(t) is integrally positive,
s 1+1
lim inf l-+az I
p(s)ds = 2a > 0,
t+1
and hence there is a T2 > 0 such that T3 > 0 such that
s
p(s)ds 2 a for all t 2 Tt. Moreover, there is a
f1
I+1
OS
s a+1
since
e(s)d.s <
f
2
for all 2 2 T3
e(s)ds + o as t + e. Let To = max(T, T’, T,, T3). Since x(t)~ K and To Z ‘Z we
f
have
i’(T,+k,x(T,+k))-
V(To,x(To))6
-k$3~F~+;+1~(s)ds
+;~~~~~‘l+k)d~
j=O
Therefore V(To + k, x(To + k)) --B- co as k -+ 00, which contradicts M I; V(t, x(t)). Now suppose that x(t) does not approach E as t + co. Then for some E > 0, there exists a sequence (t,,J such that tm + co as M + 00 and x&J E NC(s,E) n K. We can assume that N(2c, E) c U, where U is the domain of the definition ofthe ~eud~Liapunov function H(r, x). We assume condition (a). For the above E > 0, there is a A > 0 such that H(t,x) > I for x E !V(E, E) n K, and for this I there is an q > 0 such that 0 c q < E and H(t, x) < A/2 for x E N(q, E) n K. From the above, there are sequences {tk) and {CL}such that tk + cc as k + 00, tk < f; < tk+ 1, dist(x(t;), E) = 8, dist(x(rk), E) = q and q < dist(x(t), E) < E
for tk < t < t;.
For r E [tk, t;], q LT;dist(x(t), E), that is, x(t) E N’(q, E) n K and hence there is a 6 = S(q) > 0 such that W(x(t)) 2 6 for tk 5 t 5 t;. On the other hand, H(t;, x(t;)) > 1 and x(tk)E N(& E) implies H(fk,x(tk) $ A/2. Thus we have A/2 < H&x@;)) - H(tk,x(tk)). Therefore, by condition (a), t; - tk > p(n/2,x) = B > 0 if k is sufficiently large. +P Let [p] be an integer such that fl= [B ] - r, 0 5 r < 1. Since lim inf p(s)ds = 2b > 0 r-m f t 1+1
and s
t
e(s)ds + 0 as t + 03, c+B p(s)& 2 b fr
+1
and
1. YOSHIZAWA
522
if t is sufhciently large, and hence
J
t*+ J 3
t*+3
p(s)ds 2 ‘b
and
e(s)ds 5 ;
1k
1k
if k is su~~ient~y large. Since we can assume that ti is sufficiently large and we have V(Cj+
=
1
,X(tj+ 1))-
jY(t,+
1, x(tj+
v(tj, x(rj)) l))
-
V(tj
+
X(tj
8,
+
B)) + v(tj + b x(cj+ 8))-
VtjT
'('j))
we have
< =JOD rWs -ji, J
v(tk+,,x(tk+l))
-
v(fl~x(tl))
~+~~(s)~(x(~~)d~ tj
tx
+ i
ff”‘e(s)ds
j=l
fj
cc
= <
y(s)ds -
II
Then V(tk+ r , x(&+ i)) + - co as k + 00, which contradicts the boundedness from below of F’(t,x(t)). Thus we can conclude that x(t) approaches E as t -+ 00. The same argument can be applied to the case where condition (b) is satisfied. Remark 1
If there exists a continuous function q(r) 2 0 such that [0, CJZ)and d(t,x(t))
J
’ q(s)ds is unifo~Iy
~ontinuo~
on
i q(t), then condition (a) is satisfied, where
IQ, x(r)) = l$nl&
(H(t + 6, x(r + 6)) - H(t, x(t))}.
Remark 2 The condition corresponding
to LaSalle’s condition in [7] is as follows: (c)Given any a > 0 and any solution x = x(t) of (3) which remains in Kin the future, there exist T(a, x) > 0 and /I(@,x) > 0 such that
+ks, ww
>a
for
a>T(a,x)
and
t>O
implies t > fi(a,x). This condition implies condition (a) for a pseudo-Liapunov function H(x) = dist(x,E). It is clear that H(x) is a pseudo-Liapunov function for E and R Take T(a,x) and #I(a,x) in condition (c). Then, if a > T&x), t > 0 and a c H(x(a + t)) - ~(x(u)), then t must be greater
than /&X,X). In fact, suppose t 5 /3(a, x). Then
H(x(a + t)) - H(x(a)) S Ix(a + t) - x(a)1 =
f(s,x(s))ds
Ii”D
+‘f(s,x(s))ds
I; a. But
5 a, which is a contradiction.
Therefore t > &a, x)_ Example 1
Consider the equation x” + h(t, x, X’)IX’pX’ + g(x) = q(t, x, x’)
Attractivity
in non-autonomous
systems
523
or an equivalent system
x’ = Y, Y' = where LY2 continuous i. there such
w,x9 Y)lYl”Y -
g(x) + q(t,x9 Y)
(5)
0, h(t, x, y) is continuous on I x R x R, g(x) is continuous on R and q(t,x,y) is on I x R x R. We assume that exists an integrally positive function p(t) and a continuous function k(x, y) 2 0 that h(t,x,y) 2 p(t)k(x,y), X
ii. G(x) 2 0, where G(x) =
s0
g(u)du, m
iii. [q(t, x, y)l 5 e(t) for (x,y)~ R2, where e(t) is continuous and
e(s)ds < 03. s 0 t
Now consider a function
U(t, x, y) = e- E(f) {G(x) + y2/2 + l}, where E(r) =
e(s)ds. s 0
Then we have
%k x,Y) I -
ewt,x, Y)lYl” + 2s 0,
s m
where E. =
e(s)ds. Since e -Eoy2/2 5 U(t, x, y) and tit,, 5 0, y(t) is bounded as long as a
solution (x(l),:~(t)) of (5) exists. As x’(t) = y(t), each solution of (5) can be continued for all future time. (I) Suppose that k(x,y) > 0 if y # 0 and G(x) + co as (xl + 00. Then for every solution (x(t), y(t)) of (5), G(x(t)) + a constant and y(t) + 0 as t + co. Moreover, in addition, if xg(x) > 0 for x # 0, x(t) ---,c for some constant c as t + co. In this case, it is clear that the solutions of (5) are uniformly bounded. Let (x(t),y(t)) be a solution of (5) defined for t 2 to. Then there is an m > 0 such that Ix(t)1 2 ml!(t)1 5 m for all t 2 to. Then V(x, y) = G(x) + y2/2 is a Liapunov function for (5) and the set K = {(x, y);
I4 6 m,lyl S 4,
and
J&(x,Y) 5 -
PW(X,Y)IY~"Y~
+ meW.
(6)
H(y) = y2 and E = {(x, y), y = O}. Then W(x, y) is positive Let WX,Y) = Mx,~)lyl”+~, definite on K with respect to E, and H(y) is a pseudo-Liapunov function for E and K which satisfies
Since the solution is bounded and e E L1 [0, co), condition (a) in Theorem 1 is satisfied. Thus it follows from Theorem 1 that y(t) + 0 as t + 03. On theother hand, since U(t, x(t),y(t)) is nonincreasing and 0 s U(t, x(t),y(t)), U(t, x(t),y(t)) -+ a constant and consequently G(x(t)) + a constant as I -+ cc. If xg(x) > 0 (x # 0), x(t) + c for some constant c as t + 00 since the positive limit set is connected. (II) Suppose that k(x, y) = k*(y), where k*(y) > 0 for y # 0, and that Ig(x)l is bounded for all x. Thus we have the same conclusion as in (I). In this case, we have
%dx,Y) S
-pW*Wlyl”y2
+
e(t)lyl.
For the solution (x(t),y(r)) of (5) defined for t 2 to, there is an m > 0 such that ly(t)l 5 m for all t 2 to. Then W(y) = PCy)(y( 3c2 is positive definite on K = {(x, y); 1x1c 00, IyI 5 m} with respect to the set E = {(x, y); y = O}. Clearly H(y) = y2 is a pseudo-Liapunov function for E and K, and we have (7). Since Ig(x)l is bounded for all x, we have the conclusion by the same argument as in (I).
T. YOSHUAWA
524
(III) suppose that G(x) + 00 as 1x14 00 and h(t, x, y) satisfies condition (c) for any compact set K and that R- = { (x,O);x < 01 and R+ = {(x, 0);~ > 0} are connected components in E - {(O,O)},wh ere E = {(x, y);yk(x,y) = 0). Then we have the same conclusion as in (I). In this case, we have (6), and k(x, y)(~r+~ 1s . positive definite on K = {(x, y);lxl 5 m,lyl 5 m} with respect to E = {(x, y),yk(x,y) = 0). Since we assume that h satisfies condition (c), the vector field of (5) satisfies condition (c) for the compact set K. Therefore we can see that (x(t), y(t)) + E as t + co. Let Q be the positive limit set of (x(t), y(t)). Then R c E. If (0,O) E Q for any 6 > 0 there exists a point (x(t,), y(Q) which is in the b-neighborhood of the origin, but (0,O) is eventually uniformly stable by Theorem 17.4 in [ 121, and hence (x(t), y(t)) + (0,O) as t -+ 00. If (O,O)$Q 0c E - {(O,O)}. H owever, R n # C#J, where R = {(x, y); y = 0}, because if n n R = 4, Iy(t)l >= c for some constant c > 0 and thus Ix(t)\ + oc as t + co which contradicts the,boundedness of x(t). Therefore Rc R- u R+ by out assumption since R is connected. Hence Rc R, that is, y(t) + 0 as t + 00. The remainder is the same as in (I). Under the assumptions in (I) with x&x) > 0 for x # 0, if we assume the growth condition on h, that is, for any bounded continuous function (x(t), y(t)) on (I), there exists a sequence of positive numbers (~3 and a constant d > 0 such that sk+ I - Sk2 d and that h(t, x(t), y(t)) 3 0 on [sI, Sk+ d] for all k or
-
I/@+ 1)
W,
x(Q,
Y(0W
= 00,
then we can see that (x(t), y(t)) + (0,O) as t + co by the same argument as in [8]. In a special case where q(t, x, y) s 0, the null solution of (5) is equiasymptotically stable, because uniform stability and attractivitl unply equiasymptotic stability. In the case (III), the growth condition is automatically satisfied, because of condition (c). Another growth condition is 1 i=O
T s
W, x(W(t)W
5 B
for some B > 0 and all T > 0 and (x(t), y(t)) in a compact set. This condition is somewhat easily checked in applications, but it satisfies actually the above growth condition which is weaker [8]. Theorem
2
Let V(t, x) be a Liapunov function for (3) and a given set Kin D. Suppose that there exists a continuous function e(r) 2 0 and a T > 0 such that
e(t)dt < 00 and that
Y3)k x) 5 - p(W(x) + e(t)
s 03
for all t 2 T and x E X, where p(t) >=0 is continuous,
p(t)dt = co and W(x) is positive
definite on Kwith respect to a closed set E c D. Suppose tOhatthere exists a pseudo-Liapunov function H(x) for E and K. Moreover, suppose that for any solution x(t) of (3) which remains in Kin the future, there exist c > 0 and continuous functions q(t) 2 0 and r(t) 2 0 such that
4)
5 PW,
I
oc
r(t)dt < 00
0
and
Ifcw I 4(t) + 40
for all large t
or
-r(t) - 4W S If(xW)
for all large t.
Attractivity in non-autonomous
systems
525
Then every solution of (3) which remains in K in the future approaches E as t + co. Proof. Let x(t) be a solution of (3) such that x(t) E Kfor t 2 to. By the same argument as in the proof of Theorem 1, if we assume that x(t) does not approach E as t + cc and that d(x(t)) 5 q(t) + r(t) for all large t, there exist 1 > 0 and sequences {tk}, {t;} such that tk + co as k + 03, tk < fi < fk+ 1, H(X(t;)) > 3. and H(x(&)) 5 l/2. Moreover, there iS a b > 0 such that w(x(t)) 2 6 for tk 5 t 5 CL. Since we can assume that tr is sufficiently large, we have
However, we have 1 j <
H(X(Cj))-
H(x(fj))
5
6:’q(s)ds + j.; r(s)ds
and hence
Therefore we have
w;, W) s
- w19 x(h))
-~16~:jpods+~~(e(s)+.i(s))ds
5 -
i
dcl:jq(s)ds
+ l; (e(s) + r(s))ds
j=l &d-
(2
f@ls)
+ l" (44
+ y(s))ds
j=l s
-ticA,2+6c
i ~~r(+fs j=l
+ ~~(44
+ y(s))ds.
Thus V(&, x(Q) + - co as k + co, which contradicts the boundedness from below of V(t,x(t)). Thus we can conclude that x(t) approaches E as t + co. Example 2
Consider the system x’ = - u(t)x + q(t)y + cl(t) Y'
= -4(W -
(8)
P(~)Y + e,(t)
where all functions in (8) are continuous, u(t) 2 0, p(t) 2 0 and Setting E(t) =
(lel(s)l + le2(s)l)ds and E. =
(Iel(
m lei(t))dt < co (i = 1,2). s0
+ le2(s)l)ds, consider a function
u(t, x, y) = emE(‘)(x2+ y2 + 1). Then emEo(x2 + y2 + 1) 6 U(t,x,y) 5 x2 + y2 + 1 and Ij&t,x,y) 5 0. Therefore the solutions of (8) are uniformly bounded. Let (x(t), y(t)) be a solution of (8) defined for t 2 to.
T. YOWZAWA
526
Then there exists an m > 0 such that )x(t)( S m and Iy(t)l i m for t 2 to. If we consider a function V(x, y) = $x2 + y’),
&(x9y) 5 - 4Qx2-
P(~)Y~ +
el@)x+ e2WY.
Thus V(x, y) is a Liapunov function for (8) and the set K = {(x, y); (x1 S m, ly( S m}. (I) Suppose that p(t) is integrally positive and
Theny(t)+Oast+zo. In this case, we have l&,(x,y) s -p(t)y’ + 2mlez(t)l,
where
H(y) = y2. Set
s
;
IdW s is uniformly continuous on [0, cc).
+ m(lel(t)l + le,(t)l)
and If(y(t)) 5 2mzlq(t)l
q*(t) = 2m’lq(t)l + 2mle,(t)l.
uniformly continuous on [0, co), and hence condition Thus the conclusion follows from Theorem 1. (II) Suppose that u(t) and p(r) are integrally positive and
Then
(a) in Theorem
. q*(s)& is J0 1 is satislied.
‘)q(s)lds is uniformly continuous s0
on [0, co). Then (x(t),y(t)) + (0,O) as t --) 03. In this case, we have also I& (x,y) 5 -a(t)x’ + m(le,(t)l + le,(t)l), and for H(x) = x2, fi(x(t)) I 2m21q(t)l + 2mlel(t)l. Thus x(t) + 0 as t + m by Theorem 1. (III) Suppose that there is an a > 0 such that jq(t)l 5 ap(t) for t >= 0 and assume that xl p(t)dt = co. Then y(t) + 0 as t + cm. s0 In this case, we have rfs, (x,y) $ -p(t)y’ + m(lel(t)l + lez(t)l) and, for H(y) = y’, Let q*(t) = 2m21q(t)l and r(r) = 2mlez(t)l. Then fi(y(r)) 6 2mZ14@)l f We&K q*(t)/2am2 5 p(t) and H(y(t)) 2 q*(t) + r(t). Thus the conclusion follows from Theorem 2. (IV) Suppose that
s0
cop(t)dt = co and
I0
m Iq(t)(dt < co. Then y(t) + 0 as
c +
CL).
Co
In this case, letting r(t) = 2m21q(t)l + 2m(e2(t)l, we have ti(y(t))
5 r(t) and
s0
r(t)dt < co.
Thus the conclusion follows from Theorem 2. Now we shall discuss the convergence of solutions of (3). To do this, we consider the system associated to (3) x’ = f‘(t, x) Y' = f(4Y).
1
(9)
For a given set Kin D, let V(t, x, y) be a Liapunov function for (9) and the set S = K x K. Let E in R” be a non-empty closed set. Then a scalar function W: D x D + [0,03) is said to be positive definite on K with respect to E if for any E > 0 there exists a 8(s) > 0 such that W(x,y) 2 8(c) whenever (x, y)~ S and x - YE NC (E,E). Also, we say that a scalar function H(x, y) is a pseudo-Liapunov function for E and K, if there exists an open neighborhood U of E and H(x, y) is defined and continuous for (x, y) E S such that x - y E II and if H(x, y) has the property that i. for any E > 0, there is a I > 0 such that H(x, y) > ). for (x, y)~ S such that x - YE NC 6, E), ii. for any d > 0, there is an q > 0 such that H(x, y) -E 2/2 for (x, y)E S such that x - YEWLO Then, corresponding to Theorem 1, we have the following theorem which can be proved by the same idea as in the proof of Theorem 1. Theorem
3
Let l’(t, x, y) be a Liapunov function for (9) and the set K x K, where K is a given set in D.
Attractivity in non-autonomous
systems
527 r+1
Suppose that there exists a continuous function e(r) 2 0 and a T > 0 such that sf
e(s)ds
+Oast+coandthat
for all t 1 T and (x, y) E K x K, where p(t) is integrally positive and W(x, y) is positive definite on K with respect to a closed set E c R”. Suppose that there exists a pseudo-Liapunov function H(x, y) for E and K. Assume that (d) for any CI> 0 and any pair of solutions (x = x(t), y = y(t)) of (3) which remain in Kin the future, there exist T = T(a, x, y) > 0 and /I = B(a, x, y) > 0 such that if a > T, t > 0 and a < H(x(a + t), y(a + t)) - H(x(a),y(a)), then t > j?, or (e) for any a > 0 and any pair of solutions (x = x(t), y = y(t)) of (3) which remain in Kin the future, there exist T = T(a, x, y) > 0 and /I = /I(a,x,y)>Osuchthatifa>T,t>Oand H(x(a + t), y(a + t)) - H(x(a), y(u)) < -a, then t > /3. Then every pair of solutions {x(t),y(t)} of (3) which remain in K in the future satisfies x(t) - y(t) -+ E as t + co. Remark 3
The condition corresponding to condition (c) is as follows: (c*) Given any a > 0 and any pair of solutions (x = x(t), y = y(t)) of (3) which remain in K in the future, there exist T(a, x, y) > 0 and /?(a,x, y) > 0 such that - f(s, y(s))]ds > a for a > T(a, x, y), t > 0 implies t > /l(a,x, y).
IJ”0
+* [f(s, x(s))
Condition (c*) implies condition (d) for a pseudo-Liapunov function H(x,y) = dist(x - y, E). For example, if If(t,x) - f(t,y)l is bounded for all t when x - y is bounded, any pair of solutions (x(t),y(t)) which remain in K in the future and whose difference is bounded satisfies x(t) - y(t) + E as t + 00. This is a generalization of a result obtained by Swick [lo]. Now we shall see an example considered by Murakami [9] and by Swick [lo]. Example 3
Consider an equation x” + p(t)f(x’)
+ b(t)x = e(t)
(10)
and an equivalent system x' = y, y' = -p(t)f(y)
- b(t)x + e(t)
(11)
where all functions are continuous. We assume that i. equation (10) has a solution x0(t) which exists for all t 2 0 and whose derivative x&(t) is bounded for t 2 0, ii. p(t) is integrally positive, iii. f(y) is strictly monotone increasing for all y, iv. 0 c bl 6 b(t) S b2, and either b’(t) $ 0 or b’(t) 10. Then for every pair of solutions of (lo), the difference of their derivatives tends to zero as t -+ m.
To see this, consider the associated system Y, y’ = -p(tlf(y) U’ = U, U‘ = -p(t)f(v)
X’ =
- b(t)x + e(t) - b(t)u + e(t).
(12)
We consider the case where b’(t) 5 0. Consider a function v(t, x,y,u, u) = b(t)(x - u)’ + (y - u)~.
(13)
528
T. YOSHIZAWA
Then we have q12,wYJv)
s -2P(~)Lw
-_fI~)lCv- 450.
(14)
Since x,(t) exists for all t 2 0 and xb(t) = ye(t) is bounded for all t 2 0, it follows from (13) and (14) that all solutions of (10) exist in the future and their derivatives are bounded, because the difference of any pair of solutions of (11) is bounded, that is, system (11) is distancebounded by Theorem 15.4 in [12]. Let (x(t), y(t)) and (u(t), u(t)) be a pair of solutions of (11) defined for t 2 t,,. Then there is an m > 0 such that b(t)1 5 m, Iv(t)15 m for all t 2 to. Let K= {(x,y)~R’: 1x1< co, Jy( $ m} and E= ((x,y)eR2: y=O}. Then V(t,x,y,u,o) is a Liapunov function for (11) .and K x K and W(x, y, U,u) = [f(y) - f(u)](y - II) is positive definite on K with respect to E. If we set HCy, u) = (y - u)~, H is a pseudo-Liapunov function for E and K and we have H(y(t), u(t)) 6 26,(x(t) - u(t)))y(t) - u(t)1 5 a constant, because Ix(t) - u(t)! and Iy(t) - u(t)1 are bounded. Therefore, by Theorem 3, y(t) - u(t) + 0 as t -P co. For the case where b’(t) 2 0, consider a function I/ = (x - u)~ + 0, - ~)~/b(t). REFERENCES 1. E. A. Barbashin and N. N. Krasovskii, On global stability of motion, Do/cl. AU. Nauk SSSR, 86,453-456 (1952). 2. T. A. Burton, Stability theory for functional differential equations, Dons Am. Math. Sot. 255,263-275 (1979). 3. R. D. Driver, Existence and stability of solutions of a delay-differential system, Archs rot. Mech. Anal., 10, 401426 (1962). 4. L. Hatvani, Attractivity theorems for non-autonomous systems of differential equations, Acta Sci. Math. (Szeged) 40,271-283 (1978). 5. J. P. LaSalle, The extent of asymptotic stability, Proc. Natn. Acod. Sci. USA 46, 363-365 (1960). 6. 1. P. LaSalle, Stability theory and invariance principle. In Dynamical Systems. An International Symposium, Vol. 1, pp. 211-222. Academic Press, New York (1976). 1. J. P. LaSalle, Stability of nonautonomous systems, Nonlinear Anal. I, 83-91 (1976). 8. S. Murakami, Asymptotic behavior of solutions of ordinary differential equations, Tohoku Math. .I. 34,559-574 (1982). 9. S. Murakami, Asymptotic behavior of solutions of some differential equations, J. Math. Anal. Appl. to be published. 10. K. E. Swick, Convergence of solutions of nonlinear differential equations, Ann. Mar. Pura Appl. 14,1- 126 (1977). 11. T. Yoshizawa, Asymptotic behavior of solutions of a system of differential equations, Contrib. Difirential Equations 1, 371-387 (1963). 12. T. Yoshizawa, Stability Theory by Liapunods Second Method. The Mathematical Society of Japan, Tokyo (1966). 13. T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, Vol. 14. Springer, Berlin (1975). 14. T. Yoshizawa, Asymptotic Behavior o/’ Solutions in nonautonomous Systems, Trends in Theory and Practice of nonlinear Difirenriol Equarions, pp. 553-562, Marcel Dekker, New York (1984). 15. T. Yoshizawa, Asymptotic behaviors of solutions of differential equations, in Proceedings of the Colloquium on the Qualitative Theory of Differential Equations, Szeged, August 27-31, 1984, to be published.