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Invariance principle for autonomous delay difference systems
Nonlinear Analysis, Theory, Methods&Applications, Vol. 24, No. 5, pp. 735-745, 1995 Copyright© 1995ElsevierScienceLtd Printed in Great Britain. All rightsreserved 0362-546X/95 $9.50+ .00
Pergamon
0362-546X(94)EOO66-P INVARIANCE PRINCIPLE FOR AUTONOMOUS DELAY DIFFERENCE SYSTEMS SHUNIAN Z H A N G Department o f Applied Mathematics, Shanghai Jiaotong, Shanghai 200240, People's Republic o f China (Received 16 June 1993; received in revised form 8 December 1993; received for publication 30 March 1994) Key words and phrases: Autonomous delay difference systems, invariance principles, Lyapunov functionals, Lyapunov-Razumikhin functions, asymptotic stability, asymptotic constancy.
1. I N T R O D U C T I O N
It is known that since the 1960s mathematicians have established the so-called invariance principles for a variety of dynamical systems, especially for differential systems (cf. [1] and the references therein). Most of those works are motivated by LaSalle's work in 1968 (cf. [2]), which dealt with oridinary differential equations by means of Lyapunov functions. In [3] (also cf. [4]) Hale "naturally" generalized LaSalle's work to functional (or delay) differential equations by means of Lyapunov functionals instead of Lyapunov functions. However, his method is not always practical, since to construct a suitable Lyapunov functional for a given system is rather difficult (cf. [5, p. 96]) though there are many n-dimensional systems for which appropriate Lyapunov functionals can be constructed. In 1983, Haddock and Terjeki [5] developed the invariance principle for autonomous functional differential equations, more specifically, they introduced Lyapunov-Razumikhin functions rather than Lyapunov functionals. It is the opinion of the author that Lyapunov functions are, in general, easier to construct and less complicated to verify as compared with Lyapunov functionals. Nevertheless, everything has two aspects. For some systems it is difficult to get much information by just using Lyapunov functions. The purpose of this work is to establish the corresponding invariance principles for autonomous delay difference systems in both approaches: in terms of Lyapunov functionals, and Lyapunov-Razumikhin functions. Consider the following delay difference systems x(n + 1) = f(xn),
n e Z +,
(1)
where Z + denotes the nonnegative integer set, x.(s) = x ( n + s ) ,
fors=-r,-r+
1. . . . . - 1 , 0
with some integer r > O , f : X --* R k is continuous, and X = {~0:{-r, - r + 1. . . . . O} -~ Rk}, R k is the k-dimensional Euclidean space. For each ~0 e X, we define the norm of ¢ as I1~011 = maxll~0(s)l: s = - r , - r where 1-I is any norm in R k. 735
+ 1. . . . . 01,
736
SHUNIAN ZHANG
We always assume that f(tp) is well defined for any ~ e X. Consequently, for any given ~p e X and no e Z +, there exists a unique solution of (1), denoted by X(no, ~), which is defined for all integers n _> no and satisfies Xno(no, ~) = (D.
Since system (1) is autonomous, without loss of generality we may assume n o = 0, and simply denote the corresponding solution of (1) as x(tp)(.). By the uniqueness o f solutions, for any n, m ~ Z + we have Xn(Xm(~9)) = Xn+m(~9). D e f i n i t i o n 1. Let ~o ~ X, and x(~a)(.) be the corresponding solution of (1). ~, ~ X is said to be
in the to-limit set t)(~o) of tp if there exists a sequence Inil o f nonnegative integers such that ni ~ oo and IIxn,(~) - ~,ll -" 0 as i --, oo. D e f i n i t i o n 2. Let M C X. M is said to be positively invariant (relative to (1)) if for each ~p ~ M and all n ~ Z +, Xn((a) ~ M . For the sake of convenience in establishing our results, we define a mapping T: X ~ X via
(1) as follows Tip = Xl(tp), T 2 ~ = T(T(a) = T(xl((o)) = xl(x~(~a)) = x2(~P), •
.
.
,
,
.
Tn+l(a = T(Tn(a) = Xn+l((o ) . . . . .
where x(~p) is the corresponding solution of (1) to tp. It is easy to see that T is continuous by the continuity of f((a). In fact, for given e > 0 and (a* e X. Sincef(tp) is continuous at tp = (a*, there exists ~5 > 0 (we may assume J < e) such that If(tp) - f(tp*)[ < e
if I1~ - ~*11 < ~.
Now when He - tp*[[ < ~5, we have IIT~ - T~*ll = IIx~<~) - xl(~*)ll = max{Ix(e)(1
-< maxlll~
+ s) - x(¢*)(1
- ~*ll, If(~) -f(~*)ll
+ s)l: s = -r .....
0)
< E.
This shows that T is continuous at tp*.
With the above-defined mapping T we can reformulate definitions 1 and 2 in the following alternative forms. D e f i n i t i o n 1'. Let tp ~ X. ~, e X is said to be in the w-limit set t)((o) of ¢ if there exists {ni), n i
E
Z
+
such that
Tni{o --~ ¢/
and
ni ~
oo
as i ~ oo.
D e f i n i t i o n 2'. Let M C X. Relative to (1), or to T, M is said to be positively invariant if T(M) C M.
The following lemmas are important and very useful in establishing our main results. The proofs are (essentially) given in [1] and thus omitted here.
lnvariance principle
737
LEMMA 1. Every to-limit set ~(~o) is closed and positively invariant. LEMMA 2. If ~p e X, and x(tp)(') is defined and bounded for n _> - r , then (i) the set K = [x~(tp): n e Z+I is precompact; (ii) f~(~p) is nonempty, positively invariant, and compact; (iii) x~(~p) -~ f~(tp) as n ~ oo. 2. I N V A R I A N C E
PRINCIPLES
In this section, first we wish to discuss the designated properties of O(¢) in l e m m a 2 by means of L y a p u n o v functionals. Definition 3. Let G C X. W = W ( n , (o) is said to be a Lyapunov functional of (1) on G if W : Z + x G ~ R is continuous and A(1)W(n,~p) ---- W ( n + 1,x~(tp)) - W(n,~o) <_ 0 for all ¢peG andneZ+. Clearly, it follows that Ao) W(n, x,(q0) = W ( n + 1, x,+~(¢)) - W ( n , x,(tp)). In particular, if W ( n , tp) - W(tp), then
A ( , W(~p) = W(x~(~o)) - W ( ~ ) , and a ( , W(x.(~o)) = W(x,+~(¢)) -
W(Xn(~) ).
The condition Ao)W(n, ~p) _< 0 ensures that along the solution of (1) which remains in G, W is nonincreasing. Now for a given Lyapunov functional W = W(~p) on G, define E = [tp ~ (~: Ao)W(tp) = 01. Let M be the maximal positively invariant subset in E, and for c ~ R, denote W - ~ ( c ) = 1~0 e X : W(q,) = cl. By using almost the same arguments as in [1], we can show the following result, which is called the invariance principle by Lyapunov functionals. TrlEOREM 1. Let W be a Lyapunov functional of (1) on G, and x(-) be a solution of (1) which is defined and bounded for n _> - r and such that x~ e G for n e Z +. Then for some constant c there holds x~ ~ M tq W-l(c) as n --, o o . Proof. Let ~ = Xo. By assumptions that Ao) W ( x , ( ~ ) ) <<.0 and x~(¢) is bounded, W(xn(tp)) is nonincreasing with n and is bounded from below. Thus, W(xn(cp)) ~ c with some constant c as n ---~ oo.
Let ~, e ~(tp). Then there is a sequence [ni] such that ni ~ ~ and xni(q0 ~ ~, as i ~ ~ . Since W is continuous, we have W(xni(~p) ) ~ W ( ~ ) = c and, hence, f~(tp) C W - l ( c ) .
as i --* oo
738
SHUNIAN ZHANG
However, f](~0) is positively invariant, ~u e f2(~o) implies xl(gt) e fl(~o). Thus, W(xl(qt)) = c and Ao) W(~) = W(x~(g)) - W(~,) = c - c = 0. Therefore, q / e E, i.e. t~(~0) C E and, hence, a((o) C M. Since from lemma 2 we know that x.(tp) -, f~(~u), we conclude x.(~o) -+ M N W-l(c)
•
as n ~ oo.
Now we establish the invariance principle by Lyapunov (-Razumikhin) functions and first introduce the following definition. Definition 4. Let V : R ~ R be a continuous function which is called a Lyapunov (-Razumikhin) function. Define
Ao)v(~o)
=
v(f(¢))
-
v(~o(o)).
For given G C X, define E v ( G ) = {~o e G:
max
V(x,(to)(s)) =
$= --r,...,O
max
V(~o(s)), n e Z+I.
$= --r,...,O
Let M y ( G ) be the maximal positively invariant subset w.r.t. (1) in Ev(G). Namely, for each ~ M y ( G ) , there is a solution x(¢)(.) of (1) through ¢ such that x,(~0) e O for all n e Z +, and max
V(x~(¢,)(s))=
S = --r, ...,0
max
V(~(s)).
$ = --r,...,O
THEOREM 2. Assume that there exist a Lyapunov function V = V(x) and a closed set G C X positively invariant w.r.t. (1) such that V(x(~o)(1)) _<
max
for ~0 e G.
V(~o(s))
(2)
S= -r, ...,0
Then for any ~0 e G if x(~0)(.) is bounded for n _> - r , then
f~(~o)C M y ( G ) C Ev(G). Therefore, x~(~o) --, My(G) as n --*oo.
Proof. Suppose ~o e G is such that x(~o)(.)is bounded for n _> -r. Then since G is positively invariant x,(~o) e G for n e Z + and by l e m m a 2, ~2(~) is nonempty. Let W(n, (o) =
max
V(x~(~o)(s))=
S= -r,...,0
max
V(x(~o)(n + s)).
s= --r,...,0
By condition (2) and x.(~) e G for n e Z +, we have V(x(xn+l(~0)(1))) _<
max
V(x.+~(~o)(s)),
$= --r,...,0
and
V(X(Xn((O)(1)) ) <-
max
V(x.(~o)(s));
S= --r,...,O
i.e.
V(x(~o)(n + 2)) _<
max S=
--r,...,0
V(x(to)(n + 1 + s)),
Invariance principle
739
and V(x(tp)(n + 1)) _<
V(x((o)(n + s)).
max $ = --r~... ,0
Hence, W ( n + 1, Xl((p) ) :
max
V(x(xl(tp))(n + 1 + s))
s= -r,...,O
=
max
V(x((p)(n + 2 + s))
S= --r,...,O
=
max
V(x(q~)(n + 1 + s))
s= -r+ 1,..., 1
<
max
V(x(fo)(n + 1 + s))
s = -r,...,0
<
max
V(x(¢)(n + s))
S= --r,...,0
= W ( n , 6o),
n ~ Z+.
Thus, A W(1)(n, tp) < 0,
n ~ Z +.
Since W is bounded below along this solution, there exists a limit lim W ( n , ~) = lim i~oo
max
V(x,(~o)(s)) = c ~ R .
n-*oo s=-r,...,O
Let ~, ~ £)(tp), we claim that ~ ~ E v ( G ) and so f~(tp) C E v ( G ) . In fact, there exists [ni], n i e Z + such that ni -~ oo and x.~((p) -~ ~ as i ~ oo. This implies lim[ n ~
max
V(Xn~(~o)(s))]=
$=-r,...,O
max
V(~(s))=c.
s=-r,...,O
Since G is closed, ~, ~ G. This shows that D(tp) C G. By l e m m a 2 D(~p) is positively invariant, thus xn(~') C D(tp) C G for n ~ Z +. Therefore, max
V(x.(~,)(s)) = c =
s = -r,...,O
max
V(~(s)).
s= -r,...,O
By definition 4, ~, ~ E v ( G ) and f~(~0) C E v ( G ) . It follows f r o m lemma 2 that x,(tp) ~ ~(tp) C M y ( G ) C E v ( G ) as n ~ oo. Therefore, xn(tp) ~ M y ( G )
as n --* oo.
•
R e m a r k 1. It can be easily seen f r o m the p r o o f of theorem 2 that if ~ e X is such that x(~)(.) is bounded for n _> - r and x,(~) e G for n e Z ÷, then the above conclusion remains valid under the assumption that G is closed but not necessarily positively invariant. 3. A P P L I C A T I O N S
We discuss in this section the asymptotic stability and the asymptotic constancy of solutions of (1) as applications of the obtained invariance principles. First of all, the important remark 1 in Section 2 enables us to derive the following useful result.
740
SHUNIAN
ZHANG
TtmoRm,i 3. A s s u m e f ( 0 ) = 0. I f there exist a L y a p u n o v function V: R k - , R ÷ and a constant fl > 0 such that (i) V(O) = O, V(x) > 0 for 0 ;e Ixl -< B; (ii) V(x((o)(1)) < max v(tp(s)) for any 0 ~ II ll -< B, S= --r,...,O
then the solution x = 0 o f (1) is asymptotically stable.
Proof. First, we show the stability. Clearly, if ~0 = 0, then it follows f r o m (1) and f ( 0 ) = 0 that x(0)(1) = f ( 0 ) = 0, thus, Xl(0) = 0, and x(0)(2) = f(xl(O)) = 0. In this manner, we arrive at x(0)(n) = 0
for all n e Z +,
that is x(0)(.) = 0. In what follows we m a y assume that ~p ~ 0. For any given e > 0 (e _< fl), let m=
inf V ( x ) > O . e
By the continuity o f V(x) and V(0) = 0, we m a y choose ~ > 0 (~ < fl) such that Ixl < ~ implies v ( x ) < m. T h e n for any ~o: 0 ;~ [[(o[[ < t~, V(x(tp)(1)) <
V(qT(s)) < m
max s= -r,...,O
and, hence,
Ix(~)(1)/< e. Moreover, by assumption (ii) we have
max s=
V(xl(tp)(s)) <
-r, ...,O
max s=
V(tp(s)) < m.
(3)
-r,...,O
I f [x~(~o)(0)[ = Ix((o)(1)l ;e 0, then xl(~0);~0; while if IXl( )(o)l = = o and [Xl(tp)(so - 1)[ = ] - r + 1 then we still have xl(~0) # 0. Otherwise, ]Xl(~0)(0)[ = 0 and [xt(~0)(s - 1)[ = I 0(s)l -- o f o r s = - r + 1 . . . . . o, i.e. x~(~o) = o and so x(tp)(n) = 0 for n = 1, 2 . . . . . Therefore, we m a y assume Xl(tp) ;~ 0. Obviously, Ilxl(tp)]l - B. T h e n by (ii) and (3), V(x(¢)(2)) = V(x(xt(~o))(1)) <
max
V(x~(~)(s)) <_
s = -r,...,O
max
V(~(s)) < m,
s = -r,...,O
which implies Ix(~o)(2)[ < e. By induction, we can conclude that for 0 ;e [[~oH< ~ we have
V(x(tp)(n)) <
max
V(x,_l(tp)(s))
$ = --r, . . . , 0
<-...<_
max S ........
V(tp(s)) < m 0
and, thus, Ix(~p)(n)l < e This proves that x = 0 is stable.
for n = l, 2 . . . . .
(4)
Invariance principle
741
Next, we show the asymptotic stability. Let 6o > 0 be the corresponding 6(t) to the given e = / / i n the stability. Under the assumptions we may conclude by theorem 2 and remark 1 that when Iltpl] < 6o, Xn(~O) ~ M y ( G )
as n --* 0%
C Ev(G)
where G = 1~0 e X: II~ll -< P). Clearly, to prove the asymptotic stability we need only to point out M y ( G ) = 10]. In fact, by the above arguments of deriving (4) we know that for any ¢p: 0 ~ whenever IIx.(~)ll -< t~ for n e z + we always have max
V(to(s)) >
s = -r,...,O
max
V(x.(~o)(s))
I1~11 -<
for n >_ r + 1.
s = -r,...,O
Therefore, by the definition of M y , ~o ¢ My(G). Hence, M y ( G )
= [0}.
•
2. For convenience, in dealing with concrete equations we often choose Lyapunov functions as
Remark
v(x) = Ixl.
Then condition (ii) reduces to
Ixt~0)(l)l < II~ll Example
for any 0 ~ II~011 <- 8.
(5)
1. Consider x ( n + 1) = f ( x n ) ,
(6)
n ~ Z+,
where x e R k,
If(~0)l---zll¢ll
for II~0ll---
with some constants 0 < L < 1 and fl > 0. Then we may choose V ( x ) = ]x], and (5) is obviously satisfied. Hence, by theorem 3 we conclude that the zero solution x = 0 of (6) is asymptotically stable. E x a m p l e 2. More specifically, we consider the following delay difference system u(n + 1) = av(n - rO/(1 + u2(n - r2) ), v(n + 1) = bu(n - rl)/(1 + v2(n - r3)),
n • Z+, 1
(7)
where a, b are constants, rl, r2 and r 3 are integers, lal < 1, Ibl < 1, lal + Ibl ~ 0; 0 ~< rl, r2, r 3 < r for some integer r > 0. Let
x = (u, v) r,
(~ = ((PI' (P2) T,
and
Ixl = lul + Ivl,
I1~11 =
max
[[~Pl(s)l + I~p2(s)l}.
s= -r,...,O
Choose V(x) = V(u, v)=
Ixl = lul + Ivl.
742
SHUNIAN ZHANG
Then
Ix(¢o)(1)l =
[u(~0)(1)l + Iv(¢)(1)l [acp2(-r,)/(1 + (0~(-r2))l + Ibrpl(-r,)/(1
lallcz(-r0l
+
¢0~(-r3))1
+ Ibll~l(-r0l
< maxl[al, Ibl}(l~P2(-r~)l + I ~ ( - r ~ ) l ) _< zll~ll
f o r all ~o,
where 0 < L ---- maxllal, Ibll < 1. Therefore, it follows f r o m example 1 that the zero solution o f (7) is (globally) asymptotically stable.
R e m a r k 3. It is remarkable that in theorem 3 A(,) V = V(f(~o)) - V(~p(0)) = V(x((o)(1)) - V(¢(0)) m a y not even be nonnegative to conclude the asymptotic stability, which m u c h i m p r o v e d the k n o w n results (cf., say, [6]) in the case o f a u t o n o m o u s equations. We emphasize here that even for such a simple example 1 or its special case, example 2, we were unable to show the desired conclusion before by the k n o w n results. This is one o f the motivations to establish the invariance principles for delay difference systems. N o w we discuss the conditions for solutions o f (1) tending to constant limits. For L y a p u n o v function V = V(x), we define
K v = {¢P • X : V(~o(s)) = V(¢(O)) for s = - r . . . . . 0}. THEOREM 4. A s s u m e that there exist a L y a p u n o v function V = V(x) and a positively invariant, closed set G C X w.r.t. (1) such that for any ~0 • G whenever x(fp)(.) is b o u n d e d for n > - r , lim { max
V(xn(rp)(s))}
exists.
n ~oO $ ~ - - r , . . . ~ O
Then ~((o) C K v implies that
V(x(~o)(n)) = V(x.(~p)(0)) --, const,
as n ~ oo.
P r o o f . Let ~0 • G and x(~o)(') be b o u n d e d for n > - r . By assumption we m a y assume lim { max n~eo
V(xn(tP)(s))} = c
for some c • R.
$=--r,...DO
Suppose that Q((0)C K v but V ( x ( ¢ ) ( n ) ) ~ " const, as n ~ oo. T h e n since V(x(~o)(n)) is b o u n d e d for n > - r , there must be some sequence [nil and some real n u m b e r a # c such that ni ~ oo and V(x(~o)(ni)) ~ a as i ~ oo. By l e m m a 2 [xni(~P)} is p r e c o m p a c t and by assumption G is positively invariant and closed, hence, there exist some subsequence o f {n~}, denoted by In~} again, and some g / • G such that Xni(~o) ~ ~ as i ~ oo. Therefore, ~, • ~(~p) C Kv. By the definition o f K v , we have for some b • R,
v(~(s)) = v(~,(o)) = b,
s = - r . . . . . O.
lnvariance principle
743
Noting that xn~(~p)(0) ~ ~,(0) as i ~ oo, and the continuity o f V we derive V(x~,(~p)(0)) -- V(x(q~)(ni)) --" a = V(~,(0))
as i - ,
oo.
Thus, V(q/(s)) = V(~(0)) = a,
s = - r . . . . . 0.
O n the other hand, max
V(x.,(~o)(s))
~
c
=
S= --r,...,O
max
V(qt(s)) = a.
as n ~
oo.
-r,...,O
s=
This contradicts the fact a # c. Therefore, whenever Q(g0 C K v we have V(x(~o)(n)) -* const.
•
In the following we establish a simple criterion for ~(~p) C K v . T H E O ~ 5. Assume that there exists a L y a p u n o v function V = V ( x ) and a positively invariant, closed set G C X w.r.t. (1) such that: (i) V(x(~p)(1)) _< max V(~o(s)) for any (0 e G; S=
--r,...,O
(ii) for cp e G, V(x(tp)(1)) =
max
V(~o(s)) implies ~o e K v .
S ~ --r, ... ,0
T h e n for any ~p e G we have ~ ( ¢ ) C K v whenever x(~p)(.) is b o u n d e d for n >_ - r . Therefore, V(x(tp)(n)) ~ const,
as n ~ ~ .
P r o o f . Let ¢ e G be such that x(¢)(.) is b o u n d e d for n _> - r . Let ~v e ~ ( ¢ ) . Clearly, g / e G since G is closed and positively invariant. By theorem 2 ~ ~ E v ( G ) , hence, there is some c e R such that max s=
V(x(~,)(n + s)) =
max
-r,...,O
V(x,(gJ)(s)) =
s = -r,...,O
max
for n e Z +.
V(g,(s)) = c
(8)
s = -r,...,O
In particular, if n = r + 2, then we have max
V(x(~u)(r + 2 + s)) =
S = --r, . . . , 0
max
V(w(s)) = c.
S = --r, ...,0
Suppose that for some s o e { - r . . . . . 0} V(x(~u)(r + 2 + So)) =
max
V(x(qt)(r + 2 + s)) =
s = -r,...,O
max
V(W(s)) = c.
(9)
s = -r,...,O
Let s* = r + So + 1. Then 1 < s* < r + 1, it follows f r o m (9), and (8) with n = s* that V(x(qO(s* + 1)) = c =
max
V(x~,(q/)(s)),
S= --r,...,O
i.e. V(x(xs,(~u))(1))=
max
V(xs,(~u)(s)).
S= --r,...,O
Hence, by assumption (ii) (note that xs,(~v) we conclude that
e
G since g~ e G and G is positively invarariant),
Xs4~) e gv.
744
SHUNIAN ZHANG
Thus, for s = - r . . . . . 0.
V(xs,(Ct)(s)) = V(x~.(~u)(O))
This and (8) with n = s* imply that max S=
V(xs,(~u)(s)) = V(xs.(~)(0)) = c.
--r,...,O
Since - r _< - s * + 1 ___ 0, we have V(x(tu)(1)) = V(xs.(~u)(-s* + 1)) = c =
max S=
V(~u(s)).
-r,...,0
By (ii) again, this implies that ~/~ K V. Therefore, ~((0) C Kv. T o complete the p r o o f it suffices to mention that in this case the assumptions o f theorem 2 are certainly satisfied. F r o m the p r o o f o f theorem 2 we k n o w that whenever x((0)(.) is b o u n d e d for r/_> - r , lira [ max fl'-~
exists.
V(xn(~)(s))l = c
S=--r,...,O
Therefore, the desired conclusion follows f r o m theorem 4.
•
R e m a r k 4. It is notable that the L y a p u n o v functions V in theorem 5 (as well as in theorems 2
and 4) are not necessarily nonnegative. This brings out m u c h convenience in applications. E x a m p l e 3. Consider the scalar equation 0
x(n + 1) = ax(n) + b x ( n - r) - b x ( n ) x ( n - r) ~
[x(n + s) - x(n)l 2,
(10)
S ~ --r
where n • Z +, a > 0, b > 0 are constants, and a + b = 1. Let G = {(0 • X : 0 _< ~o(s) _< 1, s = - r . . . . . 0}. Then it is easy to show that G is positively invariant w.r.t. (10). Clearly, it suffices to show that 0 < x(~o)(l) _< 1 for ¢ • G. In fact, if (0 • G, then Ileal[ --- 1 and x((0)(1) = a(0(0) + b ( 0 ( - r )
I
o
1 - cp(0) ~ $=
1
[(0(s) - (0(0)12 > a(0(0) > 0,
--r
and
b)ll(01[
x((0)(1) _< all(oil + bll(0H = (a +
= 11(011~ 1.
Thus, o _~ x((0)O) -< 1.
C h o o s e V(x) = x. Then for any ~ e G we have as above that V(x((0)(1)) = x((0)(1) _<
11(oil =
max
v((0(s)).
$ = --r, ..., 0
O n the other hand, if (0 e G and V(x((0)(1)) =
max
V((0(s)), then x((0)(1) = H(0[I, i.e.
S= --r,...,O 0
(0(0) - b(0(O) + b ( 0 ( - r ) - b(0(O)(0(-r) ~ $=
[(0(s) - (o(0)12 = --r
11(011.
(11)
Invariance principle
745
I f (0 = 0, t h e n o b v i o u s l y (0 ~ K v. H e n c e , we need o n l y to c o n s i d e r the case o f (0 ;~ 0. S u p p o s e (0(0) < 11(oil ~ 0. T h e n the l e f t - h a n d side o f (11) < all(011 + oil(011 = 11(oil, hence, the e q u a l i t y (11) c a n n o t h o l d . T h e r e f o r e , (0(0) = 11(011. H o w e v e r , t h e n it clearly follows f r o m (11) t h a t 0 ( 0 ( - r ) = (0(0) and (0(0)(0(-r) ~ [(0(s) - (0(0)] 2 = 0. $=
--r
T h e r e are t w o p o s s i b l e cases: (i) if ( 0 ( - r ) = (0(0) = 0, t h e n (0 = 0, a n d c e r t a i n l y (0 e K v ; (ii) if (0(-r) = (0(0) ;~ 0, then ~s°= _r[(0(s) - (0(0)] 2 = 0 implies t h a t (0(s) = (0(0)
for s = - r . . . . . 0.
This m e a n s t h a t (0 e K e. T h e r e f o r e , in a n y case, the a s s u m p t i o n s o f t h e o r e m 5 are satisfied. Hence, V(x((0)(n)) = x((0)(n) ~ const,
as n -+ ~ .
R e m a r k 5. It is easy to see t h a t x(n) = c for a n y c e R is a s o l u t i o n o f (10) if a + b = 1. C O N C L U D I N G REMARKS It is w o r t h m e n t i o n i n g t h a t we a r e n o t expected to o b t a i n all the p a r a l l e l results for d e l a y d i f f e r e n c e systems to the k n o w n results for d e l a y d i f f e r e n t i a l systems, b e c a u s e there are essential differences b e t w e e n discrete m o d e l s a n d c o n t i n u o u s m o d e l s . F o r instance, even f o r the simple d e l a y d i f f e r e n c e systems o f the f o r m x(n + 1) - x(n) = - h ( x ( n ) ) + h(x(n - 1)), if h(u) = u 2 a n d c h o o s e x ( - l ) = l a n d x(0) = 2, t h e n it is easily seen t h a t t h e c o r r e s p o n d i n g s o l u t i o n is [ - 1 , 2 , - 1 , 2 . . . . }; while if h ( u ) = u 3 which is strictly increasing, a n d c h o o s e x ( - 1 ) = 1 a n d x ( 0 ) = 0, t h e n the s o l u t i o n is {1,0, 1 , 0 . . . . 1. N o t e t h a t n o n e o f t h e m is a s y m p t o t i c a l l y c o n s t a n t as n -+ ~ . This is a distinct c o n t r a s t to the relevant results m e n t i o n e d in [5]. I n short, t h e r e are quite a few o p e n p r o b l e m s f o r d e l a y difference systems to be i n v e s t i g a t e d further. W e m a y discuss these in a s u b s e q u e n t p a p e r . Acknowledgement--The author wishes to thank the referee for making valuable suggestions and comments on the original manuscript. REFERENCES 1. LASALLE J. P., The Stability of Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia (1976). 2. LASALLE J. P., Stability theory for ordinary differential equations, J. diff. Eqns 4, 57-65 (1968). 3. HALE J. K., Sufficient conditions for stability and instability of autonomous functional differential equations, J. diff. Eqns 1,452-482 (1965). 4. HALE J. K., Theory of Functional Differential Equations. Springer, New York (1977). 5. H A D D O C K J. R. & TERJEKI J., Liapynov-Razumikhin functions and invariance principle for functional differential equations, J. diff. Eqns 48, 95-122 (1983). 6. ELAYDI S. & ZHANG SHUNIAN, Stability and periodicity of difference equations with finite delay. Funkcialaj Ekvacioj (to appear).
Pergamon
0362-546X(94)EOO66-P INVARIANCE PRINCIPLE FOR AUTONOMOUS DELAY DIFFERENCE SYSTEMS SHUNIAN Z H A N G Department o f Applied Mathematics, Shanghai Jiaotong, Shanghai 200240, People's Republic o f China (Received 16 June 1993; received in revised form 8 December 1993; received for publication 30 March 1994) Key words and phrases: Autonomous delay difference systems, invariance principles, Lyapunov functionals, Lyapunov-Razumikhin functions, asymptotic stability, asymptotic constancy.
1. I N T R O D U C T I O N
It is known that since the 1960s mathematicians have established the so-called invariance principles for a variety of dynamical systems, especially for differential systems (cf. [1] and the references therein). Most of those works are motivated by LaSalle's work in 1968 (cf. [2]), which dealt with oridinary differential equations by means of Lyapunov functions. In [3] (also cf. [4]) Hale "naturally" generalized LaSalle's work to functional (or delay) differential equations by means of Lyapunov functionals instead of Lyapunov functions. However, his method is not always practical, since to construct a suitable Lyapunov functional for a given system is rather difficult (cf. [5, p. 96]) though there are many n-dimensional systems for which appropriate Lyapunov functionals can be constructed. In 1983, Haddock and Terjeki [5] developed the invariance principle for autonomous functional differential equations, more specifically, they introduced Lyapunov-Razumikhin functions rather than Lyapunov functionals. It is the opinion of the author that Lyapunov functions are, in general, easier to construct and less complicated to verify as compared with Lyapunov functionals. Nevertheless, everything has two aspects. For some systems it is difficult to get much information by just using Lyapunov functions. The purpose of this work is to establish the corresponding invariance principles for autonomous delay difference systems in both approaches: in terms of Lyapunov functionals, and Lyapunov-Razumikhin functions. Consider the following delay difference systems x(n + 1) = f(xn),
n e Z +,
(1)
where Z + denotes the nonnegative integer set, x.(s) = x ( n + s ) ,
fors=-r,-r+
1. . . . . - 1 , 0
with some integer r > O , f : X --* R k is continuous, and X = {~0:{-r, - r + 1. . . . . O} -~ Rk}, R k is the k-dimensional Euclidean space. For each ~0 e X, we define the norm of ¢ as I1~011 = maxll~0(s)l: s = - r , - r where 1-I is any norm in R k. 735
+ 1. . . . . 01,
736
SHUNIAN ZHANG
We always assume that f(tp) is well defined for any ~ e X. Consequently, for any given ~p e X and no e Z +, there exists a unique solution of (1), denoted by X(no, ~), which is defined for all integers n _> no and satisfies Xno(no, ~) = (D.
Since system (1) is autonomous, without loss of generality we may assume n o = 0, and simply denote the corresponding solution of (1) as x(tp)(.). By the uniqueness o f solutions, for any n, m ~ Z + we have Xn(Xm(~9)) = Xn+m(~9). D e f i n i t i o n 1. Let ~o ~ X, and x(~a)(.) be the corresponding solution of (1). ~, ~ X is said to be
in the to-limit set t)(~o) of tp if there exists a sequence Inil o f nonnegative integers such that ni ~ oo and IIxn,(~) - ~,ll -" 0 as i --, oo. D e f i n i t i o n 2. Let M C X. M is said to be positively invariant (relative to (1)) if for each ~p ~ M and all n ~ Z +, Xn((a) ~ M . For the sake of convenience in establishing our results, we define a mapping T: X ~ X via
(1) as follows Tip = Xl(tp), T 2 ~ = T(T(a) = T(xl((o)) = xl(x~(~a)) = x2(~P), •
.
.
,
,
.
Tn+l(a = T(Tn(a) = Xn+l((o ) . . . . .
where x(~p) is the corresponding solution of (1) to tp. It is easy to see that T is continuous by the continuity of f((a). In fact, for given e > 0 and (a* e X. Sincef(tp) is continuous at tp = (a*, there exists ~5 > 0 (we may assume J < e) such that If(tp) - f(tp*)[ < e
if I1~ - ~*11 < ~.
Now when He - tp*[[ < ~5, we have IIT~ - T~*ll = IIx~<~) - xl(~*)ll = max{Ix(e)(1
-< maxlll~
+ s) - x(¢*)(1
- ~*ll, If(~) -f(~*)ll
+ s)l: s = -r .....
0)
< E.
This shows that T is continuous at tp*.
With the above-defined mapping T we can reformulate definitions 1 and 2 in the following alternative forms. D e f i n i t i o n 1'. Let tp ~ X. ~, e X is said to be in the w-limit set t)((o) of ¢ if there exists {ni), n i
E
Z
+
such that
Tni{o --~ ¢/
and
ni ~
oo
as i ~ oo.
D e f i n i t i o n 2'. Let M C X. Relative to (1), or to T, M is said to be positively invariant if T(M) C M.
The following lemmas are important and very useful in establishing our main results. The proofs are (essentially) given in [1] and thus omitted here.
lnvariance principle
737
LEMMA 1. Every to-limit set ~(~o) is closed and positively invariant. LEMMA 2. If ~p e X, and x(tp)(') is defined and bounded for n _> - r , then (i) the set K = [x~(tp): n e Z+I is precompact; (ii) f~(~p) is nonempty, positively invariant, and compact; (iii) x~(~p) -~ f~(tp) as n ~ oo. 2. I N V A R I A N C E
PRINCIPLES
In this section, first we wish to discuss the designated properties of O(¢) in l e m m a 2 by means of L y a p u n o v functionals. Definition 3. Let G C X. W = W ( n , (o) is said to be a Lyapunov functional of (1) on G if W : Z + x G ~ R is continuous and A(1)W(n,~p) ---- W ( n + 1,x~(tp)) - W(n,~o) <_ 0 for all ¢peG andneZ+. Clearly, it follows that Ao) W(n, x,(q0) = W ( n + 1, x,+~(¢)) - W ( n , x,(tp)). In particular, if W ( n , tp) - W(tp), then
A ( , W(~p) = W(x~(~o)) - W ( ~ ) , and a ( , W(x.(~o)) = W(x,+~(¢)) -
W(Xn(~) ).
The condition Ao)W(n, ~p) _< 0 ensures that along the solution of (1) which remains in G, W is nonincreasing. Now for a given Lyapunov functional W = W(~p) on G, define E = [tp ~ (~: Ao)W(tp) = 01. Let M be the maximal positively invariant subset in E, and for c ~ R, denote W - ~ ( c ) = 1~0 e X : W(q,) = cl. By using almost the same arguments as in [1], we can show the following result, which is called the invariance principle by Lyapunov functionals. TrlEOREM 1. Let W be a Lyapunov functional of (1) on G, and x(-) be a solution of (1) which is defined and bounded for n _> - r and such that x~ e G for n e Z +. Then for some constant c there holds x~ ~ M tq W-l(c) as n --, o o . Proof. Let ~ = Xo. By assumptions that Ao) W ( x , ( ~ ) ) <<.0 and x~(¢) is bounded, W(xn(tp)) is nonincreasing with n and is bounded from below. Thus, W(xn(cp)) ~ c with some constant c as n ---~ oo.
Let ~, e ~(tp). Then there is a sequence [ni] such that ni ~ ~ and xni(q0 ~ ~, as i ~ ~ . Since W is continuous, we have W(xni(~p) ) ~ W ( ~ ) = c and, hence, f~(tp) C W - l ( c ) .
as i --* oo
738
SHUNIAN ZHANG
However, f](~0) is positively invariant, ~u e f2(~o) implies xl(gt) e fl(~o). Thus, W(xl(qt)) = c and Ao) W(~) = W(x~(g)) - W(~,) = c - c = 0. Therefore, q / e E, i.e. t~(~0) C E and, hence, a((o) C M. Since from lemma 2 we know that x.(tp) -, f~(~u), we conclude x.(~o) -+ M N W-l(c)
•
as n ~ oo.
Now we establish the invariance principle by Lyapunov (-Razumikhin) functions and first introduce the following definition. Definition 4. Let V : R ~ R be a continuous function which is called a Lyapunov (-Razumikhin) function. Define
Ao)v(~o)
=
v(f(¢))
-
v(~o(o)).
For given G C X, define E v ( G ) = {~o e G:
max
V(x,(to)(s)) =
$= --r,...,O
max
V(~o(s)), n e Z+I.
$= --r,...,O
Let M y ( G ) be the maximal positively invariant subset w.r.t. (1) in Ev(G). Namely, for each ~ M y ( G ) , there is a solution x(¢)(.) of (1) through ¢ such that x,(~0) e O for all n e Z +, and max
V(x~(¢,)(s))=
S = --r, ...,0
max
V(~(s)).
$ = --r,...,O
THEOREM 2. Assume that there exist a Lyapunov function V = V(x) and a closed set G C X positively invariant w.r.t. (1) such that V(x(~o)(1)) _<
max
for ~0 e G.
V(~o(s))
(2)
S= -r, ...,0
Then for any ~0 e G if x(~0)(.) is bounded for n _> - r , then
f~(~o)C M y ( G ) C Ev(G). Therefore, x~(~o) --, My(G) as n --*oo.
Proof. Suppose ~o e G is such that x(~o)(.)is bounded for n _> -r. Then since G is positively invariant x,(~o) e G for n e Z + and by l e m m a 2, ~2(~) is nonempty. Let W(n, (o) =
max
V(x~(~o)(s))=
S= -r,...,0
max
V(x(~o)(n + s)).
s= --r,...,0
By condition (2) and x.(~) e G for n e Z +, we have V(x(xn+l(~0)(1))) _<
max
V(x.+~(~o)(s)),
$= --r,...,0
and
V(X(Xn((O)(1)) ) <-
max
V(x.(~o)(s));
S= --r,...,O
i.e.
V(x(~o)(n + 2)) _<
max S=
--r,...,0
V(x(to)(n + 1 + s)),
Invariance principle
739
and V(x(tp)(n + 1)) _<
V(x((o)(n + s)).
max $ = --r~... ,0
Hence, W ( n + 1, Xl((p) ) :
max
V(x(xl(tp))(n + 1 + s))
s= -r,...,O
=
max
V(x((p)(n + 2 + s))
S= --r,...,O
=
max
V(x(q~)(n + 1 + s))
s= -r+ 1,..., 1
<
max
V(x(fo)(n + 1 + s))
s = -r,...,0
<
max
V(x(¢)(n + s))
S= --r,...,0
= W ( n , 6o),
n ~ Z+.
Thus, A W(1)(n, tp) < 0,
n ~ Z +.
Since W is bounded below along this solution, there exists a limit lim W ( n , ~) = lim i~oo
max
V(x,(~o)(s)) = c ~ R .
n-*oo s=-r,...,O
Let ~, ~ £)(tp), we claim that ~ ~ E v ( G ) and so f~(tp) C E v ( G ) . In fact, there exists [ni], n i e Z + such that ni -~ oo and x.~((p) -~ ~ as i ~ oo. This implies lim[ n ~
max
V(Xn~(~o)(s))]=
$=-r,...,O
max
V(~(s))=c.
s=-r,...,O
Since G is closed, ~, ~ G. This shows that D(tp) C G. By l e m m a 2 D(~p) is positively invariant, thus xn(~') C D(tp) C G for n ~ Z +. Therefore, max
V(x.(~,)(s)) = c =
s = -r,...,O
max
V(~(s)).
s= -r,...,O
By definition 4, ~, ~ E v ( G ) and f~(~0) C E v ( G ) . It follows f r o m lemma 2 that x,(tp) ~ ~(tp) C M y ( G ) C E v ( G ) as n ~ oo. Therefore, xn(tp) ~ M y ( G )
as n --* oo.
•
R e m a r k 1. It can be easily seen f r o m the p r o o f of theorem 2 that if ~ e X is such that x(~)(.) is bounded for n _> - r and x,(~) e G for n e Z ÷, then the above conclusion remains valid under the assumption that G is closed but not necessarily positively invariant. 3. A P P L I C A T I O N S
We discuss in this section the asymptotic stability and the asymptotic constancy of solutions of (1) as applications of the obtained invariance principles. First of all, the important remark 1 in Section 2 enables us to derive the following useful result.
740
SHUNIAN
ZHANG
TtmoRm,i 3. A s s u m e f ( 0 ) = 0. I f there exist a L y a p u n o v function V: R k - , R ÷ and a constant fl > 0 such that (i) V(O) = O, V(x) > 0 for 0 ;e Ixl -< B; (ii) V(x((o)(1)) < max v(tp(s)) for any 0 ~ II ll -< B, S= --r,...,O
then the solution x = 0 o f (1) is asymptotically stable.
Proof. First, we show the stability. Clearly, if ~0 = 0, then it follows f r o m (1) and f ( 0 ) = 0 that x(0)(1) = f ( 0 ) = 0, thus, Xl(0) = 0, and x(0)(2) = f(xl(O)) = 0. In this manner, we arrive at x(0)(n) = 0
for all n e Z +,
that is x(0)(.) = 0. In what follows we m a y assume that ~p ~ 0. For any given e > 0 (e _< fl), let m=
inf V ( x ) > O . e
By the continuity o f V(x) and V(0) = 0, we m a y choose ~ > 0 (~ < fl) such that Ixl < ~ implies v ( x ) < m. T h e n for any ~o: 0 ;~ [[(o[[ < t~, V(x(tp)(1)) <
V(qT(s)) < m
max s= -r,...,O
and, hence,
Ix(~)(1)/< e. Moreover, by assumption (ii) we have
max s=
V(xl(tp)(s)) <
-r, ...,O
max s=
V(tp(s)) < m.
(3)
-r,...,O
I f [x~(~o)(0)[ = Ix((o)(1)l ;e 0, then xl(~0);~0; while if IXl( )(o)l = = o and [Xl(tp)(so - 1)[ = ]
max
V(x~(~)(s)) <_
s = -r,...,O
max
V(~(s)) < m,
s = -r,...,O
which implies Ix(~o)(2)[ < e. By induction, we can conclude that for 0 ;e [[~oH< ~ we have
V(x(tp)(n)) <
max
V(x,_l(tp)(s))
$ = --r, . . . , 0
<-...<_
max S ........
V(tp(s)) < m 0
and, thus, Ix(~p)(n)l < e This proves that x = 0 is stable.
for n = l, 2 . . . . .
(4)
Invariance principle
741
Next, we show the asymptotic stability. Let 6o > 0 be the corresponding 6(t) to the given e = / / i n the stability. Under the assumptions we may conclude by theorem 2 and remark 1 that when Iltpl] < 6o, Xn(~O) ~ M y ( G )
as n --* 0%
C Ev(G)
where G = 1~0 e X: II~ll -< P). Clearly, to prove the asymptotic stability we need only to point out M y ( G ) = 10]. In fact, by the above arguments of deriving (4) we know that for any ¢p: 0 ~ whenever IIx.(~)ll -< t~ for n e z + we always have max
V(to(s)) >
s = -r,...,O
max
V(x.(~o)(s))
I1~11 -<
for n >_ r + 1.
s = -r,...,O
Therefore, by the definition of M y , ~o ¢ My(G). Hence, M y ( G )
= [0}.
•
2. For convenience, in dealing with concrete equations we often choose Lyapunov functions as
Remark
v(x) = Ixl.
Then condition (ii) reduces to
Ixt~0)(l)l < II~ll Example
for any 0 ~ II~011 <- 8.
(5)
1. Consider x ( n + 1) = f ( x n ) ,
(6)
n ~ Z+,
where x e R k,
If(~0)l---zll¢ll
for II~0ll---
with some constants 0 < L < 1 and fl > 0. Then we may choose V ( x ) = ]x], and (5) is obviously satisfied. Hence, by theorem 3 we conclude that the zero solution x = 0 of (6) is asymptotically stable. E x a m p l e 2. More specifically, we consider the following delay difference system u(n + 1) = av(n - rO/(1 + u2(n - r2) ), v(n + 1) = bu(n - rl)/(1 + v2(n - r3)),
n • Z+, 1
(7)
where a, b are constants, rl, r2 and r 3 are integers, lal < 1, Ibl < 1, lal + Ibl ~ 0; 0 ~< rl, r2, r 3 < r for some integer r > 0. Let
x = (u, v) r,
(~ = ((PI' (P2) T,
and
Ixl = lul + Ivl,
I1~11 =
max
[[~Pl(s)l + I~p2(s)l}.
s= -r,...,O
Choose V(x) = V(u, v)=
Ixl = lul + Ivl.
742
SHUNIAN ZHANG
Then
Ix(¢o)(1)l =
[u(~0)(1)l + Iv(¢)(1)l [acp2(-r,)/(1 + (0~(-r2))l + Ibrpl(-r,)/(1
lallcz(-r0l
+
¢0~(-r3))1
+ Ibll~l(-r0l
< maxl[al, Ibl}(l~P2(-r~)l + I ~ ( - r ~ ) l ) _< zll~ll
f o r all ~o,
where 0 < L ---- maxllal, Ibll < 1. Therefore, it follows f r o m example 1 that the zero solution o f (7) is (globally) asymptotically stable.
R e m a r k 3. It is remarkable that in theorem 3 A(,) V = V(f(~o)) - V(~p(0)) = V(x((o)(1)) - V(¢(0)) m a y not even be nonnegative to conclude the asymptotic stability, which m u c h i m p r o v e d the k n o w n results (cf., say, [6]) in the case o f a u t o n o m o u s equations. We emphasize here that even for such a simple example 1 or its special case, example 2, we were unable to show the desired conclusion before by the k n o w n results. This is one o f the motivations to establish the invariance principles for delay difference systems. N o w we discuss the conditions for solutions o f (1) tending to constant limits. For L y a p u n o v function V = V(x), we define
K v = {¢P • X : V(~o(s)) = V(¢(O)) for s = - r . . . . . 0}. THEOREM 4. A s s u m e that there exist a L y a p u n o v function V = V(x) and a positively invariant, closed set G C X w.r.t. (1) such that for any ~0 • G whenever x(fp)(.) is b o u n d e d for n > - r , lim { max
V(xn(rp)(s))}
exists.
n ~oO $ ~ - - r , . . . ~ O
Then ~((o) C K v implies that
V(x(~o)(n)) = V(x.(~p)(0)) --, const,
as n ~ oo.
P r o o f . Let ~0 • G and x(~o)(') be b o u n d e d for n > - r . By assumption we m a y assume lim { max n~eo
V(xn(tP)(s))} = c
for some c • R.
$=--r,...DO
Suppose that Q((0)C K v but V ( x ( ¢ ) ( n ) ) ~ " const, as n ~ oo. T h e n since V(x(~o)(n)) is b o u n d e d for n > - r , there must be some sequence [nil and some real n u m b e r a # c such that ni ~ oo and V(x(~o)(ni)) ~ a as i ~ oo. By l e m m a 2 [xni(~P)} is p r e c o m p a c t and by assumption G is positively invariant and closed, hence, there exist some subsequence o f {n~}, denoted by In~} again, and some g / • G such that Xni(~o) ~ ~ as i ~ oo. Therefore, ~, • ~(~p) C Kv. By the definition o f K v , we have for some b • R,
v(~(s)) = v(~,(o)) = b,
s = - r . . . . . O.
lnvariance principle
743
Noting that xn~(~p)(0) ~ ~,(0) as i ~ oo, and the continuity o f V we derive V(x~,(~p)(0)) -- V(x(q~)(ni)) --" a = V(~,(0))
as i - ,
oo.
Thus, V(q/(s)) = V(~(0)) = a,
s = - r . . . . . 0.
O n the other hand, max
V(x.,(~o)(s))
~
c
=
S= --r,...,O
max
V(qt(s)) = a.
as n ~
oo.
-r,...,O
s=
This contradicts the fact a # c. Therefore, whenever Q(g0 C K v we have V(x(~o)(n)) -* const.
•
In the following we establish a simple criterion for ~(~p) C K v . T H E O ~ 5. Assume that there exists a L y a p u n o v function V = V ( x ) and a positively invariant, closed set G C X w.r.t. (1) such that: (i) V(x(~p)(1)) _< max V(~o(s)) for any (0 e G; S=
--r,...,O
(ii) for cp e G, V(x(tp)(1)) =
max
V(~o(s)) implies ~o e K v .
S ~ --r, ... ,0
T h e n for any ~p e G we have ~ ( ¢ ) C K v whenever x(~p)(.) is b o u n d e d for n >_ - r . Therefore, V(x(tp)(n)) ~ const,
as n ~ ~ .
P r o o f . Let ¢ e G be such that x(¢)(.) is b o u n d e d for n _> - r . Let ~v e ~ ( ¢ ) . Clearly, g / e G since G is closed and positively invariant. By theorem 2 ~ ~ E v ( G ) , hence, there is some c e R such that max s=
V(x(~,)(n + s)) =
max
-r,...,O
V(x,(gJ)(s)) =
s = -r,...,O
max
for n e Z +.
V(g,(s)) = c
(8)
s = -r,...,O
In particular, if n = r + 2, then we have max
V(x(~u)(r + 2 + s)) =
S = --r, . . . , 0
max
V(w(s)) = c.
S = --r, ...,0
Suppose that for some s o e { - r . . . . . 0} V(x(~u)(r + 2 + So)) =
max
V(x(qt)(r + 2 + s)) =
s = -r,...,O
max
V(W(s)) = c.
(9)
s = -r,...,O
Let s* = r + So + 1. Then 1 < s* < r + 1, it follows f r o m (9), and (8) with n = s* that V(x(qO(s* + 1)) = c =
max
V(x~,(q/)(s)),
S= --r,...,O
i.e. V(x(xs,(~u))(1))=
max
V(xs,(~u)(s)).
S= --r,...,O
Hence, by assumption (ii) (note that xs,(~v) we conclude that
e
G since g~ e G and G is positively invarariant),
Xs4~) e gv.
744
SHUNIAN ZHANG
Thus, for s = - r . . . . . 0.
V(xs,(Ct)(s)) = V(x~.(~u)(O))
This and (8) with n = s* imply that max S=
V(xs,(~u)(s)) = V(xs.(~)(0)) = c.
--r,...,O
Since - r _< - s * + 1 ___ 0, we have V(x(tu)(1)) = V(xs.(~u)(-s* + 1)) = c =
max S=
V(~u(s)).
-r,...,0
By (ii) again, this implies that ~/~ K V. Therefore, ~((0) C Kv. T o complete the p r o o f it suffices to mention that in this case the assumptions o f theorem 2 are certainly satisfied. F r o m the p r o o f o f theorem 2 we k n o w that whenever x((0)(.) is b o u n d e d for r/_> - r , lira [ max fl'-~
exists.
V(xn(~)(s))l = c
S=--r,...,O
Therefore, the desired conclusion follows f r o m theorem 4.
•
R e m a r k 4. It is notable that the L y a p u n o v functions V in theorem 5 (as well as in theorems 2
and 4) are not necessarily nonnegative. This brings out m u c h convenience in applications. E x a m p l e 3. Consider the scalar equation 0
x(n + 1) = ax(n) + b x ( n - r) - b x ( n ) x ( n - r) ~
[x(n + s) - x(n)l 2,
(10)
S ~ --r
where n • Z +, a > 0, b > 0 are constants, and a + b = 1. Let G = {(0 • X : 0 _< ~o(s) _< 1, s = - r . . . . . 0}. Then it is easy to show that G is positively invariant w.r.t. (10). Clearly, it suffices to show that 0 < x(~o)(l) _< 1 for ¢ • G. In fact, if (0 • G, then Ileal[ --- 1 and x((0)(1) = a(0(0) + b ( 0 ( - r )
I
o
1 - cp(0) ~ $=
1
[(0(s) - (0(0)12 > a(0(0) > 0,
--r
and
b)ll(01[
x((0)(1) _< all(oil + bll(0H = (a +
= 11(011~ 1.
Thus, o _~ x((0)O) -< 1.
C h o o s e V(x) = x. Then for any ~ e G we have as above that V(x((0)(1)) = x((0)(1) _<
11(oil =
max
v((0(s)).
$ = --r, ..., 0
O n the other hand, if (0 e G and V(x((0)(1)) =
max
V((0(s)), then x((0)(1) = H(0[I, i.e.
S= --r,...,O 0
(0(0) - b(0(O) + b ( 0 ( - r ) - b(0(O)(0(-r) ~ $=
[(0(s) - (o(0)12 = --r
11(011.
(11)
Invariance principle
745
I f (0 = 0, t h e n o b v i o u s l y (0 ~ K v. H e n c e , we need o n l y to c o n s i d e r the case o f (0 ;~ 0. S u p p o s e (0(0) < 11(oil ~ 0. T h e n the l e f t - h a n d side o f (11) < all(011 + oil(011 = 11(oil, hence, the e q u a l i t y (11) c a n n o t h o l d . T h e r e f o r e , (0(0) = 11(011. H o w e v e r , t h e n it clearly follows f r o m (11) t h a t 0 ( 0 ( - r ) = (0(0) and (0(0)(0(-r) ~ [(0(s) - (0(0)] 2 = 0. $=
--r
T h e r e are t w o p o s s i b l e cases: (i) if ( 0 ( - r ) = (0(0) = 0, t h e n (0 = 0, a n d c e r t a i n l y (0 e K v ; (ii) if (0(-r) = (0(0) ;~ 0, then ~s°= _r[(0(s) - (0(0)] 2 = 0 implies t h a t (0(s) = (0(0)
for s = - r . . . . . 0.
This m e a n s t h a t (0 e K e. T h e r e f o r e , in a n y case, the a s s u m p t i o n s o f t h e o r e m 5 are satisfied. Hence, V(x((0)(n)) = x((0)(n) ~ const,
as n -+ ~ .
R e m a r k 5. It is easy to see t h a t x(n) = c for a n y c e R is a s o l u t i o n o f (10) if a + b = 1. C O N C L U D I N G REMARKS It is w o r t h m e n t i o n i n g t h a t we a r e n o t expected to o b t a i n all the p a r a l l e l results for d e l a y d i f f e r e n c e systems to the k n o w n results for d e l a y d i f f e r e n t i a l systems, b e c a u s e there are essential differences b e t w e e n discrete m o d e l s a n d c o n t i n u o u s m o d e l s . F o r instance, even f o r the simple d e l a y d i f f e r e n c e systems o f the f o r m x(n + 1) - x(n) = - h ( x ( n ) ) + h(x(n - 1)), if h(u) = u 2 a n d c h o o s e x ( - l ) = l a n d x(0) = 2, t h e n it is easily seen t h a t t h e c o r r e s p o n d i n g s o l u t i o n is [ - 1 , 2 , - 1 , 2 . . . . }; while if h ( u ) = u 3 which is strictly increasing, a n d c h o o s e x ( - 1 ) = 1 a n d x ( 0 ) = 0, t h e n the s o l u t i o n is {1,0, 1 , 0 . . . . 1. N o t e t h a t n o n e o f t h e m is a s y m p t o t i c a l l y c o n s t a n t as n -+ ~ . This is a distinct c o n t r a s t to the relevant results m e n t i o n e d in [5]. I n short, t h e r e are quite a few o p e n p r o b l e m s f o r d e l a y difference systems to be i n v e s t i g a t e d further. W e m a y discuss these in a s u b s e q u e n t p a p e r . Acknowledgement--The author wishes to thank the referee for making valuable suggestions and comments on the original manuscript. REFERENCES 1. LASALLE J. P., The Stability of Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia (1976). 2. LASALLE J. P., Stability theory for ordinary differential equations, J. diff. Eqns 4, 57-65 (1968). 3. HALE J. K., Sufficient conditions for stability and instability of autonomous functional differential equations, J. diff. Eqns 1,452-482 (1965). 4. HALE J. K., Theory of Functional Differential Equations. Springer, New York (1977). 5. H A D D O C K J. R. & TERJEKI J., Liapynov-Razumikhin functions and invariance principle for functional differential equations, J. diff. Eqns 48, 95-122 (1983). 6. ELAYDI S. & ZHANG SHUNIAN, Stability and periodicity of difference equations with finite delay. Funkcialaj Ekvacioj (to appear).