On Dynamical Stability in General Dynamical Systems

Journal of Mathematical Analysis and Applications 263, 455᎐478 Ž2001. doi:10.1006rjmaa.2001.7620, available online at http:rrwww.idealibrary.com on

On Dynamical Stability in General Dynamical Systems1 Desheng Li Department of Mathematics and Information Science, Yantai Uni¨ ersity, Yantai 264005, Shandong, People’s Republic of China E-mail: [email protected] Submitted by Helene ´` Frankowska Received July 7, 2000

This paper is concerned with dynamical stability of general dynamical systems. We discuss invariance properties of some limit sets, investigate connections between various notions and definitions related to stability and attraction properties, and establish existence results for invariant uniform attractors. We also give a characterization of asymptotic stability of general dynamical systems via uniformly 䊚 2001 Academic Press unbounded Lyapunov functions.

1. INTRODUCTION General dynamical systems ŽGDSs., which are sometimes referred to as general control systems or set-valued dynamical systems, are used to describe multi-valued differential equations Žincluding differential inclusions. w4, 17x and control systems w3, 12, 13, 18, 22, 23x as well as economic flow w6x. They have been widely studied in the literature. In this paper, we are mainly concerned with some dynamical stability properties of GDSs defined on a complete locally compact metric space X. Our work is organized as follows. Section 2 is concerned with some preliminary works. First, we give some definitions and auxiliary results. Then we discuss invariance properties of ␻-limit sets and prolongation limit sets. Invariance properties of limit sets play important roles in describing dynamical stabilities of a system. Although they have been investigated by many authors for GDSs Žsee, for instance, w7, 12, 18, 23x, etc.., we find that there is still something to be 1

Supported by the National Natural Sciences Foundation of China Ž10071066.. 455 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

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further clarified. In this section, we prove that ␻-limit sets and prolongation limit sets are negatively invariant and positively weakly invariant; however, they need not be positively invariant in the general case Žcounter-examples will be given.. We also show that through each point of a ␻-limit set or prolongation limit set, there passes a complete trajectory which is contained in the limit set. This result extends some basic knowledge concerning the limit sets for differential inclusions Žsee Filippov w8x. to GDSs. In recent years, a lot of notions and definitions concerning stability and attraction properties have appeared in the literature. In Section 3, we investigate connections between some of these different notions and definitions. In particular, we prove that asymptotic eventual stability implies uniform attraction. It has long been recognized in the classical theory of differential equations that asymptotic stability is equivalent to uniform asymptotic stability. This basic fact was extended more recently by Clarke et al. w5x to differential inclusion x⬘Ž t . g f Ž x Ž t ... As a consequence of our results we conclude that for GDSs, asymptotic eventual stability and asymptotic stability are equivalent to uniform asymptotic eventual stability and uniform asymptotic stability, respectively. Although we have shown in Section 2 that in general, ␻-limit sets need not be positively invariant, inspired by some ideas in one of our recent works w14x, we prove in Section 4 that if a compact subset M of X attracts itself, then its ␻-limit set is positively invariant. Based on this fact, we establish some existence results for invariant Ži.e., both negatively and positively invariant. uniform attractors. The methods that rely on Lyapunov functions have become a powerful and popular technique in the stability analysis of differential inclusions and control systems in recent years Žsee w1, 2, 5, 7, 15, 16, 20᎐22x, etc., and references therein .. Based on the results obtained in previous sections, in Section 5, we show that asymptotic stability is equivalent to the existence of suitable Lyapunov functions which are uniformly unbounded. In w7x, the authors characterized the asymptotic stability of general dynamical systems in terms of Lyapunov functions. However, they required a much stronger continuity assumption for the systems in space variable x; moreover, the Lyapunov function given therein may not be uniformly unbounded. In applications, uniformly unbounded Lyapunov functions are of particular interest. Unfortunately, usually it is not easy to formulate such a Lyapunov function when describing local versions of notions concerning asymptotic stability properties. One alternative approach for formulating uniformly unbounded Lyapunov functions can be found in Tsinias w22x. But using that approach one needs to take account the entering time of the trajectories into neighborhoods of the stable set M and thus has to do much

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careful mathematical analysis. Our method here makes use of the concept of K0⬁ functions and seems to be simpler and more direct. 2. PRELIMINARIES AND INVARIANCE PROPERTIES OF SOME LIMIT SETS In this paper X will always be used to denote a complete locally compact metric space with metric d Ž⭈, ⭈ ., and F is a general dynamical system defined on X. For any A, B ; X, we define d H Ž A, B . s sup d Ž x, B . ,

␦ H Ž A, B . s max Ž d H Ž A, B . , d H Ž B, A . . ,

xgA

where d Ž x, B . s inf y g B d Ž x, y .. d H Ž⭈, ⭈ . and ␦ H Ž⭈, ⭈ . are called the Hausdorff semidistance and Hausdorff distance in X, respectively. Let Q s X = Rq, where Rqs w0, ⬁.. Let 2 X be the set of all subsets of X. DEFINITION 2.1 w12x. A mapping F: Q ª 2 X is said to be a Žautonomous. general dynamical system ŽGDS in short. if the following axioms hold: Ž1. For any Ž x, t . g Q, F Ž x, t . is a closed nonempty subset of X. Ž2. F Ž x, 0. s x, ᭙ x g X. Ž3. F Ž F Ž x, t ., s . s F Ž x, t q s ., ᭙ x g X, s, t g Rq. Ž4. F Ž x, t . is continuous in t Žfor fixed x ., in the sense of Hausdorff distance. Ž5. F Ž x, t . is upper semicontinuous in x Ži.e., continuous in x in the sense of Hausdorff semidistance. Žfor fixed t ., uniformly in t on any compact interval 0 F t F T - ⬁. Remark 2.2. Axioms Ž4. and Ž5. imply that F Ž x, t . is jointly upper semicontinuous in Ž x, t .; see Roxin w18, Lemma 4.1x. The following notations and conventions will be used throughout the paper. First, for simplicity, we will always identify a single-point set  a4 with the point a. For any subset A of X and r ) 0, S Ž A, r . s  y g X : d Ž y, A . - r 4 . Let I ; Rq. We denote by F Ž A, I . the set DŽ x, t .g A=I F Ž x, t .; in particular, F Ž A. s F Ž A, Rq. . For x g X, F Ž x . was called by some authors

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the funnel of F through x in the literature; see, for instance, w7, Definition 1.9x. For sequences x n g X and t n g Rq, we also use the notation lim sup F Ž x n , t n . to denote the set defined by y g lim sup F Ž x n , t n . iff every neighborhood of y intersects F Ž x n , t n . for infinitely many values of n. Assume that Ž x n , t n . ª Ž x, t .. Let F be a GDS. Then by upper semicontinuity, we have lim sup F Ž x n , t n . ; F Ž x, t . . We will denote by CX the set of all nonempty compact subsets of X. In the following argument, all of the notions and results are stated with respect to a given GDS F defined on a locally compact metric space X. DEFINITION 2.3. A trajectory ␥ of F on w a, b x ; R1 is a mapping ␥ : w a, b x ª X satisfying ␥ Ž t 2 . g F Ž␥ Ž t 1 ., t 2 y t 1 . for any t 1 , t 2 g w a, b x, t1 F t 2 . One easily checks that if ␥ 1 and ␥ 2 are trajectories of F on w a, b x and w b, c x, respectively, with ␥ 1Ž b . s ␥ 2 Ž b ., then the concatenation ␥ of ␥ 1 and ␥ 2 defined by

␥ Ž t. s

½

␥ 1Ž t . ,

t g w a, b x ;

␥2Ž t . ,

t g w b, c x

is a trajectory of F on w a, c x. We have the following basic facts w12, 18x: PROPOSITION 2.4.

For any A g CX and T ) 0. F Ž A, w0, T x. is compact.

PROPOSITION 2.5. Assume A g CX and T ) 0. Then for any ␧ ) 0, there exists a ␦ ) 0 such that d H Ž F Ž S Ž A, ␦ . , w 0, T q ␦ x . , F Ž A, w 0, T x . . - ␧ . THEOREM 2.6. E¨ ery trajectory is continuous. Furthermore, let y g F Ž x, t 1 y t 0 ., where t 0 F t 1. Then there is a trajectory ␥ of F on w t 0 , t 1 x such that ␥ Ž t 0 . s x, ␥ Ž t 1 . s y. THEOREM 2.7 ŽBarbashin’s Theorem.. Let ␥n be a sequence of trajectories of F on w t 0 , t 1 x with ␥nŽ t 0 . ª x 0 . Then there is a subsequence ␥n i and a trajectory ␥ 0 such that ␥n i con¨ erges uniformly on w t 0 , t 1 x to ␥ 0 . DEFINITION 2.8. A set A is said to be Ž1. positively invariant, if F Ž A, t . ; A

᭙ t g Rq ;

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Ž2. positively weakly invariant, if F Ž A, t . l A / ⭋

᭙ t g Rq ;

Ž3. negatively invariant, if A ; F Ž A, t .

᭙ t g Rq.

If A is both negatively and positively invariant, i.e., F Ž A, t . s A

᭙ t g Rq,

then we say that A is invariant. PROPOSITION 2.9. Assume that A g CX. If A is negati¨ ely in¨ ariant, then for any x g A, there is a trajectory ␥ of F on Žy⬁, 0x which lies in A with ␥ Ž0. s x. Proof. We only need to prove that for any x g A, there exists a trajectory ␥ of F on wy1, 0x which lies in A such that ␥ Ž0. s x. Indeed, if this is true, then we can get by induction a sequence of trajectories ␥n Ž n s 1, 2, . . . ., where ␥n is defined on wyn, yn q 1x and contained in A, such that

␥ 1 Ž 0 . s x, ␥n Ž yn q 1 . s ␥ny1 Ž yn q 1 .

for n G 2.

Now one obtains a trajectory ␥ : Žy⬁, 0x ª X of F with ␥ Ž0. s x immediately by setting

␥ Ž t . s ␥n Ž t .

when t g w yn, yn q 1 x .

For the above purpose, we first show that for any finite number of t i g wy1, 0x Ž i s 1, . . . , n., there is a trajectory ␥ such that

␥ Ž 0 . s x,

␥ Ž y1 . , ␥ Ž t i . g A Ž i s 1, . . . , n . .

Ž 2.1.

Without loss of generality, we can assume that t 1 G t 2 G ⭈⭈⭈ G t n . By virtue of the negative invariance of A, there is a x 1 g A such that x g F Ž x 1 , yt1 .. It follows by Theorem 2.6 that there is a trajectory ␥ 1 of F on w t 1 , 0x such that

␥ 1Ž t1 . s x 1 ,

␥ 1 Ž 0 . s x.

Similarly, since x 1 g A, we can find a x 2 g A and a trajectory ␥ 2 of F on w t 2 , t 1 x such that

␥ 2 Ž t2 . s x 2 ,

␥ 2 Ž t1 . s x 1 .

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By induction, we obtain a finite number of points x i g A Ž i s 1, . . . , n q 1. and trajectories ␥ i Ž i s 1, . . . , n q 1. such that

␥ i Ž t i . s x i , ␥ i Ž t iy1 . s x iy1

for i s 1, . . . , n q 1,

where t 0 s 0, x 0 s x, and t nq1 s y1. Define ␥ : wy1, 0x ª X as

␥ Ž t . s ␥i Ž t .

when t g w t i , t iy1 x , i s 1, . . . , n q 1.

Then ␥ is a trajectory of F on wy1, 0x with the desired property Ž2.1.. Now taking a countable dense subset  t i 4 of wy1, 0x, for each n g N, we conclude that there exists a trajectory ␥n of F satisfying

␥n Ž 0 . s x,

␥n Ž y1 . , ␥n Ž t i . g A Ž for i s 1, . . . , n . . Ž 2.2. By the compactness of A, it can be assumed that ␥nŽy1. ª x 0 g A. By virtue of Theorem 2.7, there is a subsequence ␥n i that converges to some trajectory ␥ uniformly on wy1, 0x. Due to Ž2.2., we have ␥ Ž 0 . s x,

␥ Ž y1 . , ␥ Ž t i . g A Ž i s 1, 2, . . . . .

Since  t i 4 is a dense subset of wy1, 0x and ␥ is continuous, we conclude immediately that ␥ lies in A on wy1, 0x. The proof is complete. The conclusion of the following proposition can be found in Kloeden w12, Remark 3.2x and Roxin w18, Theorem 7.1x. The interested reader can also give a simple proof by an argument fully analogous to that above. PROPOSITION 2.10. Assume that A g CX. If A is positi¨ ely weakly in¨ ariant, then for any x g A, there is a trajectory ␥ of F on w0, ⬁. which lies in A with ␥ Ž0. s x. We emphasize that the positi¨ e weak in¨ ariance of A is sufficient to guarantee the conclusion in the above proposition. Now we turn to the invariance properties of prolongation limit sets and ␻-limit sets. DEFINITION 2.11. Let x g X, A be a subset of X. Ž1. The prolongation limit set JqŽ x . of x is defined as Jq Ž x . s  y g X : ᭚ x n ª x, t n ª ⬁, yn g F Ž x n , t n . such that yn ª y 4 . Ž2. The ␻-limit set ␻ Ž A. of A is defined as

␻ Ž A . s  y g X : ᭚ t n ª ⬁, yn g F Ž A, t n . such that yn ª y 4 . Remark 2.12. Let x g X, A ; X. One easily checks that Jq Ž x . s

F F Ž S Ž x, 1rn . ,

n, ⬁ . . ,

Ž 2.3.

ngN

␻ Ž A. s

F F Ž A, tG0

t , ⬁. . .

Ž 2.4.

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PROPOSITION 2.13. Let x g X, A be a subset of X. If there exist ␦ ) 0, T g Rq, and W, V g CX such that F Ž S Ž x, ␦ . , T , ⬁ . . ; W ,

F Ž A, T , ⬁ . . ; V ,

Ž 2.5.

then JqŽ x ., ␻ Ž A. g CX; moreo¨ er, they are negati¨ ely in¨ ariant and positi¨ ely weakly in¨ ariant. Proof. The nonemptyness and compactness are direct consequences of Ž2.3. ᎐ Ž2.5.. In the sequel we prove that ␻ Ž A. and JqŽ x . are negatively invariant and positively weakly invariant. Let ␾ g ␻ Ž A.. Then there exist sequences x n g A, t n g Rq with t n ª ⬁ and ␾n g F Ž x n , t n . such that ␾n ª ␾ . Let t g Rq be given arbitrarily. We first prove that there exists ␺ g ␻ Ž A. such that ␾ g F Ž ␺ , t .; hence ␻ Ž A. is negatively invariant. Disregarding a finite number of terms, it can be assumed that t - t n for all n g N. Since ␾n g F Ž x n , t n . s F Ž F Ž x n , t n y t ., t ., we can find for every n a point ␺n g F Ž x n , t n y t . such that ␾n g F Ž ␺n , t .. By Ž2.5., taking a subsequence if necessary, we may assume that ␺n ª ␺ . Then ␺ g ␻ Ž A.. Finally we conclude that

␾ s lim ␾n g lim sup F Ž ␺n , t . ; F Ž ␺ , t . .

Ž 2.6.

Now we check that F Ž ␾ , t . l ␻ Ž A. / ⭋; thus ␻ Ž A. is positively weakly invariant. For each n, take a

␺n g F Ž ␾ n , t . .

Ž 2.7.

Then since ␾n g F Ž x n , t n ., we see that ␺n g F Ž x n , t n q t .. Due to Ž2.5., taking a subsequence if necessary, we can assume that ␺n ª ␺ . Clearly ␺ g ␻ Ž A.. On the other hand, passing to the limit in Ž2.7., one finds that

␺ s lim ␺n g lim sup F Ž ␾n , t . ; F Ž ␾ , t . . This completes the proof of the desired result. Replacing ‘‘␻ Ž A.’’ with ‘‘JqŽ x .’’ and ‘‘ x n g A’’ with ‘‘ x n ª x’’ in the above argument, we obtain a proof for negative invariance and positive weak invariance of JqŽ x . immediately. The proof is complete. In the case of dynamical systems Ži.e., single-valued dynamical systems., it is well known that ␻ Ž A. and JqŽ x . are also positively invariant and hence are invariant. Situations become quite different in the case of GDSs. Here we give an example to show that in the general case, ␻ Ž A. and JqŽ x . may not possess the positive invariance property.

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EXAMPLE 2.14. Let X s R1. Define F: X = Rqª 2 X as follows: for any Ž x, t . g X = Rq, F Ž x, t . s xeyt

if x - 0,

F Ž x, t . s w x, x q t x

if x G 0.

A straightforward check shows that F satisfies axioms Ž1. ᎐ Ž5. in Definition 2.1 and hence is a GDS on X. Since F Ž x, t . converges to 0 as t ª ⬁ when x - 0, one concludes that for any x - 0, Jq Ž x . s ␻ Ž x . s  0 4 . Unfortunately  04 is not positively invariant under F. Remark 2.15. In the proof of the negative invariance of ␻ Ž A. in Proposition 2.13, the relation lim sup F Ž ␺n , t . ; F Ž ␺ , t . in Ž2.6. which is due to the upper semicontinuity of F in Ž x, t . plays a key role. Instead of upper semicontinuity Žaxiom Ž5. in Definition 2.1., we assume that F has the following Property K w7x. For any x g X and any sequence  x n4 converging to x, for any fixed t, lim sup F Ž x n , t . > F Ž x, t . . Then under the hypothesis in Proposition 2.13, one can show that ␻ Ž A. is positively invariant. Indeed, let ␾ g ␻ . Then there exist sequences x n g A, t n g Rq with t n ª ⬁ and ␾n g F Ž x n , t n . such that ␾n ª ␾ . Let t ) 0. Assume that ␺ g F Ž ␾ , t .. Since F Ž ␾ , t . ; lim sup F Ž ␾n , t ., there exist subsequences ␺n i and ␾n i , ␺n i g F Ž ␾n i , t . such that ␺n i ª ␺ . Observing that

␺n i g F Ž ␾ n i , t . ; F Ž F Ž x n i , t n i . , t . s F Ž x n i , t n i q t . , one sees that ␺ g ␻ Ž A.. This proves the desired result. The same remark holds true for JqŽ x .. It should be pointed out that for a GDS F, due to the upper semicontinuity of F in x, assuming Property K amounts to saying that F is continuous in x Žsee Elaydi and Kaul w7, Remark 2.2x.. As a direct consequence of Propositions 2.9, 2.10, and 2.13, we have PROPOSITION 2.16. Assume the hypotheses in Proposition 2.13. Then for any ␾ g JqŽ x . Ž ␻ Ž A.., there exists a complete trajectory ␥ Ž i.e., a trajectory defined on the whole line R1 . contained in JqŽ x . Ž ␻ Ž A.. with ␥ Ž0. s ␾ . The same conclusion as in Proposition 2.16 concerning some limit sets for differential inclusions which stimulates our considerations here can be found in Filippov w8, p. 130, Lemma 4x.

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3. ASYMPTOTIC STABILITIES AND ATTRACTION PROPERTIES In this section we discuss relations between some asymptotic stability and attraction properties. In particular, we prove that the notions of asymptotic stability and asymptotic eventual stability are equivalent to that of uniform asymptotic stability and uniform asymptotic eventual stability, respectively. Let X be a complete locally compact metric space. Then for any M g CX, there exists ␧ ) 0 such that the closure S Ž M, ␧ . of SŽ M, ␧ . is compact Žsee Roxin w18x.. For any V ; X, we denote by int V the interior of V. Let V, M ; X. We say that V is a neighborhood of M; by this we mean that M ; int V. Let F be a GDS defined on X. DEFINITION 3.1. Let M be a subset of X, x g X, and let V be a subset of X. We say that Ž1. M attracts x, if ᭙␧ ) 0, ᭚T ) 0 such that F Ž x, w T, ⬁.. ; SŽ M, ␧ .; Ž2. M equi-attracts x, if ᭙␧ ) 0, ᭚␦ , T ) 0 such that F Ž SŽ x, ␦ ., w T, ⬁.. ; S Ž M, ␧ .; Ž3. M attracts V, if ᭙␧ ) 0, ᭚T ) 0 such that F Ž V, w T, ⬁.. ; S Ž M, ␧ .. We first state the following basic fact. PROPOSITION 3.2. Let V be a subset of X. Suppose that there exist W g CX and T ) 0 such that F Ž V, w T, ⬁.. ; W. Then ␻ Ž V . attracts V. Proof. The proof for such results is standard in the theory of dynamical systems Žsee w9, 10, 24x etc... Here we give it just for the reader’s convenience. Assume the contrary. Then there exist a ␧ 0 ) 0 and sequences x n g V, t n g Rq with t n ª ⬁ and yn g F Ž x n , t n . such that d Ž yn , ␻ Ž V . . G ␧ 0 for each n. Since W is compact and F Ž V, w T, ⬁.. ; W, there is a subsequence yn i converging to some point y 0 g W. By Definition 2.11, one sees that y 0 g ␻ Ž V .. Thus for i sufficiently large, we have d Ž yn i , ␻ Ž V .. - ␧ 0 , which yields a contradiction and hence proves the desired result. Replacing ‘‘␻ Ž V .’’ and ‘‘ x n g V ’’ with ‘‘JqŽ x .’’ and ‘‘ x n ª x’’ in the proof of Proposition 3.2, one gives a proof for the following proposition: PROPOSITION 3.3. Let x g X. Suppose that there exist W g CX and ␦ , T ) 0 such that F Ž SŽ x, ␦ ., w T, ⬁.. ; W. Then JqŽ x . equi-attracts x.

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Let M be a subset of X. We denote by AŽ M ., A eq Ž M ., and A uŽ M . the regions of attraction, equi-attraction, and uniform attraction of M, respectively. They are defined as A Ž M . s  x g X : M attracts x 4 , A eq Ž M . s  x g X : M equi-attracts x 4 , and A u Ž M . s  x g X : M attracts a neighborhood V of x 4 . Clearly A uŽ M . ; A eqŽ M . ; AŽ M .. PROPOSITION 3.4. Let M be a subset of X. Then AŽ M . is positi¨ ely in¨ ariant. If we assume that M g CX and AŽ M . contains a neighborhood of M, then AŽ M . is open. Proof. Assume that x g AŽ M ., t ) 0. Let y g F Ž x, t .. Then for any s G 0, F Ž y, s . ; F Ž F Ž x, t . , s . s F Ž x, t q s . , from which it follows that M attracts y. Hence y g AŽ M .. This proves the positive invariance of AŽ M .. Now assume that M g CX and AŽ M . contains a neighborhood U of M. Take a ␧ ) 0 sufficiently small such that SŽ M, ␧ . ; U. Suppose x 0 g AŽ M .. Since M attracts x 0 , there exists T ) 0 such that F Ž x 0 , T . g SŽ M, ␧ .. By upper semicontinuity of F in x, there exists ␦ ) 0 such that F Ž SŽ x 0 , ␦ ., T . ; SŽ M, ␧ .. Since SŽ M, ␧ . ; AŽ M ., we deduce immediately that M attracts each point in S Ž x 0 , ␦ .. Thus SŽ x 0 , ␦ . ; AŽ M .. Therefore AŽ M . is open. Remark 3.5. One easily sees by the definition that A uŽ M . is an open subset of X. If we assume that M g CX and A eq Ž M . contains a neighborhood of M, then by an argument similar to that in the proof of Proposition 3.4, we can show that A eq Ž M . is open. We do not know whether A eq Ž M . and A uŽ M . are positively invariant or not in the general case. Remark 3.6. We claim that M attracts each compact subset K of A eq Ž M .. Indeed, let ␧ ) 0 be given arbitrarily; then for any x g K, one can find r x , Tx ) 0 such that F Ž SŽ x, r x ., w Tx , ⬁.. ; SŽ M, ␧ .. Due to the compactness of K, there exists a finite number of x denoted by x i Ž i s 1, . . . , n. such that K ; D1 F iF n SŽ x i , r x .. Let T s max 1 F i F n Tx . i i Then F Ž K , T , ⬁ . . ; S Ž M, ␧ . . This proves the claim.

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As A uŽ M . ; A eqŽ M ., it follows immediately that M attracts each compact subset K of A uŽ M .. If A eq Ž M . is open, then for any x g A eqŽ M ., we can take a ␦ ) 0 sufficiently small such that S Ž x, ␦ . ; A eq Ž M . and is compact. Since M attracts S Ž x, ␦ . , one concludes that A eq Ž M . s A uŽ M .. PROPOSITION 3.7. Assume that M g CX, x g X. Then x g A eq Ž M . iff there exist W g CX and ␦ ) 0 such that F Ž SŽ x, ␦ .. ; W and JqŽ x . ; M. Proof. Assume that there is a W g CX such that F Ž S Ž x, ␦ .. ; W. Then by Proposition 3.3, JqŽ x . g CX and equi-attracts x. Therefore if JqŽ x . ; M, x g A eq Ž M .. Conversely, if x g A eq Ž M ., then M equi-attracts x. Take a ␧ sufficiently small such that the closure U of U [ S Ž M, ␧ . is compact. Let ␦ , T ) 0 be such that F Ž S Ž x, ␦ ., w T, ⬁.. ; U. We may assume that ␦ is sufficiently small such that S Ž x, ␦ . is compact. By virtue of Proposition 2.5, we know that F ŽS Ž x, ␦ . , w0, T x. is compact. Now we set W s F ŽS Ž x, ␦ . , w0, T x. j U. Then F Ž SŽ x, ␦ .. ; W. We also show that JqŽ x . ; M. Let ␧ ) 0. Then there exist ␦ , T ) 0 such that F Ž SŽ x, ␦ ., w T, ⬁.. ; SŽ M, ␧ ., which implies that JqŽ x . ; S Ž M, ␧ . . Since ␧ is arbitrary, we conclude that JqŽ x . ; M. The proof of the proposition is complete. DEFINITION 3.8. Assume M g CX. We say that M is Ž1. Lyapunov stable, if ᭙␧ ) 0, ᭚␦ ) 0 such that F Ž SŽ M, ␦ .. ; SŽ M, ␧ .; Ž2. eventually Lyapunov stable, if ᭙␧ ) 0, ᭚␦ , T ) 0 such that Ž Ž F S M, ␦ ., w T, ⬁.. ; SŽ M, ␧ .. Clearly Lyapunov stability implies eventual Lyapunov stability. Also, one easily sees that if M g CX attracts a neighborhood V of M, then it is eventually Lyapunov stable. The following proposition indicates that in general, eventual Lyapunov stability is rather a weaker notion than that of Lyapunov stability. PROPOSITION 3.9. M g CX is Lyapuno¨ stable iff it is positi¨ ely in¨ ariant and e¨ entually Lyapuno¨ stable. Proof. We only need to show that positive invariance plus eventual Lyapunov stability implies Lyapunov stability. Let ␧ ) 0. If M is eventually Lyapunov stable, then we can take ␦ 1 , T ) 0 such that F Ž S Ž M, ␦ 1 . , T , ⬁ . . ; S Ž M, ␧ . .

Ž 3.1.

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By virtue of Proposition 2.5, there exists a ␦ 2 ) 0 such that d H Ž F Ž S Ž M, ␦ 2 . , w 0, T x . , F Ž M, w 0, T x . . F d H Ž F Ž S Ž M, ␦ 2 . , w 0, T q ␦ 2 x . , F Ž M, w 0, T x . . - ␧ . Assume that M is positively invariant. Then F Ž M, w0, T x. ; M. We have d H Ž F Ž S Ž M, ␦ 2 . , w 0, T x . , M . F d H Ž F Ž S Ž M, ␦ 2 . , w 0, T x . , F Ž M, w 0, T x . . - ␧ .

Ž 3.2.

Now we take ␦ s minŽ ␦ 1 , ␦ 2 .. Then Ž3.1. and Ž3.2. imply F Ž SŽ M, ␦ .. ; SŽ M, ␧ .. The proof is complete. Remark 3.10. Concepts concerning eventual stability properties were first introduced by La Salle and Rath in 1963 for ordinary differential equations. They were extended to nonautonomous general dynamical systems by Kloeden w13x. It should be pointed out that the notions of eventual Žstrong. stability and eventual uniform Žstrong. stability given in w13x reduce to Lyapunov stability defined as in Definition 3.8 in the case of autonomous systems. DEFINITION 3.11. Assume M g CX. We say that M is Ž1. asymptotically stable Žabsolutely asymptotically stable, uniformly asymptotically stable. if it is Lyapunov stable and AŽ M . Žresp. A eq Ž M ., A uŽ M .. is a neighborhood of M; Ž2. asymptotically eventually stable Žabsolutely asymptotically eventually stable, uniformly asymptotically eventually stable. if it is eventually Lyapunov stable and AŽ M . Žresp. A eq Ž M ., A uŽ M .. is a neighborhood of M. Remark 3.12. The notion of absolute asymptotic stability comes from Tsinias w22x. A simple fact is that absolute asymptotic stability and absolute asymptotic eventual stability are equivalent to uniform asymptotic stability and uniform asymptotic eventual stability, respectively. Indeed, if A eq Ž M . is a neighborhood of M, then by Remark 3.5, it is open. Furthermore, by Remark 3.6, A uŽ M . s A eqŽ M .. Thus A uŽ M . is a neighborhood of M. Conversely, if A uŽ M . is a neighborhood of M, then since A uŽ M . ; A eq Ž M ., one concludes that A eqŽ M . is a neighborhood of M. Hence, the conclusion holds true. In w7x, a set M g CX is said to be uniformly asymptotically stable if it is Lyapunov stable and the set A˜u Ž M . s  x g X : Jq Ž x . ; M 4

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is a neighborhood of M. Note that A˜uŽ M . may contain all of the points whose prolongation limit sets are empty. Therefore in our opinion, A˜uŽ M . may be too large in the general case. If we replace A˜uŽ M . by the set

 x g X : ᭚W g CX , ␦ ) 0 such that F Ž S Ž x, ␦ . . ; W and Jq Ž x . ; M 4 , then by Proposition 3.7, the notion of uniform asymptotic stability defined as above coincides with ours in the terminology of Definition 3.11. Now we state and prove the following theorem, which extends in some ways the results obtained in w7, Corollary 2.13x and w11, Theorem 4.14x. THEOREM 3.13. equi¨ alent: Ž1. Ž2. Ž3.

Assume that M g CX. Then the following assertions are

M is e¨ entually Lyapuno¨ stable. M ; AŽ M . and A eq Ž M . s AŽ M .. M ; A eq Ž M ..

Proof. Ž1. « Ž2.. Assume that M is eventually stable. Let ␧ ) 0; then there exist ␦ , T ) 0 such that F Ž S Ž M, ␦ . , T , ⬁ . . ; S Ž M, ␧ . .

Ž 3.4.

Ž3.4. implies that M attracts itself and hence M ; AŽ M .. Let x 0 g AŽ M .. Then there is a t 0 ) 0 such that F Ž x 0 , t 0 . g S Ž M, ␦ .. By upper semicontinuity of F in x, we can pick a r 0 ) 0 sufficiently small such that F Ž SŽ x 0 , r 0 ., t 0 . ; SŽ M, ␦ .. By Ž3.4., one concludes immediately that F Ž SŽ x 0 , r 0 ., w T q t 0 , ⬁.. ; SŽ M, ␧ .. This shows that x 0 g A eqŽ M .. Hence Ž2. holds. Ž2. « Ž3.. This is obvious. Ž3. « Ž1.. Let ␧ ) 0. Since M ; A eq Ž M ., for any x g M, there exists r x , Tx ) 0 such that F Ž S Ž x, r x . , Tx , ⬁ . . ; S Ž M, ␧ . . M being a compact set, there exist a finite number of x g M, denoted by x i Ž1 F i F n., such that M ; V [ D1 F iF n S Ž x i , r x i .. Let T s max Tx i : i s 1, . . . , n4 . As V is an open neighborhood of M, we can pick a ␦ ) 0 sufficiently small such that SŽ M, ␦ . ; V. Now we have F Ž SŽ M, ␦ ., w T, ⬁.. ; SŽ M, ␧ .. Therefore M is eventually Lyapunov stable. As a direct consequence of Remark 3.6 and Theorem 3.13, we have THEOREM 3.14. Let M g CX. Assume that M is e¨ entually Lyapuno¨ stable. Then A uŽ M . s AŽ M . pro¨ ided that AŽ M . is open.

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Proof. Assume M g CX is eventually Lyapunov stable. Then by Theorem 3.13, A eq Ž M . s AŽ M .. Furthermore, if AŽ M . is open, one concludes immediately by Remark 3.6 that A uŽ M . s AŽ M .. COROLLARY 3.15. Assume that M g CX is asymptotically Ž e¨ entually . stable. Then it is uniformly asymptotically Ž e¨ entually . stable with A uŽ M . s AŽ M .. COROLLARY 3.16. Assume that M g CX. Then M is asymptotically Ž e¨ entually . stable iff it is uniformly asymptotically Ž e¨ entually . stable.

4. EXISTENCE OF UNIFORM ATTRACTORS In Section 2, we established a result showing negative invariance for the ␻-limit set ␻ Ž A. of a subset A of X. We do not know whether ␻ Ž A. is invariant in the general case. ŽTo guarantee positive invariance for ␻ Ž A., we imposed on F the Property K assumption in Section 2.. Inspired by some ideas in one of our recent works w14x, in this section we prove that if a compact set attracts itself, then its ␻-limit set is invariant. Based on this fact and some basic knowledge in previous sections, we establish some existence results for uniform attractors. Let F be a GDS defined on a complete locally compact metric space X. We have PROPOSITION 4.1. ant; i.e.,

Assume M g CX attracts itself. Then ␻ Ž M . is in¨ ariFŽ ␻Ž M . , t. s ␻Ž M .

᭙ t G 0.

To prove Proposition 4.1, we need the following lemma. LEMMA 4.2. in¨ ariant.

Assume that V ; X is negati¨ ely in¨ ariant. Then F Ž V . is

Proof. For any t G 0, we observe that F Ž F Ž V . , t . s F Ž F Ž V , Rq . , t . s F Ž V , t , ⬁ . . ; F Ž V . . Therefore F Ž V . is positively invariant. In the following, we prove that F Ž V . is also negatively invariant. Let t G 0 be fixed. We need to show that F Ž V . ; F Ž F Ž V ., t .. Assume y g F Ž V .. Let x g V and s g Rq be such that y g F Ž x, s .. Two cases may occur. Case 1. s F t. Since V is negatively invariant, we can take a x 0 g V such that x g F Ž x 0 , t y s .. Now we have y g F Ž x 0 , t .. Noting that V s F Ž V, 0. ; F Ž V ., we conclude that y g F Ž F Ž V ., t ..

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Case 2.

469

s ) t. When this occurs, we have y g F Ž x, s . s F Ž F Ž x, s y t . , t . .

Therefore we can find at least one x 0 g F Ž x, s y t . ; F Ž V . such that y g F Ž x 0 , t ., hence y g F Ž F Ž V ., t .. The proof is complete. Proof of Proposition 4.1. Since M g CX, we can take a ␧ ) 0 sufficiently small such that S Ž M, ␧ . is a compact set. Assume that M attracts itself. Then there exists T ) 0 such that F Ž M, w T, ⬁.. ; S Ž M, ␧ . . By Propositions 2.13 and 3.2, ␻ Ž M . is a nonempty compact negatively invariant set and attracts M. Note that ␻ Ž M . is necessarily contained in M. By Lemma 4.2, we see that F Ž ␻ Ž M .. is invariant. Observing F Ž F Ž ␻ Ž M . . , t . s F Ž ␻ Ž M . , t , ⬁ . . ; F Ž M, t , ⬁ . .

Ž 4.1.

for any t g Rq, we deduce that ␻ Ž M . attracts F Ž ␻ Ž M ... It follows that F Ž ␻ Ž M .. ; ␻ Ž M . Žby the invariance of F Ž ␻ Ž M ..., which implies the positive invariance of ␻ Ž M .. Hence ␻ Ž M . is invariant. We now introduce the concepts of uniform attractor and global uniform attractor, as in the case of dynamical systems. DEFINITION 4.3. A set A g CX is said to be a uniform attractor of F, if it is invariant and A uŽ A . is a neighborhood of A. If A is a uniform attractor with A uŽ A . s X, then we say that A is a global uniform attractor of F. Remark 4.4. If A is a uniform attractor, by Remark 3.6, we know that it attracts compact subsets of A uŽ A .. Take a ␧ ) 0 sufficiently small such that S Ž A , ␧ . ; A uŽ A .. Then A attracts S Ž A , ␧ . . Therefore A is eventually Lyapunov stable. Thanks to Proposition 3.9, we deduce that A is Lyapunov stable. As a consequence, we have PROPOSITION 4.5. asymptotically stable.

If A is a uniform attractor, then it is uniformly

In applications, asymptotically stable sets are of particular interest. In some situations, it is not difficult to find a compact subset of X which attracts a neighborhood of itself. We show that this implies the existence of a uniform attractor. We have THEOREM 4.6. Assume that there is a M g CX that attracts a neighborhood of itself. Then F has a uniform attractor A ; M with A uŽ A . s A uŽ M .. Proof. Take a ␦ ) 0 such that B [ S Ž M, ␦ . g CX and M attracts B. Invoking Proposition 4.1, ␻ Ž B . is invariant Žnote that B attracts itself..

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Clearly ␻ Ž B . ; M. Set A s ␻ Ž B .. Since A attracts B, it is a uniform attractor. It is easy to see that if M attracts V ; X, then A also attracts V, from which we conclude immediately that A uŽ A . s A uŽ M .. DEFINITION 4.7. F is said to be uniformly strongly dissipative on X, if there exists a M g CX such that A uŽ M . s X. As a particular case of Theorem 4.6, we have THEOREM 4.8. Assume that F is uniformly strongly dissipati¨ e on X. Then F has a global uniform attractor A.

5. A LYAPUNOV FUNCTIONAL DESCRIPTION OF ASYMPTOTIC STABILITY It is well known that Lyapunov functions constitute a classical and useful tool in stability analysis. In this section we characterize asymptotic stability by suitable versions of uniformly unbounded Lyapunov functions. Let X be a complete locally compact metric space as in the previous sections. DEFINITION 5.1. Let ⍀ be a subset of X. A function ␣ : ⍀ ª Rq is said to be uniformly unbounded Žor radially unbounded. on ⍀, if for any ␮ ) 0, there is a compact subset K ; ⍀ such that

␣ Ž x. ) ␮

᭙x g ⍀ _ K.

DEFINITION 5.2. Let ⍀ be an open subset of X, and let M be a compact subset of ⍀. A continuous function ␣ : ⍀ ª Rq is said to be a K0⬁ function of M on ⍀, if it satisfies Ž1. Ž2.

␣ Ž x . s 0 m x g M; ␣ is uniformly unbounded on ⍀.

PROPOSITION 5.3. Let ⍀ be an open subset of X, and let M be a compact subset of ⍀. Suppose ␣ is a K0⬁ function of M on ⍀. Then for ᭙␧ ) 0, there exists ␦ ) 0 such that

␣ Ž x. - ␧

᭙ x g S Ž M, ␦ . .

Proof. If not, there would exist a ␧ 0 ) 0 and a sequence x n g ⍀ with d Ž x n , M . ª 0 such that ␣ Ž x n . G ␧ 0 . Since ⍀ is open, we can find a r ) 0 sufficiently small such that S Ž M, r . is a compact subset of ⍀. Disregarding a finite number of terms, we may assume that x n g SŽ M, r .. Now one concludes that there is a subsequence x n i that converges to some point

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x 0 g M. By continuity of ␣ , we have ␣ Ž x 0 . G ␧ 0 ) 0, which yields a contradiction. In the remaining part of this section, we always assume that the following standing assumption holds. Standing Assumption 5.4. For any open subset ⍀ of X, there exists a countable family of compact subsets ⍀ n Ž n g N . of ⍀, where ⍀ n ; int ⍀ nq 1 for each n, such that ⍀ s Dng N ⍀ n . It is easy to check that X s R m satisfies Standing Assumption 5.4. LEMMA 5.5. Let ⍀ be an open subset of X. M ; ⍀ is a nonempty compact set. Then there exists a K0⬁ function ␣ of M on ⍀. Proof. Let ⍀ n Ž n g N . be a family of compact subsets of ⍀ as in Standing Assumption 5.4. Then we also have Dng N int ⍀ n s ⍀. It follows that for any compact subset K of ⍀, there exists a ⍀ n such that K ; int ⍀ n . In particular, M ; int ⍀ n for some n. Discarding a finite number of terms, we may assume that M ; int ⍀ 1. Take a r ) 0 such that ⍀ 0 [ S Ž M, r . is compact with ⍀ 0 ; int ⍀ 1. Let Z be the set of all integer numbers. For each k g Z, k - 0, define ⍀ k s S Ž M, rr Ž 2 < k < . . . Then we obtain a sequence ⍀ k Ž k g Z . of compact sets. Note that ⍀ k l ⭸ ⍀ kq1 s ⭋

᭙k g Z,

Ž 5.1.

where, and hereafter, ⭸ A denotes the boundary of a set A ; X. Invoking w19, Sect. 2.2, Theorem 1x, for every k g Z, there exists a continuous function ␴ k : ⍀ ª w0, 1x such that

␴k Ž x . s 0 Ž x g ⍀ k . ,

␴ k Ž x . s 1 Ž x g ⭸ ⍀ kq1 . .

Ž 5.2.

Take a sequence r k g Ž0, ⬁. Ž k g Z . with r k F r kq1 for ᭙k g Z,

lim r k s 0.

kªy⬁

Ž 5.3.

Let Hk s ⍀ k _ ⍀ ky1 for k g Z. Since ⍀ ky1 ; int ⍀ k , one easily sees that ⭸ Hk s ⭸ ⍀ k . Now we define ␣ : ⍀ ª Rq as

¡0

␣ Ž x. s

~

r ky 1 q ␴ ky1 Ž x . Ž r k y r ky1 .

when x g M ; when x g Hk with k F 0;

Ž rky 1 q k y 1 . q ␴ ky1 Ž x . Ž rk y rky1 q 1 .

¢

when x g Hk with k ) 0.

We first check the continuity of ␣ . Indeed, if x g Hk with k F 0, then by the definition of ␣ , we see that

␣ Ž x . s r ky 1 q ␴ ky1 Ž x . Ž r k y r ky1 . F r ky1 q Ž r k y r ky1 . s r k .

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By Ž5.3., we deduce that lim k ªy⬁ sup x g ⍀ k ␣ Ž x . s 0, which implies directly that ␣ is continuous at any point x g M. Now consider x g ⍀ _ M. Let k g Z be such that x g Hk . If x g int Hk , then by the definition of ␣ , it is clear that ␣ is continuous at x. Thus we assume that x g ⭸ Hk s ⭸ ⍀ k . Therefore by Ž5.2., lim yg⍀ k , yªx

␴ ky 1 Ž y . s 1,

lim yg⍀ kq1 , yªx

␴ k Ž y . s 0.

If k ) 0, we have lim ygH k , yªx

s

␣ Ž y. lim

yg⍀ k , yªx

lim ygH kq1 , yªx

s

Ž rky 1 q k y 1 . q ␴ ky1 Ž y . Ž rk y rky1 q 1 . s rk q k,

␣ Ž y. lim

ygH kq1 , yªx

Ž rk q k . q ␴ k Ž y . Ž rkq1 y rk q 1 . s rk q k.

One concludes from the above two equations immediately that ␣ is continuous at x. A similar argument applies to show the continuity of ␣ at x in the case where k F 0. Now we check that ␣ satisfies the other conditions in Definition 5.2 and hence is a K0⬁ function of M on ⍀. If x g ⍀ _ M, say, for instance, x g Hk , then by the definition of ␣ , ␣ Ž x . G r ky1 ) 0. Consequently, x g M iff ␣ Ž x . s 0. Since for any k ) 0 and x g ⍀ _ ⍀ k , we have ␣ Ž x . G r k q k, one sees that ␣ is uniformly unbounded on ⍀. The proof of the lemma is complete. We recall that a function L: ⍀ ; X ª R1 is said to be upper semicontinuous, if for any x g ⍀, lim sup L Ž y . F L Ž x . . yg⍀ , y/x yªx

Let F be a GDS on X, and let ⍀ be an open subset of X. We say that F is uniformly strongly bounded on ⍀, if for any x g ⍀, there exist a neighborhood V ; ⍀ of x and a compact subset W of ⍀ such that F Ž V . ; W. One easily checks that F is uniformly strongly bounded on an open set ⍀ ; X, iff for any compact subset V of ⍀, there exists a compact subset W of ⍀ such that F Ž V . ; W. The main result in this section is the following theorem.

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THEOREM 5.6. Let M g CX. Then M is asymptotically stable iff there exist an open neighborhood ⍀ of M and a uniformly unbounded upper semicontinuous function L: ⍀ ª Rq, such that Ž1. Ž2.

LŽ x . s 0 for any x g M. If x g ⍀ _ M, then for any t ) 0, LŽ x . ) LŽ y .

᭙ y g F Ž x, t . .

Ž 5.4.

Proof. We first prove the ‘‘only if’’ part. Suppose that M is asymptotically stable; then AŽ M . is a positively invariant open subset of X. Let ⍀ s AŽ M .. We show that there is a uniformly unbounded upper semicontinuous function L: ⍀ ª Rq satisfying Ž1. and Ž2. in the theorem. By Remark 3.6 and Theorem 3.14, we know that M attracts each compact subset of ⍀. We claim that F is uniformly strongly bounded on ⍀. Indeed, let V be a compact subset of ⍀; then M attracts V. Take a ␧ ) 0 sufficiently small so that S Ž M, ␧ . ; ⍀ is compact. Now there is a T ) 0 such that F Ž V, w T, ⬁.. ; SŽ M, ␧ .. Set W s F Ž V, w0, T x. j S Ž M, ␧ . . Then W ; ⍀ is compact and F Ž V . ; W. Hence the claim is true. By virtue of Lemma 5.5, there is a K0⬁ function ␣ of M on ⍀. Define ␺ : ⍀ = Rqª Rq as

␺ Ž x, t . s sup  ␣ Ž y . : y g F Ž x, t , ⬁ . . 4

᭙ Ž x, t . g ⍀ = Rq.

By uniform strong boundedness of F, we find that ␺ is well defined. We have the following basic facts: LEMMA 5.7. that

Let x 0 g ⍀. Then for any ␧ ) 0, there exist ␦ , r ) 0 such

< ␺ Ž x 0 , t . y ␺ Ž x 0 , t⬘ . < - ␧

␺ Ž x, t . - ␺ Ž x 0 , t . q ␧

᭙ t , t⬘ g Rq, < t y t⬘ < - ␦ ; ᭙ x g S Ž x 0 , r . , ᭙ t g Rq.

Ž 5.5. Ž 5.6.

Proof. Let ␧ ) 0 be given arbitrarily. By Proposition 5.3, there exists ␩ ) 0 such that

␣ Ž x . - ␧r3

᭙ x g S Ž M, ␩ . .

Ž 5.7.

Take a r 0 ) 0 sufficiently small so that V0 s S Ž x 0 , r 0 . ; ⍀ is compact. Since M attracts V0 , we can find a ␶ ) 0 such that F Ž V0 , ␶ , ⬁ . . ; S Ž M, ␩ . .

Ž 5.8.

Set T s ␶ q 1. We claim that

␺ Ž x, t . y sup  ␣ Ž y . : y g F Ž x, w t , T x . 4 - ␧r3

᭙ x g V0 . Ž 5.9.

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Indeed, if ␺ Ž x, t . s sup ␣ Ž y . : y g F Ž x, w t, T x.4 , then we are done; otherwise, we have 0 F sup  ␣ Ž y . : y g F Ž x, w t , T x . 4 - sup  ␣ Ž y . : y g F Ž x, Ž T , ⬁ . . 4 s ␺ Ž x, t . - ␧r3

Ž by Ž 5.7. ᎐ Ž 5.8. . ,

which shows that Ž5.9. still holds. Set D s F Ž V0 , w0, T x.. Then D is a compact subset of ⍀. Noting that ␣ is uniformly continuous on D, we can pick a ␳ ) 0 such that < ␣ Ž x . y ␣ Ž y . < - ␧r3

d Ž x, y . F ␳ .

᭙ x, y g D,

Ž 5.10.

Because F Ž x 0 , t . is continuous in t Žin the sense of Hausdorff distance., one can easily verify that it is uniformly continuous in t on w0, T x. Hence there exists 0 - ␦ F 1 such that for any t, t⬘ g w0, T x with < t y t⬘ < - ␦ ,

␦ H Ž F Ž x 0 , t . , F Ž x 0 , t⬘ . . - ␳ ; consequently Žfor convenience, we assume t F t⬘.,

␦ H Ž F Ž x 0 , w t , T x . , F Ž x 0 , w t⬘, T x . . s d H Ž F Ž x 0 , w t , T x . , F Ž x 0 , w t⬘, T x . . s d H Ž F Ž x 0 , w t , t⬘ . . , F Ž x 0 , w t⬘, T x . . F d H Ž F Ž x 0 , w t , t⬘ . . , F Ž x 0 , t⬘ . . F ␳ . Furthermore, by Ž5.10., we easily deduce that for any t, t⬘ g w0, T x with < t y t⬘ < - ␦ , sup  ␣ Ž y . : y g F Ž x 0 , w t , T x . 4 y sup  ␣ Ž y . : y g F Ž x 0 , w t⬘, T x . 4 F ␧r3. Combining this with Ž5.9., one concludes immediately that < ␺ Ž x 0 , t . y ␺ Ž x 0 , t⬘ . < - ␧

᭙ t , t⬘ g w 0, T x , < t y t⬘ < - ␦ . Ž 5.11.

Now we assume that t, t⬘ g Rq, < t y t⬘ < - ␦ . For convenience, we assume that t F t⬘. Recall that ␦ F 1. If t⬘ ) T, then t ) t⬘ y ␦ G T y ␦ G T y 1 s ␶ . By Ž5.7. and Ž5.8., we deduce that < ␺ Ž x 0 , t . y ␺ Ž x 0 , t⬘ . < F ␺ Ž x 0 , t . q ␺ Ž x 0 , t⬘ . - 23 ␧ . This, together with Ž5.11., completes the proof for Ž5.5.. Because F Ž x, t . is upper semicontinuous in x, uniformly in t in any compact interval Žsee Definition 2.1., we find that there exists 0 - r F r 0 such that dH Ž F Ž S Ž x0 , r . , t . , F Ž x0 , t . . - ␳

᭙ t g w 0, T x .

Ž 5.12.

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Let x g S Ž x 0 , r .. If t g w0, T x, then by Ž5.9., Ž5.10., and Ž5.12., we have

␺ Ž x, t . F sup  ␣ Ž y . : y g F Ž x, w t , T x . 4 q ␧r3 F sup  ␣ Ž y . : y g F Ž x 0 , w t , T x . 4 q 23 ␧ F ␺ Ž x 0 , t . q ␧ . On the other hand, for t ) T [ ␶ q 1, by Ž5.7. and Ž5.8., we have

␺ Ž x, t . s sup  ␣ Ž y . : y g F Ž x, t , ⬁ . . 4 F ␧r3 - ␧ . Hence Ž5.6. holds. We now continue to prove Theorem 5.6. It is clear that ␺ is decreasing in t. Since M is positively invariant, we have

␺ Ž x, t . s 0

᭙ x g M, ᭙ t g Rq.

Ž 5.13.

Note that Ž5.5. implies that ␺ is continuous in t. Define VŽ x. s

⬁

yt

H0 ␺ Ž x, t . e

dt.

Then V is well defined. Thanks to Ž5.6., one can easily check that V is upper semicontinuous. Due to Ž5.13., we see that if x g M, then V Ž x . s 0. If x g ⍀ _ M, then since ␺ is continuous in t and

␺ Ž x, 0 . G ␣ Ž x . ) 0,

Ž 5.14.

we conclude that V Ž x . ) 0. In conclusion, V Ž x . s 0 m x g M. Now assume that x g ⍀ _ M and t ) 0. We show that V Ž y . - V Ž x . if y g F Ž x, t .. We first observe that if y g F Ž x, ␶ ., then for any t g Rq, F Ž y, t , ⬁ . . ; F Ž F Ž x, ␶ . , t , ⬁ . . s F Ž x, t q ␶ , ⬁ . . . It follows that ␺ Ž y, t . F ␺ Ž x, t q ␶ . and hence

␺ Ž y, t . F ␺ Ž x, t q ␶ . F ␺ Ž x, t .

᭙ t G 0.

Ž 5.15.

We now argue by contradiction and assume that there exist a ␶ ) 0 and y g F Ž x, ␶ . such that V Ž y . s V Ž x .. By the definition of V and Ž5.15. Žas well as the continuity of ␺ in it., we must have

␺ Ž y, t . s ␺ Ž x, t q ␶ . s ␺ Ž x, t .

᭙ t G 0;

in particular, 0 - Ž by Ž 5.14. . - ␧ 0 [ ␺ Ž x, 0 . s ␺ Ž x, ␶ . s ⭈⭈⭈ s ␺ Ž x, n␶ . s ⭈⭈⭈ .

Ž 5.16.

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On the other hand, by Proposition 5.3, we can take a ␩ ) 0 such that

␣ Ž x . - ␧ 0r2

᭙ x g S Ž M, ␩ . .

Since M attracts x, there is a T ) 0 such that F Ž x, w T, ⬁.. ; SŽ M, ␩ .. It follows that

␺ Ž x, n␶ . [ sup  ␣ Ž z . : z g F Ž x, w n␶ , ⬁ . 4 F ␧ 0r2 for n sufficiently large. This contradicts Ž5.16.. Let ␾ Ž x . s ␺ Ž x, 0.. By the first relation in Ž5.14., one sees that ␾ is uniformly unbounded on ⍀. By Ž5.13., ␾ Ž x . s 0 for any x g M. If y g F Ž x, t ., then F Ž y, w0, ⬁.. ; F Ž x, w t, ⬁.. and, by consequence, ␾ Ž y . F ␾ Ž x .. Define a function L as LŽ x . s ␾ Ž x . q V Ž x .

᭙ x g ⍀.

Then L has all the required properties in the theorem. The proof of ‘‘if ’’ part. Assume that there exist an open neighborhood ⍀ of M and a uniformly unbounded upper semicontinuous function L: ⍀ ª Rq satisfying Ž1. and Ž2.. Let V be any compact subset of ⍀. Due to the upper semicontinuity of L, we deduce that L is bounded on V and hence there is a B ) 0 such that LŽ x . F B for any x g V. Since L is uniformly unbounded, there exists a compact subset W of ⍀ such that LŽ x . ) B for any x g ⍀ _ W. By Ž2., ⍀ B [  x g ⍀ : LŽ x . F B4 ; W and is positively invariant. Since V ; ⍀ B , we have F Ž V . ; ⍀ B ; W. By virtue of Proposition 3.2, we know that ␻ Ž V . attracts V. We show that ␻ Ž V . ; M, and thus M attracts V. Suppose not. Then due to the upper semicontinuity of L, there exists a x 0 g ␻ Ž V . _ M such that L Ž x 0 . s max L Ž x . ) 0. xg ␻ Ž V .

Ž 5.17.

Since ␻ Ž V . is negatively invariant, by Proposition 2.9, there is a trajectory ␥ on wy1, 0x which lies in ␻ Ž V . such that ␥ Ž0. s x 0 . Because ␥ is continuous, we see that ␥ Ž t . f M for t - 0 sufficiently small. Take a t 0 - 0 such that ␥ Ž t 0 . f M. By Ž2., we have LŽ␥ Ž t 0 .. ) LŽ x 0 ., as x 0 g F Ž␥ Ž t 0 ... This contradicts Ž5.17.. We also show that M is positively invariant. Indeed, if this is not the case, then there are x g M and t g Rq such that y f M for some y g F Ž x, t .. Thanks to Theorem 2.6, there is a trajectory ␥ of F on w0, t x such that ␥ Ž0. s x, ␥ Ž t . s y. Let t 0 s inf  ␶ : ␥ Ž s . f M for any ␶ - s F t 4 .

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Clearly ␥ Ž t 0 . g M. By Ž1., we have LŽ y . ) 0. By Ž2., we find that LŽ␥ Ž s .. G LŽ y . for any s g Ž t 0 , t x. Furthermore, by upper semicontinuity of L, we deduce that LŽ␥ Ž t 0 .. G LŽ y . ) 0, which leads to a contradiction. Since M attracts each compact subset of ⍀, we know that M attracts itself and thus it is eventually Lyapunov stable. Thanks to Proposition 3.9, we conclude that M is Lyapunov stable. This completes the proof of our theorem. Remark 5.8. Some further discussions on GDSs and their applications to differential inclusions will be reported in a forthcoming paper.

ACKNOWLEDGMENTS The author is indebted to the referee for invaluable comments and suggestions which greatly improved the quality of the paper. He also expresses his gratitude to Professors Huan Zhongdan and Huang Haiyang for helpful discussions.

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