Semidynamical Systems with Nonunique Global Backward Extensions II

Funkcialaj Ekvacioj, 27 (1984), 85-100

Semidynamical Systems with Nonunique Global Backward Extensions II; the Negative Aspects By

Saber ELAYDI (Kuwait University, Kuwait) §1.

Introduction.

In [5] the author considered the class of semidynamical systems with nonunique global backward extensions. For a semidynamical system in this class, all negative and through each point in the phase space there exists at solutions are principal least one principal negative solution that passes through it. Furthermore, in such systems there are no start points [1] and the negative escape time (McCann’s sense a for each point in the phase space of the system [5]. Furthermore, such [9] is systems arise naturally from normal control systems [7], [5]. In this paper we continue the study started in [5]. We deal here with the negative version of some important dynamical properties, such as stability, attraction, invariance and asymptotic stability. Following Roxin [10] and Kloeden [8], a weak version and a strong version are defined for each dynamical property. The main purpose of this paper is to carry over, as much as possible, basic results from the theory of dynamical systems [2] to the class of semidynamical systems with nonunique global backward extensions. Differences between the two classes will be demonstrated through the examples of section 6. We have been influenced by the work in [10] and [8] in constructing our examples and in introducing definitions and ideas. Let $(X, ¥pi)$ be a semidynamical system [1]. It will be assumed throughout this paper that (i) $X$ is locally compact, Hausdorff and first countable and (ii) $(X, ¥pi)$ has a global backward extension. A function : ( ¥ , is called a principal negative solution of $(X, ¥pi)$ if $¥sigma(-t+s)=¥sigma(-t)¥pi s$ whenever and in $R^{+}(R^{+}$ denotes the set of nonnegative reals) and . The negative solution is said to pass $a ¥ in X$ $ ¥ sigma(0)=a$ through a point if . The -negative trajectory through a is defined to be the set . The following sets are of special interest. (1) The funnel through $a¥in X[1]$ ; $¥mathrm{H}¥exists$

$)$

$¥infty$

$¥sigma$

$- infty$

$¥mathrm{O}]¥rightarrow X$

$s$

$t¥geq s$

$¥sigma$

$C_{¥sigma}^{-}(a)$

$¥sigma$

$¥sigma(-¥infty,$

$0]$

{

$F(a)=$ $b|b¥pi t=a$

$t$

for some

(2) The attainability set of $a¥in X$ at

$t¥in R^{+}$

$-t_{0}¥in R^{-};$

}.

86

S. ELAYDI $F(a, -t_{¥mathrm{Q}})=¥{b|b¥pi t_{0}=a¥}[6]$

(3) The attainability set of A

$¥subset X$

and

, [8].

$S¥subset R^{-};$

$F(¥mathrm{A}, S)=¥{b|b¥pi s=a, a¥in ¥mathrm{A}, -s¥in S¥}$

.

A point $a¥in X$ is said to be a start point if $F(a)=¥phi[1]$ . A subset A of $X$ is said to be {positively invariant} {weakly negatively invariant} {strongly negatively invariant} if, respectively, , , for some negative solution {for each through } . In section 2 we study weak and strong versions of the negative limit sets, the negative prolongation sets and the negative prolongation limit sets. Part of these notions were introduced in [3] and [4]. In section 3, negative strong stabilities are investigated. Three kinds of such stabilities are introduced; (1) negative strong Liapunov stability (2) strong -stability and (3) strong -stability, where and denotes respectively, the negative prolongation relation and the negative prolongation limit relation. We also introduce the notions of charactristic and stable characteristic . In section 4 negative weak stabilities are considered. Results similar to those in section 3 are obtained. In section 5 the notions of negative attraction and negative asymptotic stability are defined and studied. In section 6 we give several examples to illustrate some of the differences between dynamical systems and namical systems. will denote the closure of A and A will denote the For a subset A of boundary of A. For any other unexplained notation we refer the reader to [1]. $a¥in ¥mathrm{A}$

$¥{¥mathrm{A}¥pi R^{+}¥subset ¥mathrm{A}¥}$

$a$

$C_{¥sigma}^{-}(a)¥subset ¥mathrm{A}$

$¥{F(¥mathrm{A})¥subset ¥mathrm{A}¥}$

$¥mathrm{D}^{-}$

$D^{-}$

$¥mathrm{J}^{-}$

$J^{-}$

$O^{-}$

$O^{-}$

$¥mathrm{s}¥mathrm{e}¥mathrm{m}¥mathrm{i}¥mathrm{d}_{¥mathrm{d}}¥mathrm{v}$

$X,¥overline{A}$

§2.

$¥partial$

Negative sets.

Definition 2.1. Let $a¥in X$ and let be a principal solution through . Then we have the following definitions. through ; (i) The -negative . ] and its closure -negative limit set of ; (ii) The for some sequence with in { }. (iii) The negative limit set of ; $L^{-}(a)=¥{b¥in X|$ there exist sequences in with and in $X$ with $b_{i}¥in F(a, -t_{i})$ and }. (iv) The negative prolongation set of ; $D^{-}(a)=¥{b¥in X|$ there exist a sequence of principal negative solutions with and a sequence with in . $¥sigma$

$C_{¥sigma}^{-}(a)=¥sigma(-¥infty,$

$0$

$¥overline{C_{¥sigma}^{-}(a)}=K_{¥sigma}^{-}(a)$

$a$

$¥sigma$

$L_{¥sigma}^{-}(a)=$

$a$

$¥mathrm{t}¥mathrm{r}¥mathrm{a}_{¥mathrm{J}}^{¥dot{¥tau}}¥mathrm{e}¥mathrm{c}¥mathrm{t}¥mathrm{o}¥mathrm{r}¥mathrm{y}$

$¥sigma$

$a$

$¥mathrm{n}¥mathrm{e}_{¥backslash }¥sigma.¥mathrm{a}¥mathrm{t}¥mathrm{i}¥mathrm{v}¥mathrm{e}$

$b¥in X|¥sigma(-t_{i})¥rightarrow b$

$¥{t_{i}¥}$

$R^{+}$

$ t_{i}¥rightarrow¥infty$

$a$

$¥{t_{i}¥}$

$R^{+}$

$ t_{i}¥rightarrow¥infty$

$¥{b_{i}¥}$

$b_{i}¥rightarrow b$

$a$

$¥{¥sigma_{i}¥}$

$¥sigma_{i}(0)¥rightarrow a$

$¥{t_{i}¥}$

$R^{+}$

$¥sigma_{i}(-t_{i})¥rightarrow b¥}$

Sem idynamical Systems

87

(v) The negative prolongation limit set of ; $a$

there exist a sequence of principal negative solutions and a sequence in with and .

$J^{-}(a)=¥{b¥in X|$

with

(vi)

$¥sigma_{i}(0)¥rightarrow a$

$R^{+}$

$¥{t_{i}¥}$

$¥{¥sigma_{i}¥}$

$¥sigma_{i}(-t_{i})¥rightarrow b$

$ t_{i}¥rightarrow¥infty$

$K^{-}(a)=¥overline{F(a)}$

The following two results from [5] are employed throughout this paper and we state them here for the convenience of the reader. Theorem 2.2 [5; 1.8, 1.10]. Let be a semidvnamical that $alI$ $X$ negative sofutions in are principal and there is at least one principal negative solution passing through each point in X. Then the following hold: is compact whenever A (i) The attainability set and are compact in their respective spaces. (ii) The attainability set $F(¥mathrm{A}, [-t, 0])$ is connected whenever A is compact and connected and ¥ . ¥ , (where $X^{*}=$ (iii) $(X, ¥pi)$ can be extended to the semidynamical system $X$ is the one point compactification of and $¥pi^{*}(x, t)=¥pi(x, t)$ for , $(¥mathrm{X} ¥pi)$

$syste.¥prime ns¥iota¥iota ch$

$tf¥iota e$

$F(¥mathrm{A}, P)$

$¥subset X$

$P¥subset R^{-}$

$t in R^{+}$

$(X^{*},

$ X¥cup$

pi^{*})$

$¥{¥infty¥}$

$ x¥neq¥infty$

$¥pi^{*}(¥infty, t)=¥infty$

for

$t¥in R^{+}$

).

Theorem 2.3 [5, 4.2]. Let $(X, ¥pi)$ be a semidynamical system that have the same properties mentioned in Theorem 2.2. Let be a sequence of principal negative solutions for which . Then there exist a negative solution with $¥sigma(0)=a$ and a subsequence such that . of for $aff$ ¥ $¥{¥sigma_{i}¥}$

$¥sigma_{i}(0)¥rightarrow a¥in X$

$¥{¥sigma_{i_{j}}¥}$

$¥sigma$

$t in R^{+}$

$¥sigma_{i_{j}}(-t)¥rightarrow¥sigma(-t)$

$¥{¥sigma_{i}¥}$

The following statements are valid for a point (i) $L^{+}(a¥pi t)=L^{+}(a)¥pi t=L^{+}(a)$ .

Lemma 2.4.

(ii) (iii)

$D^{+}(a)¥pi t¥subset D^{+}(a);D^{+}(a)¥pi t¥subset D^{+}(a¥pi t)$ $J^{+}(a)¥pi t=J^{+}(a)¥subset J^{+}(a¥pi t)$

$a¥in X$

and

$t¥in R^{+}$

.

.

.

for some sequence with in . Let A $=¥{a¥pi(t+t_{i})|i=1,2, --¥}¥cup¥{b¥}$ . Then A is compact. This implies by , Theorem 2.2 that ¥ is compact. Since the sequence is in ¥ we may assume, without loss of generality, that . Hence $c¥in L^{+}(a)$ . Further¥ ¥ ¥ ¥ more, ¥ and thus $c¥pi t=b$ . Consequently, $b¥in L^{+}(a)¥pi t$ and ¥ ¥ ¥ ¥ ¥ . From [1] it follows that hence . Now let $L^{+}(a)$ . with Then . There exists a for some sequence on ¥ ¥ positive integer such that $t_{i}>t$ for all . Thus ¥ ¥ and . This implies that $b¥in L^{+}(a¥pi t)$ and hence $L^{+}(a)¥subset L^{+}(a¥pi t)$ . Consequently, $L^{+}(a¥pi t)=L^{+}(a)¥pi t=L^{+}(a)$ . (ii) The proof of the first part can be found in [1]. The proof of the second part is easy and will be omitted.

Proof. Let

$b¥in L^{+}(a¥pi t)$

.

Then

$(a¥pi t)¥pi t_{i}¥rightarrow b$

$¥{t_{i}¥}$

$R^{+}$

$ t_{i}¥rightarrow¥infty$

$F( mathrm{A}, -t)$

$¥{a¥pi t_{i}¥}$

$F( mathrm{A}, -t)$

$a¥pi t_{i}¥rightarrow c$

$a pi(t_{i}+t)=(a pi t) pi t_{i} rightarrow c pi t$

$L^{+}(a) pi t subset L^{+}(a)$

$L^{+}(a pi t) subset L^{+}(a) pi t$ $a¥pi t_{i}¥rightarrow b$

$i_{0}$

$(t_{i}-t)¥rightarrow¥infty$

$¥{t_{i}¥}$

$i¥geq i_{0}$

$R^{+}$

$ t_{i}¥rightarrow¥infty$

$(a pi t) pi(t_{i}-t)=a pi t_{i} rightarrow b$

$ b¥in$

S. ELAYDI

$¥mathfrak{Z}8$

So let $b¥in J^{+}(a)$ . Then there are sequences in $X$ and with such that in . Since and the set A $=¥{a_{i}¥pi t_{i}|i=1,2, --¥}¥cup¥{b¥}$ is compact, it follows from Theorem 2.2 that ¥ is compact. There exists a positive integer such that $t_{i}>t$ for all . ¥ ¥ ¥ ¥ ¥ is in Hence for . Then we may assume that . ¥ ¥ Hence $c¥pi t=b$ and consequently, $b¥in J^{+}(a)¥pi t$ . Therefore and thus $J^{+}(a)=J^{+}(a)¥pi t$ . Furthermore, and $a_{i}¥pi t_{i}=(a_{i}¥pi t)¥pi(t_{i}-t)b$ for . Hence ¥ ¥ and thus . This completes the proof of the lemma.

(iii) According to [1] $¥{a_{i}¥}$

$J^{+}(a)¥pi t¥subset J^{+}(a)$ $R^{+}$

$¥{t_{i}¥}$

.

$ t_{i}¥rightarrow¥infty$

$a_{i}¥pi t_{i}¥rightarrow b$

$a_{i}¥rightarrow a$

$F( mathrm{A}, -t)$

$i¥geq i_{0}$

$i_{0}$

$i geq i_{0} {a_{i} pi(t_{i}-t) }$

$F( mathrm{A}, -t)$

$a_{i}¥pi(t_{i}-t)¥rightarrow c$

$J^{+}(a) subset J^{+}(a) pi t$

$a_{i}¥pi t¥rightarrow a¥pi t$

$b$

$¥in J^{+}(a¥pi t)$

$i¥geq i_{0}$

$J^{+}(a) subset J^{+}(a pi t)$

Remark 2.5. Let $a¥in X$ and let be a negative solution through . Let . Then $b¥pi s=a$ for some ¥ . An associated negative solution through can be defined by letting $¥sigma_{b}(-t)=¥sigma(-t-s)$ for . Now suppose that $b¥in C^{+}(a)$ . $a ¥ pi s=b$ ¥ Then . An associated negative solution for some through can be defined by letting $a$

$¥sigma$

$b$

$s geq 0$

$¥in C_{¥sigma}^{-}(a)$

$¥sigma_{b}$

$b$

$t¥geq 0$

$s geq 0$

$¥sigma_{b}(-t)=¥left¥{¥begin{array}{l}a¥pi(s-t)¥¥¥sigma(-t+s)¥end{array}¥right.$

In either case the sets , and and , , respectively. $C_{¥sigma}^{-}(b)$

$C_{¥sigma}^{-}(b)$

$K_{¥sigma}^{-}(b)$

$b$

$¥sigma_{b}$

$K_{¥sigma}^{-}b(b)$

$¥mathrm{i}¥mathrm{i}¥mathrm{f}¥mathrm{f}0t¥geq s¥leq t.¥leq s$

$L_{¥sigma}^{-}b(b)$

will be denoted, for simplicity, by

$L_{¥sigma}^{-}(b)$

Lemma 2.6. Let $a¥in X$ and be a principal negative solution through the following statements are valid. (i) for ¥ , (ii) $b¥in D^{-}(a)ffa¥in D^{+}(b):b¥in J^{-}(a)$ iff $a¥in J^{+}(b)$ , (iii) $D^{-}(a)¥subset D^{-}(a)¥pi t¥subset D^{-}(a¥pi t)$ , (iv) $J^{-}(a)¥pi t=J^{-}(a)¥subset J^{-}(a¥pi t)$ , $¥sigma$

$L_{¥sigma}^{-}(a¥pi t)=L_{¥sigma}^{-}(a)¥pi t=L_{¥sigma}^{-}(a)=L_{¥sigma}^{-}(¥sigma(-t))$

(v)

$L^{-}(a)=L^{-}(a)¥pi t¥subset L^{-}(a¥pi t)$

$a$

Then

.

$t geq 0$

.

Then there exists a sequence in with , and such that . Using Theorem 2.2 we may assume that $c ¥ pi t=b$ . Then and . Hence and consequently, . According to [4] . Now let . Then for some sequence in with . From Remark 2.5 we have and thus . Therefore . The last statement in (i) may be verified similarly. of principal negative solu(ii) Let $b¥in D^{-}(a)$ . Then there are sequences ¥ tions and in with and . Since ¥ , it follows that $b¥in D^{+}(a)$ . Conversely, let $a¥in D^{+}(b)$ . Then there are sequences in $X$ and with in . For each there exists and a principal negative solution through . According to Remark 2.5, for each there exists an associated principal negative solution through ; that is

Proo].. (i) Let

$t_{i}¥leq t$

$b¥in L_{¥sigma}^{-}(a¥pi t)$

.

$R^{+}$

$¥{t_{i}¥}$

$¥sigma(-t_{i}+t)¥rightarrow b$

$ t_{i}¥rightarrow¥infty$

$¥sigma(-t_{i})¥rightarrow c¥in X$

$c¥in L_{¥sigma}^{-}(a)$

$L_{¥sigma}^{-}(a¥pi t)¥subset L_{¥sigma}^{-}(a)¥pi t$

$b¥in L_{¥sigma}^{-}(a)¥pi t$

$L_{¥sigma}^{-}(a)¥pi t¥subset L_{¥sigma}^{-}(a)$

$¥sigma(-t_{i})¥rightarrow d$

$¥{t_{i}¥}$

$¥sigma(-t_{i})=¥sigma_{a¥pi t}(-t_{i}-t)¥rightarrow d$

$R^{+}$

$d¥in L_{¥sigma}^{-}(a)$

$ t_{i}¥rightarrow¥infty$

$d¥in L_{¥sigma}^{-}(a¥pi t)$

$L_{¥sigma}^{-}(a¥pi t)=L_{¥sigma}^{-}(a)¥pi t=$

$L_{¥sigma}^{-}(a)$

$¥{¥sigma_{i}¥}$

$¥{t_{i}¥}$

$R^{+}$

$¥sigma_{i}(-t_{i})¥rightarrow b$

$¥sigma_{i}(0)¥rightarrow a$

$ sigma_{i}(0)= sigma_{i}(-t_{i}+t_{i})=$

$¥sigma_{i}(-t_{i})¥pi t_{i}$

$¥{b_{i}¥}$

$R^{+}$

$¥{t_{i}¥}$

$¥sigma_{i}$

$b_{i}¥rightarrow b$

$b_{i}¥pi t_{i}¥rightarrow a$

$i$

$b_{i}$

$i$

$¥sigma_{i}$

$b_{i}¥pi t_{i}$

$¥sigma_{i}(0)=$

89

Semidynamical Systems

This implies that $b¥in D^{-}(a)$ The second part of (ii) can be and proved similarly. of principal negative solu(iii) Let $b¥in D^{-}(a)$ . Then there exist sequences . By Theorem 2.2 we may assume and with in tions and ¥ $c ¥ pi t=b$ . . Hence $c¥in D^{-}(a)$ and thus $b=c¥pi t$ ¥ , where that $c ¥ in X$ $d ¥ in D^{-}(a) ¥ pi t$ $D^{-}(a) ¥ subset D^{-}(a) ¥ pi t$ with . Then there exists . Now let Consequently, $c¥pi t=d$ and $c¥in D^{-}(a)$ . $a ¥ in D^{+}(c)$ by implies . This Then it follows from (ii) that $¥in D^{+}(c)¥pi t¥subset D^{+}(c¥pi t)=D^{+}(b)$ . Hence $b¥in D^{-}(a¥pi t)$ . Consequently, Lemma2.4 that $¥sigma_{i}(-t_{i})=b_{i}$

$b_{i}¥pi t_{i}$

$¥{¥sigma_{i}¥}$

$R^{+}$

$¥{t_{i}¥}$

$¥sigma_{i}(-t)¥rightarrow b$

$¥sigma_{i}(0)¥rightarrow a$

$ in D^{-}(a) pi t$

$¥sigma_{i}(-t_{i}-t)¥rightarrow c¥in X$

$a¥pi t$

$D^{-}(a)¥pi t¥subset D^{-}(a¥pi t¥grave{)}$

.

So let $b¥in J^{-}(a)$ . Then there such that with in of principal negative solutions and are sequences . . By Theorem 2.2 we may assume that a and $b=c ¥ pi t$ and consequently, Hence $c¥pi t=b$ . Furthermore, $c¥in J^{-}(a)$ . Thus $d ¥ in J^{-}(a)$ $J^{-}(a)=J^{-}(a) ¥ pi t$ . Then from (ii) . Now let . It follows that $ ¥ in J^{+}(d) ¥ pi t=J^{+}(d)$ $a ¥ in J^{+}(d)$ . Thus (Lemma 2.4). This implies it follows that . that $d¥in J^{-}(a¥pi t)$ . Consequently, and with in (v) Let $b¥in L^{-}(a)$ . Then there are sequences $X$ $b_{i} ¥ in F(a,-t_{i})$ . Let be a principal negative solution and such that in . may /. by 2.2 we assume that Then Theorem through for each ¥ , it follows that $c¥in L^{-}(a)$ . This implies that Thus $c¥pi t=b$ . Since ¥ $b=c¥pi t$ ¥ ¥ . Let $d¥in L^{-}(a)¥pi t$ . Then there exists and hence $e¥in L^{-}(a)$ with $e¥pi t=d$. with and in Furthermore, there are sequences $e_{i}¥in F(a, -t_{i})$ such that such that $t_{i}-t>0$ . There exists a positive integer . Hence , $e_{i}¥pi t¥in F(a, -t_{i}+t)¥subset F(a¥pi t, -t_{i})$ and . For for ¥ ¥ ¥ ¥ ¥ $d¥in L^{-}(a)¥cap L^{-}(a¥pi t)$ and consequently . It and $L^{-}(a)=L^{-}(a) ¥ pi t$ . The proof of the lemma is now complete. follows that

(iv) It is known from [4] that

$J^{-}(a)¥pi t¥subset J^{-}(a)$

.

$R^{+}$

$¥{t_{i}¥}$

$¥{¥sigma_{i}¥}$

$ t_{¥dot{x}}¥rightarrow¥infty$

$¥sigma_{i}(-t_{i}-t)¥rightarrow c¥in X$

$¥sigma_{i}(-t_{i})¥rightarrow b$

$¥sigma_{i}(0)¥rightarrow$

$¥in J^{-}(a)¥pi t$

$J^{-}(a)¥subset J^{-}(a)¥pi t$

$a¥pi t$

$J^{-}(a)¥subset J^{-}(a¥pi t)$

$¥{t_{i}¥}$

$b_{i}¥rightarrow b$

$R^{+}$

$¥{b_{i}¥}$

$ t_{i}¥rightarrow¥infty$

$¥sigma_{i}$

$¥sigma_{i}(-t)¥rightarrow c¥in X$

$b_{i}$

$ sigma_{i}(-t) subset F(a)$

$¥in L^{-}(a)¥pi t$

$L^{-}(a) subset L^{-}(a) pi t$

$R^{+}$

$¥{t_{i}¥}$

$i_{0}$

$e_{i}¥rightarrow e$

$e_{i}¥pi t¥rightarrow e¥pi t=d$

$i¥geq i_{0}$

$i¥geq i_{0}$

$ t_{i}¥rightarrow¥infty$

$L^{-}(a pi t) subset L^{-}(a)$

$L^{-}(a) pi t subset L^{-}(a pi t)$

Lemma 2.7. Le? hold: following the

$a¥in X$

and

$¥sigma$

be a principal negative solution through

, (i) (ii) $L^{-}(a)=¥bigcap_{t¥geq 0}¥{K^{-}(b)|b¥in F(a, -t)¥}$ (iii) $D^{-}(a)=¥cap¥overline{¥{F(U)}|U$ is an open neighborhood of }, (iv) there exists a principal negative solution through a $L_{¥sigma}^{-}(a)=¥bigcap_{s¥geq 0}$

$a$

.

Then

$¥{¥bigcup_{t¥geq 0}¥sigma(-s-t)¥}$

$a$

$su_{¥vee}^{n}fr$

$¥mathrm{v}$

$¥cap¥{D^{-}(b)|b¥in C_{¥nu}^{-}(a)¥}$

(v) (vi) (vii)

$J^{-}(a)=$

,

$K_{¥sigma}^{-}(a)=C_{¥sigma}^{-}(a)¥cup L_{¥sigma}^{-}(a)$

$D^{-}(a)=F(a)¥cup J^{-}(a)$

that

,

,

$K^{-}(a)=F(a)¥cup L^{-}(a)$ .

Proof. (i) Let

and such that that $t_{i}>s$ for

$b¥in L_{¥sigma}^{-}(a)$

$¥sigma(-t_{i})¥rightarrow b$

$i¥geq i_{0}$

.

. Let

Hence

.

with Then there exists a sequence in . Then there exists a positive integer $¥{t_{i}¥}$

$R^{+}$

$s¥geq 0$

$¥sigma(-t_{i})=¥sigma(-t_{i}+s-s)=¥sigma(-s-(t_{i}-s))¥rightarrow b$

.

$ t_{i}¥rightarrow¥infty$

.

$i_{0}¥mathrm{S}^{¥rceil}¥mathrm{J}^{¥prime ¥mathrm{h}}$

Thus

90

S. ELAYDI

$b¥in¥bigcup_{t¥geq 0}¥sigma(-s-t)$

.

Since

$s$

was arbitrary chosen,

$b¥in¥bigcap_{s¥geq 0}$

{

$¥bigcup_{t¥geq 0}$

(?s?t). Hence

Now let and let $U$ be an {¥ ¥ ¥ $s ¥ geq 0$ open neighborhood of . Then for , . Hence there exists $r=r(s)¥geq 0$ such that $¥sigma(-s-r)¥in U$. This implies that the net $¥{¥sigma(-t)|t¥in R^{+}¥}$ is frequently in $U$. Hence there exists a subnet with such that . Consequently, . The conclusion of part (i) now follows: (ii) Proof of (ii) is similar to the proof of (i) and will be omitted. of principal negative solu(iii) Let $b¥in D^{-}(a)$ . Then there are sequences with and tions and in . Let $U$ be an open neighborhood . Hence $¥sigma_{i}(-t_{i})¥in F(U)$ . Consequently, of . Then for . $ b¥in¥cap$ ¥ ¥ ¥ ¥ Thus is an neighborhood of }. It follows that { is an open neighborhood of }. Now let $c¥in¥cap¥{¥overline{F(U)}|U$ is an open neighborhood of }. Let $¥{U_{i}|i=1,2,3, --¥}$ and $¥mathrm{I}’¥{V_{i}|i=1,2,3, --¥}$ be local bases at and , ¥ ¥ ¥ respectively. Then for each , . Let $c_{i}¥in V_{i}¥cap F(U_{i})$ . Then . Furthermore, there exists with for some and for each /. Then clearly . There exist principal negative solutions such that and $b ¥ in D^{-}(a)$ This implies that . proof completes This the (Remark 2.5). of part (iii). of principal negative solutions (iv) Let $c¥in J^{-}(a)$ . Then there are sequences and in with such that and . According to Theorem2.3 there exists a principal negative solution passing through such that for all ¥ . Let $b¥in¥nu(-s)$ for some $s¥in R^{+}$ . Then ¥ ¥ ¥ $¥nu(-s)=b$ and for , where $t_{i}>s$ whenever . $J^{-}(a) ¥ subset ¥ cap ¥ {D^{-}(b)|$ $c ¥ in D^{-}(b)$ This implies, following Remark 2.5, that . Hence . Now let $c¥in¥cap¥{D^{-}(b)|b¥in C_{¥nu}^{-}(a)¥}$ . Then from Lemma 2.6, it follows that $b¥in D^{+}(c)$ for all . For each ¥ there exists $d=¥nu(-t)$ . Thus $a¥in¥cap¥{D^{+}(c)¥pi t|t¥in R^{+}¥}¥subset¥cap¥{D^{+}(c¥pi t)|t¥in R^{+}¥}=J^{+}(c)$. This implies by Lemma 2.6 $c ¥ in J^{-}(a)$ that . The conclusion of this part now follows. Parts (v), (vi) and (vii) are simple observations and their proofs will be omitted. $L_{¥sigma}^{-}(a)¥subset¥bigcap_{s¥geq 0}$

$¥{¥bigcup_{t¥geq 0}¥sigma(-s-t)¥}$

.

$b¥in¥bigcap_{s¥geq 0}$

$ bigcup_{t geq 0} sigma(-s-t)$

$b¥in¥overline{¥bigcup_{t¥geq 0}¥sigma(-s-t)}$

$b$

$¥sigma(-t_{i})$

$¥sigma(-t_{i})¥rightarrow b$

$ t_{i}¥rightarrow¥infty$

$b¥in L_{¥sigma}^{-}(a)$

$¥{¥sigma_{i}¥}$

$¥{t_{i}¥}$

$R^{+}$

$¥sigma_{i}(0)¥in U$

$a$

$¥sigma_{i}(-t_{i})¥rightarrow b$

$¥sigma_{i}(0)¥rightarrow a$

$b¥in¥overline{F(U)}$

$i¥geq i_{0}$

$¥overline{F(U)}|U$

$D^{-}(a) subset cap { overline{F(U)}|U$

$a$

$a$

$a$

$a$

$¥mathrm{f}$

$ V_{i} cap F(U_{i}) neq phi$

$a_{i}¥in U_{i}$

$c$

$c_{i}¥rightarrow c$

$t_{i}¥geq 0$

$c_{i}¥pi t_{i}=a_{i}$

$a_{i}¥rightarrow a$

$¥sigma_{i}(0)=a_{i}$

$¥sigma_{i}$

$¥sigma_{i}(-t_{i})=c_{i}$

$¥{¥sigma_{i}¥}$

$¥{t_{i}¥}$

$R^{+}$

$ t_{i}¥rightarrow¥infty$

$¥sigma_{i}(-t_{i})¥rightarrow c$

$¥sigma_{i}(0)¥rightarrow a$

$a$

$¥nu$

$¥sigma_{i}(-t)¥rightarrow¥nu(-t)$

$t in R^{+}$

$¥sigma_{i}(-s)¥rightarrow$

$ sigma_{i}(-s-(t_{i}-s))= sigma_{i}(-t_{i}) rightarrow c$

$i¥geq i_{0}$

$i¥geq i_{0}$

$b¥in C_{¥nu}^{-}(a)¥}$

$t in R^{+}$

$b¥in C_{¥mu}^{-}(a)$

Remark 2.8. (i) It is known from [3] and [4] that the sets and $J^{-}(x)$ are closed positively and weakly negatively invariant. The same properties hold for $L^{-}(x)$ . One can also remark from Lemma 2.7 that $D^{-}(a)$ is closed. Furthermore, it follows from Theorem 2.3 that $D^{-}(a)$ is weakly negatively invariant. However $D^{-}(a)$ is not positively invariant as it can be easily seen in the examples of section 6. (ii) We observe also that if a set $H$ is strongly negatively invariant, then is weakly negatively invariant. This is an immediate consequence of Theorem 2.3. This result cannot be strengthened as is shown in Example 6.3. $L_{¥sigma}^{-}(x)$

$¥overline{H}$

Lemma 2.9. If there are nets following statements are valid.

$¥{a_{i}¥}$

and

$¥{b_{i}¥}$

in

$X$

with

$a_{i}¥rightarrow a$

and

$b_{i}¥rightarrow b$

, then the

91

Semidynam ical Systems

(i) If (ii)

for each , then for each , then

$b_{i}¥in D^{-}(a_{i})$

$ffb_{i}¥in J^{-}(a_{i})$

$b¥in D^{-}(a)_{9}$

$i$

$i$

$b¥in J^{-}(a)$

.

The proof is easy and will be omitted.

Proof.

Lemma 2.10.

For each

Proof.

,

$x¥in X$

$ L^{-}(x)¥neq¥phi$

whenever

$ J^{-}(x)¥neq¥emptyset$

and

pact.

$co.m$

¥ ¥ Suppose that and compact and $ L^{-}(x)=¥phi$ for a point $x¥in X$. $ F(x) ¥ cap J^{-}(x) ¥neq¥phi$ , then it follows from Remark 2.8 that for is closed. If Then $y¥in F(x)¥cap J^{-}(x)$ , for some principal negative solution through . Since $J^{-}(x)$ is compact and , it follows that . This implies by Lemma 2.6 that . Consequently $ L^{-}(x)¥neq¥phi$ and we thus have a contra$ F(x) ¥ cap J^{-}(x)=¥phi$ . diction. Suppose that Then there exists an open neighborhood $U$ . Let $z¥in J^{-}(x)$ . Then there are of $J^{-}(x)$ with compact such that sequences of principal negative solutions and in with such that and . Hence for some positive integer , ¥ ¥ ¥ ¥ ¥ and , . Thus ( $-¥infty$ , for . Let for ( for . Since may is compact, we assume that . ¥ ¥ Hence $b¥in D^{-}(x)$ . Since $¥partial U¥cap F(x)=¥phi$ and ¥ , we then have a contradiction. The proof of the lemma is now complete. $ J^{-}(x) neq phi$

$F(x)$

$C_{¥sigma}^{-}(y)¥subset J^{-}(x)$

$y$

$K_{¥sigma}^{-}(y)¥subset J^{-}(x)$

$ L_{¥sigma}^{-}(y)¥neq¥phi$

$ L_{¥sigma}^{-}(x)¥neq¥phi$

$¥overline{U}¥cap F(x)=¥phi$

$¥overline{U}$

$¥{¥sigma_{i}¥}$

$¥sigma_{i}(-t_{i})¥rightarrow z$

$¥sigma_{i}(0)¥rightarrow x$

$¥sigma_{i}$

$R^{+}$

$¥{t_{i}¥}$

$- infty$

$¥sigma_{i}(-s_{i})¥in¥partial U$

$ t_{i}¥rightarrow¥infty$

$i_{0}$

$ mathrm{O}] cap X-U neq phi$ $i¥geq i$

$i¥geq i_{0}$

$¥sigma_{i}(-¥infty,$

$¥partial U$

$¥mathrm{O}]¥cap U¥neq¥phi$

$i¥geq i_{0}$

$¥mathrm{O}]¥cap¥partial U¥neq¥phi$

$¥sigma_{i}$

$¥sigma_{i}(-s_{i})¥rightarrow b¥in¥partial U$

$ partial U cap J^{-}(x)= phi$

Theorem 2.11.

For each

,

$x¥in X$ $D^{-}(x)$

is connected whenever it is compact.

is a seperation of $D^{-}(x)$ , where A and are disjoint nonempty compact sets. Suppose that A. Let $y¥in B$ . Then $y¥in D^{-}(x)$ . There exists an open neighborhood $U$ of A such that is compact and . There exists a sequence of principal negative solutions such that and for some sequence . We may assume, without loss of generin ality, that and for all /. Since is connected, for each /. Thus for each , there exists such that . is compact, may assume that Since . Then $z¥in D^{-}(x)$ and we thus have a contradiction. The proof of the lemma is now compelte.

Proof. Assume that $D^{-}(x)=$ A

$¥cup B$

$B$

$ x¥in$

$¥overline{U}¥cap B=¥phi$

$¥overline{U}$

$¥sigma_{i}(0)¥rightarrow x$

$¥{¥sigma_{i}¥}$

$¥sigma_{i}(-t_{i})¥rightarrow y$

$¥{t_{i}¥}$

$¥sigma_{i}(0)¥in U$

$R^{+}$

$¥sigma_{i}(-t_{i})¥in X-¥overline{U}$

$¥sigma_{i}([-t_{i}, 0])$

$¥sigma_{i}([-t_{i}, ¥mathrm{O}])¥cap¥partial U¥neq¥phi$

$¥sigma_{i}(-s_{i})¥in¥partial U$

Theorem 2.12. ever it is compact.

$s_{i}¥in R^{+}$

$i$

$¥partial U$

$ffM$

$¥sigma_{i}(-s)¥rightarrow z¥in¥partial U$

is a connected subset

of

, then

$X$

$D^{-}(M)$

is connected wfien-

is a separation of $D^{-}(M)$ into two disjoint nonempty compact sets. Since A or $M¥subset B$. is connected, either Assume that A. There exists $x¥in M$ with $ D^{-}(x)¥cap B¥neq¥phi$ . Since it is also true ¥ A that and $D^{-}(x)$ is connected (Theorem 2. 11), we thus have a contradiction. The proof of the theorem is now complete.

Proof. Assume that $D^{-}(M)=$ A

$¥cup B$

$M¥subset D^{-}(M)$

$ M¥subset$

$ D^{-}(x) cap$

$¥neq¥phi$

$ M¥subset$

92

§3.

S. ELAYDI

Strong stability.

Definition 3.1. A subset $M$ of $X$ is said to be strongly negatively Liapunov stable if for each open neighborhood $U$ of $M$ there exists an open neighborhood of $M$ with $F(V)¥subset U$.

$V$

De finition 3.2. A subset $M$ of $X$ is said to be strongly { -stable} if $U$ $M$ of there exists an open neibhborhood $V$ of $M$ for each open neighborhood such that $¥{D^{-}(V)¥subset U¥}¥{J^{-}(V)¥subset U¥}$ . $¥{¥mathrm{D}^{-}- ¥mathrm{s}¥mathrm{t}¥mathrm{a}¥mathrm{b}1¥mathrm{e}¥}$

$¥mathrm{J}^{-}$

The semidynamical system $(X, ¥pi)$ is said to have one of the properties in definitions 3.1 and 3.2 if for every $a¥in X$, $K^{-}(a)$ possesses the property. Definition 3.3. A point $a¥in X$ is said to be of characteristic if $D^{-}(a)=K^{-}(a)$ . The system $(X, ¥pi)$ is said to be of characteristic if $D^{-}(a)=K^{-}(a)$ for each $a¥in X$. $O^{-}$

$O^{-}$

Theorem 3.4. A compact subset negatively Liapunov stable.

of $X$

$M$

is strongly

$D^{-}$

-stabfe iff it is strongly

strongly -stable. Then it follows from Lemma 2.7 that $M$ is strongly negatively Liapunov stable. Conversely, assume that $M$ is strongly negatively Liapunov stable. Suppose that $M$ is not strongly -stable. Then there $U$ $M$ $D^{-}(W) ¥ not ¥ subset U$ such open neighborhood of that exists an for all open neighbor$V$ $W$ $M$ . There exists an open neighborhood of of $M$ with . hood $ D^{-}(V)= ¥ bigcup_{a ¥ in V} ¥ cap$ is an open neighborhood of } Lemma 2.7, { and we then have a contradiction. This completes the proof of the theorem.

Proof. Assume that $M$ is

$¥mathrm{D}^{-}$

$¥mathrm{D}^{-}$

$¥overline{F(V)}¥subset U$

$¥overline{F^{¥urcorner}(W)}|W$

Theorem 3.5.

A compact subset

$¥mathrm{a}$

of $X$ is strongly

$M$

$D^{-}-$

stable

$¥subset¥overline{F(V)}¥subset U$

iff $D^{-}(M)=M$.

strongly -stable. Then $M$ is strongly negatively invariant. It follows from Lemma 2.7 that $M¥subset D^{-}(M)$ . Let $a¥in D^{-}(M)-M$. . Since $M$ is strongly Then there exists an open neighborhood $U$ of $M$ with -stable, there exists an open neighborhood $W$ of $M$ with $D^{-}(W)¥subset U$. Hence $a¥in D^{-}(M)¥subset D^{-}(W)¥subset U$ Consequently, and thus we have a contradiction. $D^{-}(M)=M$ . Conversely, assume that $D^{-}(M)=M$ and $M$ is not strongly -stable. $M$ Then it follows from Theorem 2. 1 that is not strongly negatively Liapunov stable. Then there exists an open neighborhood $U$ of $M$ whose closure is compact such that $F(W)¥not¥subset U$ for all open neighborhood $W$ of $M$. Hence there are sequences and for each . Hence for in $U$ and in such that $0]=$ each there exists a principal negative solution through such that ¥ ¥ ¥ ¥ is continuous [1], it follows . Since each and for each . that . implies This is each connected for that . We may assume that such that Hence for each there exists

Proof. Assume that $M$ is

$¥mathrm{D}^{-}$

$a¥not¥in U$

$¥mathrm{D}^{-}$

$¥mathrm{D}^{-}$

$¥overline{U}$

$¥{a_{i}¥}$

$R^{+}$

$¥{t_{i}¥}$

$a_{i}¥rightarrow a¥in M$

$F(a_{i}, -t_{i})¥not¥subset U$

$i$

$a_{i}$

$¥sigma_{i}$

$ C_{ sigma}^{-}i(a_{i}) cap(X-U) neq phi$

$ C_{¥sigma}^{-}i(a_{i})¥cap U¥neq¥phi$

$i$

$C_{¥sigma}^{-}i(a_{i})$

$i$

$s_{i}¥in R^{+}$

$i$

$¥sigma_{i}(-¥infty,$

$¥sigma_{i}$

$ C_{¥sigma}^{-}i(a_{i})¥cap¥partial U¥neq¥emptyset$

$¥sigma_{i}(-s_{i})¥in¥partial U$

$i$

93

Sem idynam ical Systems

Thus $d¥in D^{-}(M)=M$ and we thus have a contradiction. completes the proof of the theorem.

This

.

$¥sigma_{i}(-s_{i})¥rightarrow d¥in¥partial U$

Theorem 3.6.

A compact negatively invariant subset

$iffJ^{-}(M)=M$.

$M$

of

$X$

is strongly

$J^{-}$

-stabfe

strongly -stable. Let $b¥in J^{-}(M)-M$ . Then there $U$ . There is an open neighborhood exists an open neighborhood of $M$ with $W$ of $M$ with $J^{-}(W)¥subset U$ . Thus $b¥in J^{-}(M)¥subset J^{-}(W)¥subset U$ and hence we have a contradiction. Consequently, $J^{-}(M)¥subset M$. Let $a¥in M-J^{-}(M)$ . Since $M$ is compact, it follows easily from Lemma 2.9 that $J^{-}(M)$ is closed. Hence there exists an open . Let $b¥in J^{-}(a)$ . is compact and such that neighborhood $U$ whose closure such in of principal negative solutions and Then there are sequences with in . Hence there exists a sequence , and that . This is compact, we may assume that . Since $c ¥ in J^{-}(a) ¥ subset J^{-}(M)$ Consequently, and thus we then have a contradiction. implies that $M¥subset J^{-}(M)$ and hence $M=J^{-}(M)$ . Conversely, assume that $M=J^{-}(M)$ . Then $D^{-}(M)=F(M)¥cup J^{-}(M)=M$. This implies by Theorem 3.5 that $M$ is strongly -stable and consequently, $M$ is strongly -stable. The proof of the theorem is now complete.

Proof. Assume that $M$ is

$¥mathrm{J}^{-}$

$b¥not¥in U$

$a¥not¥in U$

$¥overline{U}$

$¥sigma_{i}(-t_{i})¥rightarrow b$

$¥sigma_{i}(0)¥rightarrow a$

$¥sigma_{i}(-s_{i})¥rightarrow c¥in¥partial U$

$¥mathrm{J}^{-}$

$¥mathrm{D}^{-}$

$D^{-}$

$R^{+}$

$¥{s_{i}¥}$

$¥partial U$

$¥sigma_{i}(-s_{i})¥in¥partial U$

$R^{+}$

$¥{t_{i}¥}$

$¥{¥sigma_{i}¥}$

Corollary 3.7. A compact negatively invariant subset -stabfe iff it is strongly -stabfe.

$M$

of

$X$

is strongly

$J^{-}$

Proof. This follows immediately from Theorems 3.5 and 3.6. is of characteristic Theorem 3.8. The semidynamical system $(X, ¥pi)$

relation

$S=¥{(a, b)¥in X¥times X|b¥in K^{-}(a)¥}$

$O^{-}$

is closed, negatively invariant refation on

iff

the

.

$X$

be a sequence . Let is of characteristic which converges to $(a, b)$ . Then $b_{i}¥in K^{-}(a_{i})=D^{-}(a_{i})$ , for each . It follows in is from Lemma 2.9 that $b¥in D^{-}(a)=K^{-}(a)$ and consequently, $(a, b)¥in S$. Thus $(c, d) ¥ in X ¥ times X$ with $c¥in F(a, -t)$ , $d¥in F(b, -t)$ . Suppose closed. Let $(a, b)¥in S$ and $b=d ¥ pi t$ $d ¥ not ¥ in K^{-}(c)=D^{-}(c)$ . It follows from Lemma 2.6 that . Then that $b¥not¥in D^{-}(c¥pi t)=D^{-}(a)=K^{-}(a)$ which is false. Hence is negatively invariant. Conis closed and negatively invariant. Let $b¥in D^{-}(a)$ . Then versely, assume that with in of principal negative solutions and there are sequences for each , it follows that $(a, b)¥in S$. . Since and . Hence $D^{-}(a)¥subset K^{-}(a)$ . From Remark 2.8, it follows that $D^{-}(a)=$ Thus $K^{-}(a)¥cup J^{-}(a)$ and hence $K^{-}(a)¥subset D^{-}(a)$ . This implies that $D^{-}(a)=K^{-}(a)$ and con. The proof of the theorem is now complete. sequently, is of characteristic

Proof. Assume that

$(X, ¥pi)$

$O^{-}$

$¥{(a_{i}, b_{i})¥}$

$S$

$i$

$S$

$¥not¥in D^{-}(c)¥pi t$

$S$

$S$

$¥{t_{i}¥}$

$¥{¥sigma_{i}¥}$

$(¥sigma_{i}(0), ¥sigma_{i}(-t_{i}))¥in S$

$¥sigma_{i}(-t_{i})¥rightarrow b$

$R^{+}$

$¥sigma_{i}(0)¥rightarrow a$

$¥mathrm{f}$

$b¥subset K^{-}(a)$

$O^{-}$

$a$

Theorem 3.9. If a semidynamical system of characteristic . $O^{-}$

$(X, ¥pi)$

is strongly-stable, then it is

94

S. ELAYDI

Proof. The proof is easy and will be omitted. if it is Definition 3.10. A point $a¥in X$ is said to be of stable characteristic $X$ in with and and whenever there are sequences of characteristic $b_{i} ¥ in F(x_{i})$ has a convergent subsequence. The semidynamical system , then and $(X, ¥pi)$ is of stable characteristic . if every point in $X$ is of stable characteristic $O^{-}$

$O^{-}$

$¥{b_{i}¥}$

$¥{a_{i}¥}$

$a_{i}¥rightarrow a$

$¥{b_{i}¥}$

$O^{-}$

$O^{-}$

Remark 3. 11. Let $a¥in X$ be of stable characteristic mediately from Definition 2.7 that $K^{-}(a)$ is compact.

$O^{-}$

.

Then it follows im-

Theorem 3.12. A semidynamical system $(X, ¥pi)$ is of stable characteristic is of characteristic $O^{-}and$ $K^{-}(a)$ is compact for each $a¥in X$.

$O^{-}$

if it

and that $K^{-}(a)$ is compact is of characteristic is not of stable characteristic for each $a¥in X$. Suppose that there exist $X$ ¥ and with , and in . Thus there exist sequences $U$ $K^{-}(x)$ open such neighborhood of be an has no convergent subsequence. Let there . For each for all that is compact. This implies that . Thus such that and exists a principal negative solution . Since is compact, . for Let for $z ¥ in D^{-}(x)=K^{-}(x)$ a and we then have contradiction. . Consequently . The converse is clear. This shows that is of stable characteristic

Proof. Assume that

$(X, ¥pi)$

$O^{-}$

$x¥in ¥mathrm{X}¥backslash ¥mathrm{v}¥mathrm{h}¥mathrm{i}¥mathrm{c}¥mathrm{h}$

$O^{-}$

$¥{y_{i}¥}$

$¥{x_{i}¥}$

$y_{i}¥not¥in U$

$¥overline{U}$

$y_{i} in F(x_{i})$

$¥{y_{i}¥}$

$i¥geq i_{0}$

$i¥geq i_{0}$

$¥sigma_{i}(-t_{i})=y_{i}$

$¥sigma_{i}(0)=x_{i}$

$¥sigma_{i}$

$¥sigma_{i}(-s_{i})¥in¥partial U$

$i¥geq i_{0}$

$ C_{¥sigma}^{-}i(x_{i})¥cap¥partial U¥neq¥phi$

$x_{i}¥rightarrow x$

$¥partial U$

$i¥geq i_{0}$

$¥sigma_{i}(-s_{i})¥rightarrow z¥in¥partial U$

$O^{-}$

$x$

Theorem 3.13. A semidynamical systems $(X, ¥pi)$ is of stable characteristic -stabfe and $K^{-}(a)$ is compact for each $a¥in X$. it is strongly

$O^{-}$

iff

$D^{-}$

. Suppose that for is of stable characteristic -stable. Then from Theorem 3.4 it follows some $x¥in X$, $K^{-}(x)$ is not strongly that $K^{-}(x)$ is not strongly negatively Liapunov stable. Hence there exists an open ¥ ¥ for all open neighborhoods $V$ of neighborhood $U$ of $K^{-}(x)$ such that $K^{-}(x)$ . in $U$ which converges to $a¥in K^{-}(x)$ and Then there exists a sequence for all . Let $y_{i}¥in F(x_{i})-U$. Since is of stable characteristic such that ¥ ¥ . Hence . Then has a convergent subsequence , $(X, ¥pi)$ is $y¥in D^{-}(a)¥subset D^{-}(x)=K^{-}(x)$ and thus we have a contradiction. Therefore -stable. The converse follows from Theorems 3.12 and 3.9. strongly

Proof. Assume that

$(X, ¥pi)$

$O^{-}$

$¥mathrm{D}^{-}$

$F(V) not subset U$

$¥{x_{i}¥}$

$F(x_{i})¥not¥subset U$

$O^{-}$

$x$

$i$

$¥{y_{i_{n}}¥}$

$¥{y_{i}¥}$

$y_{i_{n}} rightarrow y in X-U$

$¥mathrm{D}^{-}$

Theorem 3.14. A compact subset $M$ of $X$ is strongly negatively Liapunov stable iff every component of $M$ is strongly negatively Liapunov stable.

Suppose that $M$ is strongly negatively Liapunov stable. Then $M$ is negatively invariant. Furthermore, it follows from Theorem 3.4 and 3.5 that be a component of $M$. Then one can show that $D^{-}(M_{i})$ is $D^{-}(M)=M$. Let closed. Thus $D^{-}(M_{i})$ is compact. Then according to Theorem 2.12, $D^{-}(M_{i})$ is

Proof.

$M_{i}$

95

Semidynamical Systems

connected. This implies that $D^{-}(M_{i})=M_{i}$ . It follows from Theorems 3.4 and 3.5 is strongly negatively Liapunov stable. The converse is trivial. that $M_{i}$

§4.

Weak stability.

DeMition 4.1. Let $a¥in X$ and be a principal negative solution through . Then we have the following: (i) The -negative prolongation set of ; $D_{¥sigma}^{-}(a)=¥{b¥in X|$ there exists a sequence of principal negative solutions such that for each and for some sequence in . (ii) The a-negative prolongation limit set of ; $J_{¥sigma}^{-}(a)=¥{b¥in X|$ there exists a sequence of principal negative solutions such that for each and for some sequence in with }. $a$

$¥sigma$

$a$

$¥sigma$

$¥{¥sigma_{i}¥}$

$t¥in R^{+}$

$¥sigma_{i}(-t)¥rightarrow¥sigma(-t)$

$¥sigma_{i}(-t_{i})¥rightarrow b$

$¥{t_{i}¥}$

$R^{+}¥}$

$a$

$¥{¥sigma_{i}¥}$

$t¥in R^{+}$

$¥sigma_{i}(-t)¥rightarrow¥sigma(-t)$ $R^{+}$

$¥sigma_{i}(-t_{i})¥rightarrow b$

$¥{t_{i}¥}$

$ t_{i}¥rightarrow¥infty$

Lemma 4.2. Let $a¥in X$ and be a principal negative solution through the following statements are valid: and , (i) $¥sigma$

$J_{¥sigma}^{-}(a)=J_{¥sigma}^{-}(a)¥pi t¥subset J^{-}(a)$

Proof.

Then

.

The proof is similar to that of Lemma 2.6.

Lemma 4.3.

(i) (ii) (iii)

.

$D_{¥sigma}^{-}(a)¥subset D^{-}(a)$

$D_{¥sigma}^{-}(a)¥subset D_{¥sigma}^{-}(a)¥pi t$

(ii)

$a$

$J_{¥sigma}^{-}(a)$

$D_{¥sigma}^{-}(a)$

be a principal negative solution through . Then is closed, positively invariant and weakly negatively invariant. is closed and weakly negatively invariant. Let

$a¥in X$

and

$D_{¥sigma}^{-}(a)=C_{¥sigma}^{-}(a)¥cup J_{¥sigma}^{-}(a)$

Proof. The proof follows

$a$

$¥sigma$

.

easily from Theorem 2.3 and Lemma 4.2.

Definition 4.4. A subset $M$ of $X$ is said to be weakly negatively Liapunov stable if for each open neighborhood $V$ of $M$ there exists an open neighborhood $V$ of $M$ such that for each $a¥in V$, for some principal negative solution through . $C_{¥sigma}^{-}(a)¥subset U$

$¥sigma$

$a$

Definition 4.5. A subset $M$ of $X$ is weakly { -stable} if for each open neighborhood $U$ of $M$ there exists an open neighborhood $V$ of $M$ such that for each $a¥in V$ there exists a principal negative solution through a with . $¥{D^{-}- ¥mathrm{s}¥mathrm{t}¥mathrm{a}¥mathrm{b}1¥mathrm{e}¥}$

$J^{-}$

$¥sigma$

$¥{D_{¥sigma}^{-}(a)¥subset U¥}$

$¥{J_{¥sigma}^{-}(a)¥subset U¥}$

DeLition 4.6. A point a $¥in X$ is said to be of weakly characteristic for some principal negative solution through a.

$O^{-}$

if

stable

iff

$D_{¥sigma}^{-}(a)=K_{¥sigma}^{-}(a)$

Theorem 4.7. A compact subset it is weakly -stable. $D^{-}$

$M$

of

$X$

is weakly negatively

$Liap_{¥vee}’ inov$

96

S. ELAYDI

Proof. The proof is similar to that of Theorem 3.4 and will be omitted. -stabfe ifffor each Theorem 4.8. A compact subset $M$ of $X$ is weakly such $that¥cup¥{D_{¥sigma}^{-}(a)|a¥in M¥}=M$. there exists a principal negative solution $D^{-}$

$a¥in M$

$¥sigma_{a}$

Proof. The proof is similar to that of Theorem 3.5

and will be omitted.

Theorem 4.9. A compact negatively invariant subset $M$ of $X$ is weakly -stable such that iff for each $a¥in M$ there exists a principal negative solution $¥cup¥{J_{¥sigma_{a}}^{-1}(a)|a¥in M¥}=M$. $J^{-}$

$¥sigma_{a}$

Proof. The proof is similar to that of Theorem 3.6, $D^{-}$

Corollary 4.10. A compact negatively invariant subset -stable iff it is weakly -stabfe.

Proof. This follows immediately from Theorems 4.8 §5.

$M$

of

$X$

is weakly

$J^{-}$

and 4.9.

Attraction and saymptotic stability. Definition 5.1. For a nonempty subset $M$ of $X$ we have the following: (i) The region of weak negative attraction of $M$ ; A $w-(M)=¥{x¥in X|L^{-}(x)¥cap M¥neq¥phi¥}$ , (ii) The region of negative attraction of $M$ ; $¥mathrm{A}^{-}(M)=¥{x¥in X|L^{-}(x)¥subset M, L^{-}(x)¥neq¥phi¥}$ , (iii) The region of strong negative attraction of $M$ ; A $s-(M)=¥{x¥in X|J^{-}(x)¥subset M, J^{-}(x)¥neq¥phi¥}$ .

A point $x¥in X$ is said to be {weakly negatively {strongly negatively attracted} to $M$ if tively.

attracted} {negatively attracted}

, respec-

$¥{x¥in ¥mathrm{A}_{w}^{-}(M)¥}¥{x¥in ¥mathrm{A}^{-}(M)¥}¥{x¥in ¥mathrm{A}_{s}^{-}(M)¥}$

Deffiition 5.2. A nonempty subset $M$ of $X$ is said to be {a negative weak attractor} {a negative attractor} {a negative strong attractor} if {A $s-(M)$ } is a neighborhood of $M$, respectively. The adjective global is added to the above if the corresponding region of attraction is the whole phase space of $X$. $¥{¥mathrm{A}_{w}^{-}(M)¥}¥{A^{-}(M)¥}$

Lemma 5.3. For a nonempty subset and , (i) The sets $w-(M)$ A . (ii) $¥mathrm{A}_{w}^{-}(M)$

$¥mathrm{A}^{-}(M)$

$M$

of

$¥mathrm{A}_{s}^{-}(M)$

, the following hold: are weakly invariant.

$X$

$¥mathrm{A}_{s}^{-}(M)¥subset A^{-}(M)¥subset$

Proof.

Statement (i) follows from Lemma 2.6 and statement (ii) follows from

Lemma 2. 10. Theorem 5.4. A compact subset $M$ of $X$ is strongly negatively Liapunov stable A $s-(M)$ . iff it is negatively invariant and $ M¥subset$

97

Semidynam ,,.cal Systems

Let that $M$ is strongly negatively Liapunov stable. . There $z¥in F(M)-M$. Then there exists an open neighborhood $U$ of $M$ with exists an open neighborhood $V$ of $M$ with $F(V)¥subset U$. Thus $z¥in F(M)¥subset F(V)¥subset U$ and hence we have a contradiction. Consequently $F(M)¥subset M$ and $M$ is thus negaThen there exists . Suppose that tively invariant. Let $x¥in M$. $y¥in J^{-}(x)-M$ . . Then there exists Let $W$ be open neighborhood of $M$ with an open neighborhood $G$ of $M$ with $F(G)¥subset W$. From Lemma 2.7 it follows that and we thus have a . This implies that and $M$ is . Conversely, assume that contradiction. Hence negatively invariant. If $M$ is not strongly negatively Liapunov stable, then there ¥ ¥ for exists an open neighborhood $U$ of $M$ with compact and such that $U$ in in and all open neighborhoods $V$ of $M$. Hence there exist sequences ¥ ¥ $F(x_{i}, [-t_{i}, 0])$ is connected . Since and such that for . Let (Theorem 2.2) for each , is compact, we may assume . Since each . Then $b_{i}¥in F(x_{i}, -s_{i})$ for some . Hence $b¥in D^{-}(x)$ . Since $D^{-}(x)=F(x)¥cup J^{-}(x)$ (Lemma 2.7) and that $F(x)¥subset M$, $b¥in J^{-}(x)$ . This leads to a contradiction since $J^{-}(x)¥subset M$ for each $x¥in M$. This completes the proof of the theorem.

Proof. Assume

$z¥not¥in U$

$ x¥not¥in$

$¥mathrm{A}_{s}^{-}(M)$

$y¥not¥in¥overline{W}$

$y¥in J^{-}(x)¥subset D^{-}(x)¥subset¥overline{W}$

$D^{-}(x)¥subset¥overline{F(G)}¥subset¥overline{W}$

$M¥subset ¥mathrm{A}_{s}^{-}(M)$

$M¥subset ¥mathrm{A}_{s}^{-}(M)$

$F(V) not subset U$

$¥overline{U}$

$¥{t_{i}¥}$

$¥{x_{i}¥}$

$F(x_{i}, -t_{i}) not subset U$

$x_{i}¥rightarrow x¥in M$

$R^{+}$

$i$

$b_{i}¥in F(x_{i}, [-t_{i}, ¥mathrm{O}])¥cap¥partial U$

$ F(x_{i}, [-t_{i}, ¥mathrm{O}])¥cap¥partial U¥neq¥phi$ $s_{i}¥in R^{+}$

$i$

$¥partial U$

$b_{i}¥rightarrow b¥in¥partial U$

A subset $M$ of $X$ is said to be negatively asymptotically stable it is a negative attractor and strongly negatively Liapunov stable.

Definition 5.5.

Theorem 5.6.

Let X. Then a component componen .

$M$

of

if

be a compact negatively asymptotically stable subset of is negatively asymptotically stable iff it is an isolated

$M$

$t$

Let be a negatively asymptotically stable component of $M$. If is strongly is any other component of $M$, then according to Theorem 3. 14 the set Claim that negatively Liapunov stable. Hence is negatively invariant. . . To prove the claim, assume that there exists ¥ ¥ ¥ ¥ and thus we have a conThen and . Furthermore, tradiction. Since is a negative attractor, there exists an open neighborhood $U$ is an of . Hence with is isolated. Conversely, assume that $U$ of $M$ such isolated component of . Then there exists an open neighborhood that is compact, and $ U¥cap(X-M_{1})=¥phi$ . According to Theorem 3.14 the set is negatively strongly Liapunov stable. Hence there exists an open neighborhood $V$ of . It follows that for each $x¥in V$, with and $L^{-}(x)¥subset U$. Since , $L^{-}(x)¥subset M$ for each $x¥in V$. This implies that for $x ¥ in V$ ¥ each . Thus , and consequently, $M$ is an attractor. This completes the proof of the theorem.

Proof.

$M_{2}$

$M_{1}$

$M_{2}$

$M_{2}$

$x¥in ¥mathrm{A}^{-}(M_{1})¥cap M_{2}$

$¥mathrm{A}^{-}(M_{1})¥cap M_{2}=¥phi$

$ L^{-}(x) neq phi$

$L^{-}(x) subset M_{1}$

$L^{-}(x) subset M_{2}$

$M_{1}$

$U¥subset ¥mathrm{A}^{-}(M_{1})$

$M_{1}$

$M_{1}$

$M_{1}$

$M_{1}$

$U¥subset ¥mathrm{A}^{-}(M)$

$¥overline{U}$

$M_{1}$

$¥overline{F(V)}¥subset U$

$M_{1}$

$V¥subset ¥mathrm{A}^{-}(M)$

$L^{-}(x) subset M_{1}$

$V¥subset ¥mathrm{A}^{-}(M_{1})$

$ L^{-}(x)¥neq¥emptyset$

98

§6.

S. ELAYDI

Examples.

The following examples demonstrate some of the basic differences between dynamical systems and semidynamical systems. Example 6. 1. ically in Figure 1.

Let $X$ be the real line. A semidynamical system is defined graph-

Figure 1.

All points in $X-¥{0¥}$ are points of negative uniqueness. Notice that $L^{-}(0)=$ $¥{-1,0,1¥}$ is disconnected. Furthermore, $J^{-}(0)=L^{-}(0)$ and is thus disconnected. This is in contrast with the situation in dynamical systems. The set {0} is positively Liapunov stable, weakly negatively Liapunov stable but not strongly negatively Liapunov stable. Let $H=¥{-1,1¥}$ . Then A $w-(H)=[-2,2]$ and $¥mathrm{A}^{-}(H)=(-2,0)¥cup$ ¥ . Thus $H$ is a negative strong attractor. Notice that A $w-(H)$ is not open, not negatively invariant and weakly negatively invariant. Furthermore, $H$ is strongly negatively Liapunov stable and hence it is negatively asymptotically stable. $(0, 2)= mathrm{A}_{s}^{-}(H)$

Example 6.2. Let $X=R$ be the real line. A semidynamical system is defined graphically in Figure 2. Notice that $L^{-}(0)=R$ while for all negative through 0. Thus Lemma 2.6 cannot be improved. The set {0} is solutions globally positively asymptotically stable but it is not even weakly negatively invariant. We remark also that $F$(0, [?t,$ -s]$ ) is connected iff $s=0$ . $ L_{¥sigma}^{-}(0)=¥phi$

$¥sigma$

$ $

Example 6.3. Figure 3. Notice strongly invariant. strongly invariant

invariant for

$a¥in R^{2}$

Let $X=R^{2}$ . A semidynamical system is defined graphically in that the set $G=¥{(x, y)|-1