Forward Omega Limit Sets of Nonautonomous Dynamical Systems

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DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020065 DYNAMICAL SYSTEMS SERIES S Volume 13, Number 4, April 2020 pp. 1103–1114

FORWARD OMEGA LIMIT SETS OF NONAUTONOMOUS DYNAMICAL SYSTEMS

Hongyong Cui, Peter E. Kloeden and Meihua Yang School of Mathematics and Statistics, and Hubei Key Laboratory of Engineering Modeling and Scientiﬁc Computing Huazhong University of Science and Technology Wuhan 430074, China

Dedicated to J¨urgen Scheurle on his 65th birthday.

Abstract. The forward ω-limit set ωB of a nonautonomous dynamical system ϕ with a positively invariant absorbing family B = {B(t), t ∈ R} of closed and bounded subsets of a Banach space X which is asymptotically compact is shown to be asymptotically positive invariant in general and asymptotic negative invariant if ϕ is also strongly asymptotically compact and eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. In addition, a necessary and suﬃcient condition for a ϕ- invariant family A = {A(t), t ∈ R} in B of nonempty compact subsets of X to be a forward attractor is generalised to this context.

1. Introduction. Omega limit sets play a fundamental role in characterising the behaviour of dynamical systems. Indeed, the attractor of an autonomous dynamical system is an omega limit set. In such systems the omega limit sets are invariant. The situation is more complicated in nonautonomous dynamical systems because the behaviour now depends on the actual time and not on the the time that has elapsed since starting. This leads to new concepts of nonautonomous attractors that consist of families of compact subsets. Two kinds have been distinguished, pullback attractors which depend on information from the distant past and forward attractors which depend on the future behaviour of the system. The constituent subsets depend on time and are mapped into each other by the system, i.e., are invariant in a new nonautonomous sense. Vishik [15] also proposed another kind of nonautonomous attractor, which he called a uniform attractor. It is essentially a forward omega limit set which attracts bounded sets uniformly in initial times. No claim about its invariance was made by Vishik, but see [1,2,4,7] for a later discussion for skew product semiﬂows. In addition, the convergence uniformly in all initial times is far too strong, because forward attractor is concerned only about what happens in the distant future and not in the past.

2010 Mathematics Subject Classiﬁcation. Primary: 35B40; Secondary: 35B41, 37L30. Key words and phrases. Nonautonomous dynamical system, forward attracting set, omega limit sets, uniform attractor, asymptotic positive invariance, asymptotic negative invariance, strong asymptotic compactness, upper semi continuous dependence on parameters. HC was partially funded by China Postdoctoral Science Foundation 2017M612430. PEK and MY were partially supported by the Chinese NSF grant 11571125.

1103 1104 HONGYONG CUI, PETER E. KLOEDEN AND MEIHUA YANG

Recently, Kloeden & Yang [12] showed that the forward omega limit set ωB of a nonautonomous dynamical system ϕ is asymptotic positive invariant for nonautonomous difference equations with a positively invariant compact absorbing set B. This concept has been long known in the differential equations literature [8, 13]. Also in [9] it was shown that ωB is also asymptotic negative invariant provided ϕ is eventually continuous in its initial value inside the compact absorbing set B uniformly on bounded time sets independently of the initial time. These results were used to show the upper semi continuous dependence of ωB on parameters. These results are generalised here to a nonautonomous dynamical system ϕ with a positively invariant absorbing family B = {B(t), t ∈ R} of closed and bounded subsets of X. This requires a successive strengthening of assumptions on compactness, such as asymptotic compactness and strong asymptotic compactness, and some eventual uniform properties. In addition, a necessary and sufficient condition in [10] for a ϕ-invariant family A = {A(t), t ∈ R} in B of nonempty compact subsets of X to be a forward attractor is shown to hold when, amongst other things, the absorbing family is strongly asymptotically compact and the process is eventually continuous in its initial value uniformly on bounded time sets independently of the initial time.

2. Dissipative nonautonomous systems. Let distX (·, ·) be the Hausdorff semi- distance between nonempty closed and bounded subsets of a Banach space X and define 2 2 R≥ := {(t, t0) ∈ R : t ≥ t0}. Since forward asymptotic behaviour depends on the future of the system and ∗ ∗ not on its past, it can be restricted to initial times t0 ≥ T for some T . In 2 fact, the system need only be defined for such times, i.e., the time set R∗,≥ := 2 ∗ {(t, t0) ∈ R : t ≥ t0 ≥ T }. Consider a nonautonomous dynamical system ϕ on X defined as a process or two-parameter semi-group, i.e.,

2 Deﬁnition 2.1. A process on the state space X is a mapping ϕ : R≥ × X → X, which satisﬁes the initial value, 2-parameter semigroup and continuity properties:

(i) ϕ(t0, t0, x0) = x0 for all t0 ∈ R and x0 ∈ X; (ii) ϕ(t2, t0, x0) = ϕ (t2, t1, ϕ(t1, t0, x0)) for all t0 ≤ t1 ≤ t2 in R and x0 ∈ X; 2 (iii) the mapping (t, t0, x0) 7→ ϕ(t, t0, x0) of R≥ × X into X is continuous. We assume that the nonautonomous system ϕ is dissipative. In particular, we assume that ϕ has an absorbing family B = {B(t), t ∈ R} of closed and bounded subsets of X which is ϕ-positively invariant, i.e.,

ϕ(t, t0,B(t0)) ⊂ B(t), ∀t ≥ t0.

Assumption 1. There exists a ϕ-positive invariant family B = {B(t), t ∈ R} of closed and bounded subsets of X such that for any bounded subset D of X and every ∗ t0 ≥ T there exists a TD,t0 ≥ 0 for which

ϕ(t, t0, x0) ∈ B(t) for all t ≥ t0 + TD,t0 , x0 ∈ D. In addition, we assume that ϕ is asymptotically compact on B. FORWARD OMEGA LIMIT SETS 1105

Assumption 2. The process ϕ is asymptotically compact for the ϕ-positive invariant family B = {B(t), t ∈ R} of closed and bounded subsets of X if there is a compact subset K of X such that

lim distX (ϕ(t, t0,B(t0)),K) = 0. t→∞ 2.1. Omega limit sets. Let B = {B(t), t ∈ R} be ϕ-positive invariant family of closed and bounded subsets of X. The omega limit set starting in B(t0) at time ∗ t0 ≥ T is deﬁned as \ [ ωB(t0)(t0) := ϕ (s, t0,B(t0)).

t≥t0 s≥t

Clearly, y ∈ ωB(t0)(t0) if and only if there exist sequences tn → ∞ and xn ∈ B(t0) 0 0 such that φ(t , t , x ) → y. Note that ω (t ) ⊂ ω 0 (t ) if t ≤ t . n 0 n B(t0) 0 B(t0) 0 0 0 Lemma 2.2. Assume that ϕ is asymptotically compact for the family B = {B(t), t ∈

R}, i.e., satisﬁes Assumption2. Then the omega limit set ωB(t0)(t0) starting in

B(t0) at time t0 is nonempty and compact with ωB(t0)(t0) ⊂ K for each t0 ∈ R. Moreover, lim distX ϕ(t, t0,B(t0)), ωB(t )(t0) = 0. (2.1) t→∞ 0

Proof. Consider a sequence yn = ϕ(tn, t0, bn) ∈ ϕ(tn, t0,B(t0)), where tn → ∞. Then distX (yn,K) ≤ distX (ϕ(tn, t0,B(t0)),K) → 0 as n → ∞.

Since K is compact there is a kn ∈ K such that

kyn − knk = distX (yn,K) → 0 as n → ∞. ¯ Moreover, there is a convergent subsequence knj → k ∈ K. Hence ¯ kynj − kk ≤ distX ynj ,K → 0 as nj → ∞. ¯ Clearly, k ∈ ωB(t0)(t0). Thus ωB(t0)(t0) is nonempty.

Now let ωn be a sequence in ωB(t0)(t0). Then there exist sequences ϕ(tn, t0, bn) with bn ∈ B(t0) and tn → ∞ such that 1 kϕ(t , t , b ) − ω k < , n ∈ . n 0 n n n N Hence,

distX (ωn,K) ≤ kϕ(tn, t0, bn) − ωnk + distX (ϕ(tn, t0,B(t0)),K) 1 ≤ + dist (ϕ(t , t ,B(t )),K) → 0 as n → ∞, n X n 0 0 so distX (ωn,K) → 0. Arguing as above, there is a subsequence ωnj and aω ¯ ∈ K such that ωnj → ω¯ as nj → ∞. Moreover,ω ¯ ∈ ωB(t0)(t0), since ωB(t0)(t0) is closed.

Hence, ωB(t0)(t0) is compact.

Similarly, let ω ∈ ωB(t0)(t0). Then there exist sequences ϕ(tn, t0, bn) with bn ∈ B(t0) and tn → ∞ such that kϕ(tn, t0, bn) − ωk → 0 as n → ∞. Moreover,

distX (ω, K) ≤ kω − ϕ(tn, t0, bn)k + distX (ϕ(tn, t0, bn),K)

≤ kω − ϕ(tn, t0, bn)k + distX (ϕ(tn, t0,B(t0)),K) → 0 as n → ∞, so distX (ω, K) = 0. Hence, ω ∈ K. Since the choice of ω ∈ ωB(t0)(t0) was arbitrary, this implies that ωB(t0)(t0) ⊂ K for each t0 ∈ R. 1106 HONGYONG CUI, PETER E. KLOEDEN AND MEIHUA YANG

Finally, suppose that the forward convergence (2.1) does not hold. Then there exist an ε0 > 0 and a sequence tn → ∞ as n → ∞ such that distX ϕ(tn, t0,B(t0)), ωB(t0)(t0) ≥ 2ε0, n ∈ N.

The set ϕ(tn, t0,B(t0)) may not be compact, so the supremum deﬁning the distance distX ϕ(tn, t0,B(t0)), ωB(t0)(t0) = sup distX y, ωB(t0)(t0) y∈ϕ(tn,t0,B(t0)) need not be attained. However, for each n ∈ N there is a point yn ∈ ϕ(tn, t0,B(t0)) such that distX ϕ(tn, t0,B(t0)), ωB(t0)(t0) − ε0 ≤ distX yn, ωB(t0)(t0) ≤ distX ϕ(tn, t0,B(t0)), ωB(t0)(t0) , which means that distX yn, ωB(t0)(t0) ≥ ε0, n ∈ N.

Since yn ∈ ϕ(tn, t0,B(t0)) and

distX (yn,K) ≤ distX (ϕ(tn, t0,B(t0)),K) → 0 as n → ∞, there exists a convergent subsequence ynk → y¯ ∈ K. Moreover,y ¯ ∈ ωB(t0)(t0) by deﬁnition. However, distX ynk , ωB(t0)(t0) ≥ ε0, so distX y,¯ ωB(t0)(t0) ≥ ε0, which is a contradiction. Hence the forward convergence (2.1) must hold.

3. Asymptotic invariance. Let B = {B(t), t ∈ R} be the ϕ-positive invariant family of closed and bounded subsets of X in Assumption1. Then Assumption2 through Lemma 2.2 ensures that the omega limit set [ ωB := ωB(t0)(t0) ∗ t0≥T of B is nonempty, compact and included in K. Since the sets ωB(t0)(t0) are increasing with t0 this union is in fact a limit. −t The simple examplex ˙ = −x + e with ωB = {0} shows that the set ωB need not be invariant or even positive invariant. 3.1. Asymptotic positive invariance. Deﬁnition 3.1. A set A is said to be asymptotically positive invariant if for any monotonic decreasing sequence εp → 0 as p → ∞ there exists a monotonic increasing sequence Tp → ∞ as p → ∞ such that

ϕ (t, t0,A) ⊂ Bεp (A) , t ≥ t0, (3.1) for each t0 ≥ Tp, where Bεp (A) := {x ∈ X : distX (x, A) < εp}. The following result was proved in [12] for difference equations and in ordinary differential equations in [9]. The proof is similar, but with critical difference due to the fact that the absorbing family is not just one set or compact. It requires an additional assumption. Assumption1 implies that the compact set K in Assumption2 is absorbed into

B = {B(t), t ∈ R}. Speciﬁcally, ϕ(τ, t0,K) ⊂ B(τ) for τ ≥ TK,t0 + t0. By the 2-parameter semigroup property,

ϕ(t, t0,K) = ϕ(t, τ, ϕ(τ, t0,K)) ⊂ ϕ(t, τ, B(τ)), t ≥ τ ≥ TK,t0 + t0, FORWARD OMEGA LIMIT SETS 1107 so

distX (ϕ(t, t0,K),K) ≤ distX (ϕ(t, τ, B(τ)),K) → 0 as t → ∞. (3.2) The following stronger assumption is needed to ensure limiting dynamics remains asymptotically compact. It essentially says that compact subset K is asymptotically positive invariant. Assumption 3. The compact subset K of X in Assumption2 is such that, for every sequences t0,j ≤ tj with t0,j → ∞,

lim distX (ϕ(tj, t0,j,K),K) = 0. j→∞

Theorem 3.2. Let Assumptions1,2 and3 hold. Then ωB is asymptotically positive invariant.

Proof. For ε1 > 0 ﬁxed, we prove by contradiction that there exists t1 = t1(ε1) large enough such that

distX (ϕ(t, t0, ωB), ωB) < ε1 for t ≥ t0 ≥ t1(ε1). (3.3)

If it is not the case, then there are sequences t0,j ≤ tj ≤ t0,j + T0(t0,j, ε1) with t0,j → ∞ as j → ∞ such that

distX (ϕ(tj, t0,j, ωB), ωB) ≥ ε1, j ∈ N.

Since ωB is compact and x0 7→ ϕ(tj, t0,j, x0) is continuous for each ﬁxed j, the set ϕ(tj, t0,j, ωB) is compact, so there exists an ωj ∈ ωB ⊂ K such that

distX (ϕ(tj, t0,j, ωj), ωB) = distX (ϕ(tj, t0,j, ωB), ωB) ≥ ε1, j ∈ N.

Deﬁne yj := ϕ(tj, t0,j, ωj). Then yj ∈ ϕ(tj, t0,j,K), so

distX (yj,K) ≤ distX (ϕ(tj, t0,j,K),K) . Hence, by Assumption3,

distX (yj,K) → 0 as j → ∞. As in the proof of Lemma 2.2 it follows that there exists a convergent subsequence yjk → y¯ ∈ K. Moreover,y ¯ ∈ ωB by the deﬁnition. However, distX (yj, ωB) ≥ ε1, so distX (¯y, ωB) ≥ ε1, which is a contradiction. Thus for this ε1 > 0 there exists t1 = t1(ε1) large enough such that (3.3) holds. Repeating inductively with εp+1 < εp and tp+1(εp+1) > tp(εp), it follows that ωB is asymptotically positively invariant.

3.2. Asymptotic negative invariance. Deﬁnition 3.3. A set A is said to be asymptotic negatively invariant if for every ∗ a ∈ A, ε > 0 and T > 0, there exist tε ≥ T + T and aε ∈ A such that

kϕ (tε, tε − T, aε) − ak < ε. To show that this property holds requires some further assumptions on the future uniform behaviour of the process. Let B = {B(t), t ∈ R} be the ϕ-positive invariant family of closed and bounded subsets of X in Assumption1. First, the asymptotic compactness in Assumption2 needs to be strengthened to strongly asymptotically compact. 1108 HONGYONG CUI, PETER E. KLOEDEN AND MEIHUA YANG

Assumption 4. The nonautonomous system ϕ is strongly asymptotically compact for the ϕ-positive invariant family B = {B(t), t ∈ R} of closed and bounded subsets of X in the sense that there is a compact subset K of X such that for every sequences t0,j < tj with t0,j → ∞ nd tj − t0,j → ∞,

lim distX (ϕ(tj, t0,j,B(t0,j)),K) = 0. j→∞

An alternative version of Assumption4 would be to assume that the sets B(t) are compact rather than just closed and bounded. However, in practice, it is much harder to show that a set is compact. The current version holds when the system has a Vishik uniform attractor and the sets B(t) are uniformly bounded. Note that Assumption4 implies Assumption2. Indeed, if dist X (ϕ(tj, t0,B(t0)), K) does not converge to 0 for some sequence tj → ∞, a contradiction will occur as by positive invariance and Assumption4 we have distX (ϕ(tj, t0,B(t0)),K) = distX ϕ(tj, tj − 1, ϕ(tj − 1, t0,B(t0))),K

≤ distX (ϕ(tj, tj − 1,B(tj − 1)),K) → 0, as j → ∞.

We also need the process to be continuous in its initial state on time intervals of bounded length uniformly in the starting time.

Assumption 5. The mapping x0 7→ ϕ(t, t0, x0) is continuous in x0 ∈ B(t0) ∪ K uniformly on any time interval of ﬁnite length, i.e. for every ε > 0 and T > 0 there exists δ = δ(ε, T ) such that

kϕ(t, t0, x0) − ϕ(t, t0, y0)k < ε for kx0 − y0k < δ, x0, y0 ∈ B(t0) ∪ K (3.4)

∗ for all t ∈ [t0, t0 + T ] and t0 ≥ T .

Theorem 3.4. Let Assumptions1,3,4 and5 hold. Then ωB is asymptotic negatively invariant.

Proof. To show this let ω ∈ ωB, ε > 0 and T > 0 be given. Then there exist sequences bn ∈ B(τn) and τn < tn with τn → ∞, tn − τn → ∞, and an integer N(ε) such that 1 kϕ (t , τ , b ) − ωk < ε, n ≥ N(ε). n n n 2

Define an := ϕ (tn − T, τn, bn). Then by Assumption4, there exists a convergent subsequence anj := ϕ tnj − T, τnj , bnj → ωε ∈ K as nj → ∞. By definition, ωε ∈ ωB. From Assumption5 the process ϕ is continuous in initial conditions uniformly on finite time intervals of the same length, i.e., (3.4). Hence, for Nb(ε, T ) large enough,

1 ϕ tnj , tnj − T, anj − ϕ tnj , tnj − T, ωε < ε, nj ≥ Nb(ε, T ). (3.5) 2 By the 2-parameter semi-group property ϕ tnj , tnj − T, anj = ϕ tnj , tnj − T, ϕ tnj − T, τnj , bnj = ϕ tnj , τnj , bnj . FORWARD OMEGA LIMIT SETS 1109

Then ω − ϕ tnj , tnj − T, ωε ≤ ω − ϕ tnj , tnj − T, anj + ϕ tnj , tnj − T, anj − ϕ tnj , tnj − T, ωε = ω − ϕ tnj , τnj , bnj + ϕ tnj , tnj − T, anj − ϕ tnj , tnj − T, ωε 1 1 < ε + ε = ε. 2 2 This is the desired result.

Remark 3.5. Assumptions1-5 can be satisﬁed, for example, for non-autonomous dynamical system ϕ which is asymptotically autonomous satisfying, for every sequences t0,j ≤ tj with t0,j → ∞ and bounded set E,

sup distX (ϕ(tj, t0,j, x), φ(tj − t0,j, x)) → 0, j → ∞, (3.6) x∈E where φ is the limit semigroup, with global attractor A. In this case, the compact set K can be taken as the global attractor A, provided that the ϕ-positive invariant family B of closed and bounded subsets is uniformly bounded, i.e., ∪t≥T ∗ B(t) is du 3 bounded. For example, the reaction-diffusion equation dt − 4u + u + u = g(x, t) 2 ∗ 2 defined on R satisfies (3.6), if g ∈ Lloc(T , ∞; L (R)) is asymptotically autonomous 2 such that for some g0 ∈ L (R)

Z tj c(tj −s) 2 e kg(s) − g0k ds → 0, j → ∞, t0,j for all t0,j ≤ tj with t0,j → ∞. This condition is quite strong, but others are possible. A simple example is for the switching term g(x, t) = −g0(x) for t < 0 and g(x, t) = g0(x) for t ≥ 0.

4. Upper semi continuity in a parameter. Now consider a parameterised fam- ν ∗ ily of processes ϕ (t, t0, x0), where ν ∈ [0, ν ], on X. Assumptions1,3 and4 need to be strengthened so the parameterised family of processes is absorbing in a family B = {B(t), t ∈ R} of closed and bounded subsets of X uniformly in ν ∈ [0, ν∗] and is asymptotically compact for B with a common compact set K for all ν ∈ [0, ν∗].

Assumption 6. There exists a family B = {B(t), t ∈ R} of closed and bounded subsets of X which is ϕν -positive invariant for each ν ∈ [0, ν∗] such that for any ∗ bounded subset D of X and every t0 ≥ T there exists a TD,t0 ≥ 0 (independent of ν) for which

ν ∗ ϕ (t, t0, x0) ∈ B(t), ∀t ≥ t0 + TD,t0 , x0 ∈ D, ν ∈ [0, ν ]. Assumption 7. Each system ϕν , ν ∈ [0, v∗], satisﬁes Assumptions3 and4 for the same compact set K. In addition, the following uniform continuous convergence of the processes in the parameter is needed. 1110 HONGYONG CUI, PETER E. KLOEDEN AND MEIHUA YANG

Assumption 8. For every ε > 0 and T > 0 there exists a δ(ε, T ) > 0 such that

ν 0 ϕ (t, t0, b) − ϕ (t, t0, b) < ε, t0 ≤ t ≤ t0 + T, b ∈ B(t0) ∪ K,

∗ for |ν| < δ(ε, T ) and t0 ≥ T .

0 0 Finally, the uniform attraction of the set ωB for the system ϕ is also required.

0 Assumption 9. ωB uniformly attracts K, i.e., for every ε > 0 there exists a T (ε), ∗ which is independent of t0 ≥ T , such that

0 0 ∗ distX ϕ (t, t0,K) , ωB < ε, t ≥ t0 + T (ε), t0 ≥ T . Under the above assumptions the upper semi continuous convergence of the omega limit sets in the parameter holds.

Theorem 4.1. Suppose that Assumptions6,7,8 and9 hold. Then

ν 0 distX ωB, ωB → 0, as ν → 0. Proof. A proof by contradiction will be used. Suppose for some sequence of parameters νj → 0 that the above limit is not true, i.e., there exists an ε0 > 0 such that

νj 0 distX ωB , ωB ≥ 4ε0, j ∈ N.

νj νj Since ωB is compact, there exists ωj ∈ ωB such that

0 νj 0 distX ωj, ωB = distX ωB , ωB ≥ 4ε0, j ∈ N. (4.1)

∗ By Assumption9 there is a T = T (ε0) such that for any t0 ≥ T

0 0 distX ϕ (t0 + T, t0,K) , ωB < ε0.

Then use Assumption8 with this T to pick a νj < δ(ε0,T ) to ensure that

νj 0 ∗ ϕ (t0, t0 − T, b) − ϕ (t0, t0 − T, b) < ε0, b ∈ B(t0 − T ) ∪ K, t0 ≥ T + T.

νj Fix such a νj and use the asymptotical negative invariance of ωB to obtain the νj j ∗ existence of an ωj,T ∈ ωB ⊂ K and a tε ≥ T + T so that

νj j j ϕ tε, tε − T, ωj,T − ωj < ε0.

j Then, with t0 taken as tε above,

0 νj j j distX ωj, ωB ≤ ωj − ϕ tε, tε − T, ωj,T

νj j j 0 j j + ϕ tε, tε − T, ωj,T − ϕ tε, tε − T, ωj,T

0 j j 0 +distX ϕ tε, tε − T, ωj,T , ωB

< ε0 + ε0 + ε0 = 3ε0, which contradicts the assumption (4.1). FORWARD OMEGA LIMIT SETS 1111

5. Nonautonomous forward attractors. A ϕ-invariant family A = {A(t), t ∈ R} of nonempty compact subsets of X is called a forward attractor if it forward attracts all families D = {D(t), t ∈ R} of nonempty bounded subsets of X, i.e.,

lim distX (ϕ(t, t0,D(t0)),A(t)) = 0, (ﬁxed t0). (5.1) t→∞ A forward attractor A = {A(t), t ∈ R} is Lyapunov asymptotic stable, i.e., both forward attracting (5.1) and Lyapunov stable. Simple examples show that a forward attractor need not be unique, even when uniformly bounded [10, 12], but can be asymptotically unique [5]. It is possible to construct the component subsets of candidates for forward attractors in the same way as for a pullback attractor [10]. This is based on the observation that a ϕ-positively invariant family of nonempty compact subsets con- tains a maximal ϕ-invariant family of nonempty compact subsets. Essentially, it is formed by all the entire trajectories in the ϕ-positively invariant family. When the component subsets of the absorbing family B = {B(t), t ∈ R} are only closed and bounded but not compact, an additional assumption such as pullback asymptotic compactness of ϕ is needed to ensure that the intersection of nested subsets in (5.2) are nonempty, i.e. the sequences {ϕ(t, t0,n, b0,n)}n∈N are precompact for all b0,n ∈ B(t0,n) and t0,n → −∞. Then the following theorem adapted from [10] holds. Theorem 5.1. Suppose that a process ϕ on X has a ϕ-positively invariant family B = {B(t), t ∈ R} of nonempty closed and bounded subsets of X and is pullback asymptotically compact on B. Then ϕ has a maximal ϕ-invariant family A = {A(t), t ∈ R} in B of nonempty compact subsets of X determined by \ A(t) = ϕ (t, t0,B(t0)) for each t ∈ R. (5.2) t0≤t This pullback construction is used only inside the ϕ-positively invariant family B. It is equivalent to

lim distX (ϕ(t, t0,B(t0)),A(t)) = 0, (ﬁxed t). (5.3) t0→−∞ Theorem 5.1 does not, however, imply that the subsets given by (5.2) form a pullback attractor, since nothing has been assumed about what is happening outside of the sections B(t) of B. With the additional assumption that B is pullback absorbing, then the family A is a pullback attractor. To ensure that the family A constructed by the pullback method in (5.2) is forward attracting the set of omega limit points of solutions starting in A needs to coincide with the set of omega limit points of solutions starting in the family B, rather than to be just a proper subset. In addition, the family B should be forward absorbing.

5.1. Conditions ensuring forward convergence. Let B = {B(t), t ∈ R} be ϕ- positively invariant family of nonempty closed and bounded subsets of X and recall that for each t0 ∈ R, the forward omega limit set starting in B(t0) at times t0 is deﬁned by \ [ ωB(t0)(t0) := ϕ (s, t0,B(t0)).

t≥t0 s≥t 1112 HONGYONG CUI, PETER E. KLOEDEN AND MEIHUA YANG

In addition, [ ωB := ωB(t0)(t0). ∗ t0≥T Suppose that Assumptions1 and2 hold. Then the above sets are nonempty compact subsets of K and, by Lemma 2.2, lim distX ϕ(t, t0,B(t0)), ωB(t )(t0) = 0, (ﬁxed t0). (5.4) t→∞ 0

Since A(t0) ⊂ B(t0) and A(t) = ϕ(t, t0,A(t0)) ⊂ ϕ(t, t0,B(t0)), it follows that lim distX A(t), ωB(t )(t0) = 0, (fixed t0). (5.5) t→∞ 0 The set of omega limit points for dynamics starting inside the family of sets A = {A(t), t ∈ R} is defined by \ [ \ [ ωbA := ϕ(t, t0,A(t0)) = A(t), t0∈R t≥t0 t0∈R t≥t0 which is nonempty and compact as a family of nested compact sets. Obviously, ωbA ⊂ ωB ⊂ K. The inclusions here may be strict. The existence of omega limit points in ωB that are not in ωbA means that A cannot be forward attracting from within B. The converse also holds. The following theorem is adapted from [10]. Theorem 5.2. Suppose that Assumptions1,3,4 and5 hold and that the assumptions of Theorem 5.1 hold. Then A is forward attracting from within B, i.e., the forward convergence (5.1) holds, if and only if ωbA = ωB. Proof. (Sufficiency) Suppose that the forward convergence (5.1) does not hold. Then there is an ε0 > 0 and a sequence tn → ∞ as n → ∞ such that

distX (ϕ(tn, t0,B(t0)),A(tn)) ≥ 2ε0 for all n ∈ N.

The set ϕ(tn, t0,B(t0)) is not compact, so the supremum deﬁning the distance

distX (ϕ(tn, t0,B(t0)),A(tn)) = sup distX (y, A(tn)) y∈ϕ(tn,t0,B(t0)) need not be attained. However, arguing as in the proof of Lemma 2.2, for each n ∈ N there is point yn ∈ ϕ(tn, t0,B(t0)) such that

distX (ϕ(tn, t0,B(t0)),A(tn)) − ε0 ≤ distX (yn,A(tn))

≤ distX (ϕ(tn, t0,B(t0)),A(tn)) , which means that distX (yn,A(tn)) ≥ ε0, n ∈ N. Now kyn − ank ≥ distX (yn,A(tn)) ≥ ε0, n ∈ N, for any points an ∈ A(tn). Since the an, yn ∈ ϕ(tn, t0,B(t0)) and

distX (an,K) ∨ distX (yn,K) ≤ distX (ϕ(tn, t0,B(t0)),K) → 0 as n → ∞, there exist convergent subsequences ynj → y¯ ∈ K and anj → a¯ ∈ K. It follows that ky¯ − a¯k ≥ ε0. From the deﬁnitionsy ¯ ∈ ω (t ) ⊂ ω anda ¯ ∈ ω . Since the a and hencea ¯ B(t0) 0 B bA n were otherwise arbitrary, it follows that

distX (ωB, ωbA) ≥ distX (¯y, ωbA) ≥ ε0. FORWARD OMEGA LIMIT SETS 1113

Hence ωbA 6= ωB. (Necessity) This will also be proved by contradiction. Suppose ωbA 6= ωB. Since ωbA ⊂ ωB, there is a point x ∈ ωB \ ωbA. Without loss of generality, assume that x ∈ ω (t ) for some t ≥ T ∗ such that dist (x, ω ) ≥ 4δ for some δ > 0. B(t0) 0 0 X bA 0 0 By the forward attraction, distX φ(t, t0,B(t0)),A(t) < δ0 for all t large enough. Since x ∈ ωB(t0)(t0) indicates that there are sequences xn ∈ B(t0) and tn → ∞ such that φ(tn, t0, xn) → x as n → ∞, we have

distX (x, A(tn)) ≤ distX (x, φ(tn, t0, xn)) + distX (φ(tn, t0,B(t0)),A(tn)) < 2δ0 for tn large enough. This implies \ [ distX x, A(tm) ≤ 2δ0. n∈N m≥n Hence, \ [ distX (x, ωbA) ≤ distX x, A(tm) ≤ 2δ0, n∈N m≥n which contradicts the assumption that distX (x, ωbA) ≥ 4δ0. Remark 5.3. Generally, the existence of forward attractors is fairly hard to verify in general applications. In [6], several conditions are given for asymptotically autonomous dynamical systems to ensure the tails of the pullback attractor converge forwards and backwards to the global attractor of the limit semigroup. In this case, i.e., if limt→∞ distH (A(t), A) = 0 and limt→−∞ distH (A(t), A) = 0, where distH is the full Hausdorﬀ metric, A = {A(t)}t∈R is the pullback attractor and A is the global attractor of the limit semigroup, then the pullback attractor A is the forward attractor by Theorem 5.2 since ωbA = ωB = A. Several particular ODE examples of forward attractors are also given in [5].

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