Equi-attraction and continuity of attractors for skew-product semiflows

In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated to non-autonomous dynamical systems. To this aim proper notions of equi-attraction have to be introduced in phase spaces where the driving systems depend on a parameter. Results on the upper and lower-semicontinuity of uniform and cocycle attractors are relatively new in the literature, as a deep understanding of the internal structure of these sets is needed, which is generically difficult to obtain. The notion of lifted invariance for uniform attractors allows us to compare the three types of attractors and introduce a common framework in which to study equi-attraction and continuity of attractors. We also include some results on the rate of attraction to the associated attractors.

1. Introduction and preliminaries. The skew-product semiflow is an important tool to understand the dynamics of some non-autonomous differential equations (see [23,24,25]). It consists of a very ingenious way of tracking the non-autonomous nature of the equation into an autonomous equation in a product space. Autonomous equations are nowadays quite well known and there are many results on local and global existence and regularity of solutions, also on existence of attractors and stability under perturbations, [1,3,6,11,13,20,27].
In order to analyse the asymptotic behavior of solutions, we concentrate our attention on the attractors. These are subsets of the phase space which contain most of the important information concerning the long time behavior of solutions. There are several notions of attractor for non-autonomous problems. On one hand, the pullback and the cocycle attractors refer to an invariant family of time-dependent sets (with the pullback attraction property) containing all global bounded solutions and, consequently, all interesting structures closely related to its dynamics. On the other hand, the uniform attractor is a compact bounded subset of the phase space which attracts bounded sets in the forwards sense, but not invariant. The pullback and the uniform attractors are different in general. All these notions can be related with the aid of the concept of skew-product semiflow, see [23,24,26].
The relation between equi-attraction and continuity of attractors has not been clarified yet for uniform attractors. One of the aims of this paper is to study the relationship between equi-attraction and continuity of uniform, cocycle and pullback attractors. To this end we will treat all of these attractors in a skewproduct semiflow framework. Though the skew-product semiflows are semigroups in a product space, note that, under perturbation, the phase space changes with the parameter, which was not considered in the previous literature.
Before going further, we will introduce some necessary concepts, definitions, terminology and results which will be crucial for our analysis.
Let A and B be two subsets of the metric space (X, d). We denote by dist(A, B) the Hausdorff semidistance between two subsets, that is, dist(A, B) = sup a∈A d(a, B) and we write dist H (A, B) = dist(A, B) + dist(B, A) for the Hausdorff distance.
Skew-product semiflows appear in a natural way when studying non-autonomous differential equations. Indeed, let us consider a family of differential equations in a Banach space (X, · X ). Let f ∈ C b (R × X, X), where C b (R × X, X) is the space of all continuous and bounded maps from R × X to X with a suitable metric ρ.
Assume that {θ t : t ∈ R} is a group in C b (R × X, X) where (θ t φ)(s, x) := φ(t + s, x), for all t, s ∈ R and x ∈ X and φ ∈ C b (R × X, X). Let us denote by Σ f the closure of {θ t f : t ∈ R} in the metric space (C b (R × X, X), ρ).
For a given σ ∈ Σ f we consider the system and for x 0 ∈ X, let x(t, x 0 , σ) be the solution of (1) at time t ∈ R with nonautonomous function σ ∈ Σ f and initial condition x(0, x 0 , σ) = x 0 . We will use the notation ϕ(t, σ)x 0 := x(t, x 0 , σ), when the non-autonomous function belongs to Σ f . Note that these solutions satisfy, for all x 0 ∈ X and σ ∈ Σ η , the following properties: 3. ϕ(t + s, σ)x 0 = ϕ(t, θ s σ)ϕ(s, σ)x 0 , the cocycle property. A map ϕ : [0, ∞) × X × Σ η → X together with a group {θ t : t ∈ R} with the properties above is called a non-autonomous dynamical system (or nds for short) and is denoted by (ϕ, θ) (X,Σ f ) . However, the are situations in which we can have a set of mathematical elements {X, Σ f , ϕ, θ} satisfying the precedent properties and not being necessarily associated to a differential equation in the form described above. For this reason, we will develop now an abstract theory in which we do not need any relationship to any differential equation, although our primary interest will be related to one of them.
Thus, in an abstract way, suppose that (C, ρ) is a complete metric space with a group {θ t : t ∈ R}. Take a family {σ η : η ∈ [0, 1]} in C with the property that σ η → σ 0 , as η → 0 + . Let us assume that For η ∈ [0, 1], we denote by Γ η = {θ t σ η : t ∈ R} the orbit of σ η by {θ t : t ∈ R} on C. Let Σ η be the closure of Γ η and we denote We can now define the concept of uniform attractor for a non-autonomous dynamical system. It is the minimal closed set that attracts all solutions uniformly with respect the non-autonomous functions on Σ η and with respect to initial values x 0 in bounded subsets of X. Precisely: Definition 1.1. Suppose (ϕ η , θ) (X,Ση) is a non-autonomous dynamical system. A uniform attractor A η (if it exists) for this system is the minimal closed subset of X such that sup σ∈Ση dist(ϕ η (t, σ)B, A η ) → 0, as t → +∞, for each bounded subset B of X.
A recent characterization of a uniform attractor ( [2]) is given by the property of lift invariance. A global solution through x ∈ X and σ ∈ Σ η for the nds (ϕ η , θ) (X,Ση) is a map ξ : R → X that satisfies, for all t ≥ s, ϕ η (t − s, θ s σ)ξ(s) = ξ(t) and ξ(0) = x, Definition 1.2. We say a subset M of X is lifted invariant by the nds (ϕ η , θ) (X,Ση) if for every x ∈ M there is σ ∈ Σ η and a global solution ξ : R → X through x and σ in M . Moreover, we say that M is an isolated lifted invariant if there is a neighborhood U of M such that M is the maximal lifted invariant set on U. Proposition 3.21] The uniform attractor of the non-autonomous dynamical system (ϕ η , θ) (X,Ση) , if it exists, is the maximal bounded isolated lifted invariant set of X. Now let us define the skew-product semiflow associated to a non-autonomous dynamical system (ϕ η , θ) (X,Ση) . For t ≥ 0 and (x, σ) ∈ X × Σ η let It is easy to see that {π η (t) : t ≥ 0} is a semigroup on X × Σ η .
The existence of uniform attractor for the non-autonomous dynamical system (ϕ η , θ) (X,Ση) is equivalent to the existence of a global attractor for the skew-product semiflow, see [2,Proposition 3.1]. Furthermore, we have the relation Π X A η = A η , where A η denotes the global attractor for the skew-product, Π X is the continuous first coordinate projection of X × Σ η and A η the uniform attractor of the nonautonomous dynamical system.
There is also a relation between uniform attractors and pullback attractors which is shown by means of cocycle attractors. A non-autonomous set is a family of subsets of X which is indexed in Σ η , formally {B(σ) ⊂ X : σ ∈ Σ η }. We say the non-autonomous set is bounded, closed or compact, if every fiber B(σ) is bounded, closed or compact on X, respectively. Ση) is a non-autonomous dynamical system and {θ t : t ∈ R} is a driving group over Σ η . We say that the compact nonautonomous set {A η (σ) : σ ∈ Σ η } is a cocycle attractor for the non-autonomous dynamical system if If σ∈Ση A η (σ) is precompact and the pullback attraction of (4) is uniform on σ ∈ Σ η , then the existence of the cocycle attractor is equivalent to the existence of the global attractor for the skew-product semiflow on X × Σ η , therefore to the existence of uniform attractor, see [2,Theorem 3.11]).
The family {A η (θ t σ) : t ∈ R}, where {A η (σ) : σ ∈ Σ η } is the cocycle attractor described above, is a pullback attractor for the evolution process (5); indeed, we have that In this paper we study equi-attraction and continuity of attractors for all of these different notions of attractors. More particularly, in Section 2.1 we prove that equiattraction is equivalent to continuity of global attractors for skew-product semiflows. It is remarkable that even though the skew-product is a semigroup on a product space, this space changes depending on a parameter on the driving system, so that our results cannot be deduced directly from previous ones in the literature. In Section 2.2 we show that uniform attraction of bounded subsets (uniformly with respect to parameters) is equivalent to continuity of uniform attractors, extending the results of [7] to the case of uniform attractors. Then, in Subsection 2.3 we discuss this equivalence for cocycle attractors, extending the results on [12]. In Section 3 we relate continuity of global attractors for skew-product semiflows with continuity of uniform and cocycle attractors for non-autonomous dynamical systems. An application to a non-autonomous perturbation of an autonomous differential equation is also shown to illustrate our abstract results.
2. Equi-attraction for non-autonomous systems. This section is devoted to the definition of equi-attraction on the different settings for non-autonomous systems and relate equi-attraction with continuity of the attractor.
We start by stating the assumptions for the rest of the section. Let X be a Banach space and (C, ρ) a complete metric space with a group {θ t : t ∈ R}. Then take a family of elements on C, {σ η ∈ C : η ∈ [0, 1]}, such that it holds (2) and (3) .
It is easy to see from (2) that the family is a family of non-autonomous dynamical systems and define the skew-product semiflow π η (t) : Suppose that for each η there is a global attractor A η for the skew-product semiflow. So there exists a uniform attractor A η = Π X A η for the non-autonomous dynamical system (ϕ η , θ) (X,Ση) on X.
Then the global attractor of the skew-product semiflow A η may be expressed in the product space X × Σ η in terms of the cocycle attractor. We have that A η ⊂ A η × Σ η , for all η ∈ [0, 1], but the equality does not hold in general. However, if the non-autonomous set {A η (σ) : σ ∈ Σ η } is the cocycle attractor for the nds (ϕ η , θ) (X,Ση) , then (see [4,Theorem 3.4]
for any bounded subset B of X. Then, the global attractor for the skew-product semiflow is given by As for the other, let (x, σ) ∈ A η be given. Note that, by our definition of Σ η , there is a sequence τ k ∈ R such that θ τ k σ η → σ on Σ η . We must show that there is also a sequence Indeed, if {S η σ (t, s) : t ≥ s} denotes the evolution process defined in (5), we know that x ∈ A η (σ), which is the pullback attractor for S η σ (·, ·), at time t = 0, thus there is a global solution γ : Observe that Γ := t∈R γ(t) is bounded. So, by (7), Then, as θ τ k σ η → σ in Σ η and the nds (ϕ η , θ) (X,Ση) is uniformly continuous on compact subsets of X, we have that Hence, from (9) and (10), choosing r = −t 0 and taking Therefore, by (8), , and thanks to (11) the proof is concluded.
Hypothesis (7) is needed in order to prove that existence of cocycle attractors implies existence of global attractors for skew-product semiflows and it also plays an important role on the relation of equi-attraction and continuity.
2.1. On skew-product semiflows. Skew-product semiflows are semigroups in a product space X × Σ η , so one could think that we can just apply the known results for continuity and equi-attraction in [12]. The difference here is that the family π η acts on the phase space X × Σ η and they change with the parameter η ∈ [0, 1]. As seen before Σ η is the closure of the orbit of an element of a fixed metric space C (which in the case of the non-autonomous differential equation is C b (R × X, X)), thus possessing a common metric.
2.1.1. Equi-attraction and continuity. We prove here results of the equivalence between equi-attraction and continuity for skew-product semiflows.
Consider (Λ, d Λ ) a metric space (the parameter space). We recall that a family of subsets {C λ } λ∈Λ of the metric space (X, d) is said to be If {C λ } λ∈Λ is both upper and lower semicontinuous at λ 0 it is said to be continuous at λ 0 .
Suppose a continuity hypothesis on the skew-product semiflows, that is for any compact subset K of X and We say that the family of global attractors (12) and (13). If {A η : η ∈ [0, 1]} is equi-attracting, then this family of attractors is continuous at η = 0, that is, from the equi-attraction of the global attractors.
On the other hand, our continuity assumption (12) on the skew-product semiflows implies that For the other term of the Hausdorff distance, notice that Thanks to both inequalities the theorem is proved.
We emphasize here that it is not an easy task to define and prove a result as above when the family of semigroups is defined on phase spaces that vary with the parameter. The problem arises as it is not possible to define a suitable notion of continuity of semigroups. An important feature of skew-product semiflows is that even though the phase space changes, there is a common one such that we are able to keep fixed.
We recall that continuity of global attractors does not imply that the family is equi-attracting, so we need the following additional hypothesis on the semigroups.
In this case, we ask η∈[0,1] Σ η to be precompact in C, therefore it is bounded.
Again, as η∈[0,1] Σ η is precompact in C and Σ η is invariant over {θ t : t ≥ 0}, every sequence {σ k } k∈N (or {θ τ k σ k } k∈N ) has a convergent subsequence. Thus, in order to verify collective asymptotic compactness we may only care about the term of {π η k (t k )(x k , σ k )} that lies on X. More precisely, if for every sequences t k → ∞ and {x k } bounded in X, for which the corresponding sequence {ϕ η k (t k , σ k )x k } is also bounded in X, we have that the sequence {ϕ η k (t k , σ k )x k } admits a convergent subsequence, then the family of skew-product semiflows is collectively asymptotically compact.
The following result is the converse to Theorem 2.2. Proof. Suppose by contradiction that there exist > 0, sequences for all k ∈ N. Note at first that there exists a sequence σ k ∈ Σ η k , although, as {θ τ σ η k : τ ∈ R} is dense on Σ η k and the skew-product semiflow is continuous, we can assume that the sequence in (17) may be taken in {θ τ σ η k : τ ∈ R}.

2.1.2.
Rates of convergence. We can use the equi-attraction for semigroups to obtain a better estimate and rate of convergence of the global attractors for the skewproduct semiflows. This is a consequence of the theorem for semigroups.
Proof. Fix t ≥ 0. Remember that A η = π η (t)A η and that B = Π X B is precompact in X. Thus, we have that Since B ⊂ η∈[0,1] B × Σ η , the above inequalities ensure that Analogously, we have that In order to complete the proof, from (21) and (22) we choose ≤ ζ 0 and t = ζ −1 ( ). Then, there is a constant c > 0 such that 2.2. On uniform attractors. In this section we state a definition for equi-attraction in the sense of uniform attractors and show the equivalence between equiattraction and continuity for uniform attractors. We exploit the connection between uniform attractors and global attractors for skew-product semiflows, and also prove that continuity for attractors of skewproduct semiflow is equivalent to continuity of uniform attractors, as long as the driving group also possesses continuous attractors.
The sense of equi-attraction we are going to use for uniform attractors of a family of non-autonomous dynamical systems is defined as follows: We define this concept as uniform equi-attraction since the uniform attractor A η of (ϕ η , θ) (X,Ση) is the minimal closed set of X that uniformly (for σ ∈ Σ η ) attracts all bounded subsets of X. 1], and assume that

Uniform equi-attraction and continuity of uniform attractors. Let us consider a family of non-autonomous dynamical systems
Moreover, suppose that for all compact sets K of X and any T > 0 we have The proof of the following result follows the same steps as in Theorem 2.2. However there are some differences which require our attention. For this reason, we prefer to fully prove it. Theorem 2.9. Assume that the family of uniform attractors {A η : η ∈ [0, 1]} associated to the non-autonomous dynamical systems {(ϕ η , θ) (X,Ση) : η ∈ [0, 1]} uniformly equi-attracts bounded subsets of X, and that (25) and (26) hold. Then the family of uniform attractors is continuous at η = 0, that is Proof. First, observe that the subset B = η∈[0,1] A η is bounded in X. Therefore, From (26) there is 0 < η 0 < 1 such that, for all η ≤ η 0 , From the lifted invariance property of the uniform attractors and Lemma 2.1 it is clear that A η ⊆ σ∈Ση ϕ η (t, σ)A η , from the continuity of the nds {(ϕ η , θ)} (X,Ση) we have that Now, to prove that dist(A 0 , A η ) → 0 when η → 0, we use similar arguments. As above, we have that And we recall that, as in (28), equi-attraction implies that dist(ϕ η (t 0 , σ)A 0 , A η ) ≤ /2 for every σ ∈ Σ η and η ∈ [0, 1]. Also, the first term on the inequality can be estimated in a similar way using our continuity hypothesis.
In order to prove the converse, as done before in [7], we need more uniformity on the assumptions on the non-autonomous dynamical systems. Below we give an expected definition that is made necessary in the proof.
for every bounded subset B of X.
More precisely, compactness and invariance of Σ imply that these definitions are equivalent, as we discussed at Subsection 2.1.
The family of non-autonomous dynamical systems is uniformly bounded, so Fix t > 0 and define the sequence s k := t k −t > 0. From the collective uniform asymptotic compactness property the sequence {ϕ η k (s k , θ τ k σ η k )x k : k ∈ N} has a convergent subsequence, and assume without loss of generality that ϕ η k (s k , θ τ k σ η k )x k → b in X, when k → ∞.
It follows from the uniform asymptotic compactness and from (26), that for every t > 0 there exists b ∈ B and σ ∈ Σ 0 such that The above contradicts the fact that A 0 is the global attractor for ϕ 0 on X.

2.2.2.
Rates of convergence. One interesting remark about equi-attraction is that if we have an explicit bound for the attraction we are able to transfer it to get an upper bound on the closeness of the attractors. As expected, we have also this result in the case of uniform attractors.

Theorem 2.13. Consider a family of non-autonomous dynamical systems
Assume that there exists a strictly decreasing function ζ : for all t ≥ 0. Suppose also that where E η (t) → 0, as η → 0, for each t.
Proof. First, let us fix t ≥ 0. Note that A η ⊂ σ∈Ση ϕ η (t, σ)A η , so we must have Taking ≤ ζ 0 and letting t = ζ −1 ( ) we combine the inequalities above to deduce as it was claimed.

Recall that
The uniform equi-attraction of cocycle attractors is in a way stronger than the uniform attractors. That is made clear below, since continuity of cocycle attractors implies continuity of uniform attractors but the converse is not true in general.

2.3.1.
Equi-attraction and continuity of cocycle attractors. We recall that the family of cocycle attractors have a close relation to pullback attractors. To be more clear, if we consider, for σ ∈ Σ η , the evolution process then if S η σ has pullback attractor {A σ (t) : t ∈ R}, for all t ∈ R, if uniqueness is assumed it is not hard to see that A σ (t) = A η (θ t σ), the right one being the cocycle attractor described above.
Here we define the equi-attraction for a family of cocycle attractors {A η (σ) : σ ∈ Σ η }. Indeed it is not the same equi-attraction for evolution processes, since we are asking for a uniformity on the family of processes. This will allow us to gain some uniformity on the continuity of attractors.
for all bounded subsets D ⊂ X.
Even more, one can relate Definition 2.15 with Definition 2.8, since A η = σ∈Ση A η (σ). Recall that {θ t : t ∈ R} is a group over Σ η , which is invariant. Therefore, for all fixed t, as we vary θ −t σ on Σ η we are covering all Σ η , then we may exchange the supreme of θ −t σ at (35) by σ ∈ Σ η to obtain that the family of uniform attractors {A η : η ∈ [0, 1]} equi-attracts bounded subsets of X. The converse, however, does not need to be true, because attraction of the union σ∈Ση A η (σ) does not guarantee that the fiber of the non-autonomous set A η (σ) attracts bounded subsets by the "action of" σ.
In the same way as before, equi-pullback attraction implies continuity of cocycle attractors.
As the proof follows the same steps as those in theorems 2.5 and 2.12, we will omit them. The reader may check the paper [7] for a similar proof.

2.3.2.
Rates of convergence. As one can imagine, a rate of equi-attraction is transferred to the proximity of the cocycle attractors too. This is what is shown in the next theorem.
for all t ≥ 0. Moreover, suppose that with E η (t) → 0, as η → 0, for each t ≥ 0. Then The proof is analogous to the proof of theorems 2.6 and 2.13.
Then there exists a constant c > 0 such that 3. Relationships on the continuity of attractors. In this final section we want to relate continuity among the different kind of attractors described above. We know how existence of global attractors for skew-product semiflows are almost equivalent to existence of uniform and cocycle attractors for non-autonomous dynamical systems. Our question now is, if they all exist, wether continuity of global attractors for skew-product semiflows implies that the family of uniform attractors are continuous and vice-versa. Given a family of non-autonomous dynamical systems {(ϕ η , θ) (X,Ση) } η∈[0,1] let {π η (t) : t ≥ 0} η∈[0,1] denote the skew-product semiflow associated. Assume that for each η ∈ [0, 1] there exists a global attractor A η ⊂ X × Σ η for the {π η (t) : t ≥ 0}. This implies that A η = Π X A η is the uniform attractor for the nds (ϕ η , θ) (X,Ση) and the non-autonomous set {A η (σ)} σ∈Σ defined as A η (σ) := {x ∈ X|(x, σ) ∈ A η } is the cocycle attractor (see [4,Theorem 3.4]).
Proof. This is a straightforward consequence of Lemma 3.1 and Proposition 3.2.
Proof. Note that, from (43), we have As the families {Σ η } η∈[0,1] and {A η (θ t σ η ) : t ∈ R} η∈[0,1] are lower semicontinuous at η = 0, uniformly on t ∈ R, we are able to conclude that dist(A 0 , A η ) → 0, as η → 0, since the right hand side of (46) is controlled by the lower semicontinuity from the cocycle attractors. Now we will show an example of a non-autonomous perturbation of an autonomous dynamical system.