ON THE STRUCTURE OF THE GLOBAL ATTRACTOR FOR INFINITE-DIMENSIONAL NON-AUTONOMOUS DYNAMICAL SYSTEMS WITH WEAK CONVERGENCE

The aim of this paper is to describe the structure of global attractors for infinite-dimensional non-autonomous dynamical systems with recurrent coefficients. We consider a special class of this type of systems (the so–called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). In particular, we apply the general results obtained in our previous paper [6] to study the almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations (functional-differential equations, evolution equation with monotone operator, semi-linear parabolic equations).


Introduction
The objective of this paper is to analyze the well-known Seifert's problem for several types of infinite-dimensional non-autonomous dynamical systems with weak convergence. To be more precise, consider a differential equation (1) x = f (t, x), where f ∈ C(R × R n , R n ). Assume that the right-hand side of (1) satisfies hypotheses ensuring existence, uniqueness and extendability of solutions of (1), i.e., for all (t 0 , x 0 ) ∈ R × R n there exists a unique solution x(t; t 0 , x 0 ) of equation (1) with initial data t 0 , x 0 , and defined for all t ≥ t 0 .
Then, we can establish the following interesting problem.
Seifert's Problem (see [15] for more details): Suppose that equation (1) is dissipative and the function f is almost periodic (with respect to time). Does equation (1) possess an almost periodic solution?
Fink and Fredericson [15] and Zhikov [26] established that, in general, even when equation (1) is scalar, the answer to Seifert's question is negative.
In our previous paper [6], we included several comments concerning some aspects related to this problem, and some relevant references dealing with it. In addition, we showed that if equation (1) is weak convergent (i.e., there exists a positive number L such that lim t→+∞ |ϕ(t, x 1 , g) − ϕ(t, x 2 , g)| = 0 for all |x i | ≤ L (i = 1, 2) and g ∈ H(f )), and f is pseudo recurrent with respect to the time variable (in particular, f is recurrent, almost automorphic, Bohr almost periodic or quasi periodic), then, equation (1) admits a unique pseudo recurrent (respectively, recurrent, almost automorphic, Bohr almost periodic, quasi periodic) solution. If this solution is Lyapunov stable, then the Levinson center (the compact global attractor) is a minimal almost periodic set. If it is not Lyapunov stable, then the Levinson center contains a minimal almost periodic set, but it is not minimal (this means, in particular, that equation (1) admits a family (more than one) of solutions which are bounded on R). In [7] we generalize this result to the case of difference equations.
In this paper we will carry out a similar analysis to prove analogous results for the following three classes of differential equations: -Functional differential equations (FDEs) with finite delay. -Evolution equations x + Ax = f (t) with monotone (generally speaking nonlinear) operator A. -Semi-linear parabolic equations x + Ax = F (t, x) with linear (unbounded) operator A.
We present our results in the framework of general non-autonomous dynamical systems (cocycles) and we apply our abstract theory mainly developed in [6] to the three classes of differential equations mentioned previously.
In order not to be repetitive with our previous papers on this topic, especially [6,7], we will skip to recall preliminary definitions and results which are necessary for our analysis and refer the reader to these papers. However, to make easier the reading, we have included some of this material in Appendix A at the end of this paper.
The paper is organized as follows.
Section 2 is devoted to the study of asymptotic behavior of non-autonomous FDEs with finite delay. In particular, we give a description of the structure of the compact global attractor for weak convergent FDEs (Theorem 2.5). We study the almost periodic and asymptotically almost periodic solutions (Subsection 2.1), uniformly compatible (by the character of recurrence with the right-hand side) solutions of strict dissipative equations (Subsection 2.2), convergence and weak convergence for functional-differential equations (FDEs) with finite delay, and also the problem of existence of almost periodic solutions of uniformly dissipative FDEs are studied (Subsection 2.3).
In Sections 3 and 4 we present some results about convergence and/or weak convergence of two classes of infinite-dimensional differential equations with unbounded operators: evolution equations x + Ax = f (t) with monotone operator (generally speaking non-linear) A, and semi-linear equation x + Ax = F (t, x) with linear (unbounded) part A, respectively.

Functional differential equations (FDEs) with finite delay
Let us first recall some notions and notations concerning functional differential equations (see [16] for more details). Let r > 0, C([a, b], R n ) be the Banach space of all continuous functions ϕ : [a, b] → R n equipped with the sup-norm. If [a, b] = [−r, 0], then we set C r := C([−r, 0], R n ). Let τ ∈ R, A ≥ 0 and u ∈ C([τ − r, τ + A], R n ). We will define u t ∈ C r for all t ∈ [τ, τ + A] by the equality u t (θ) := u(t + θ), −r ≤ θ ≤ 0. Consider a functional differential equation where f : R × C r → R n is continuous.
Let us set H(f ) := {f s : s ∈ R}, where f s (t, ·) = f (t + s, ·) and by bar we denote the closure in the compact-open topology on C(R × C r , R n ).
Along with equation (3) let us consider the family of equations where g ∈ H(f ).
A function f ∈ C(R × C r , R n ) (respectively, equation (3)) is called regular (see [24]), if for every v ∈ C r and g ∈ H(f ), equation (4) admits a unique solution passing through v at the initial moment t = 0.
Below, in this section, we suppose that equation (3) is regular.
2. Taking into account item 1. in this remark, we will use below the notions of "solution" and "trajectory" for equation (3) as synonym concepts.
2.1. Weak convergent FDEs with finite delay. Consider a differential equation where f ∈ C(Y × C r , R n ), and (Y, R, σ) is a dynamical system.
Following [24], the function f ∈ C(Y × C r , R n ) (respectively, equation (5)) is said to be regular, if for all u ∈ C r and y ∈ Y , equation (5) admits a unique solution ϕ(t, u, y) passing through the point u ∈ C r at the initial moment t = 0 and defined on R + .
Thus, the triplet C r , ϕ, (Y, R, σ) is a cocycle (non-autonomous dynamical system) which is associated to (generated by) equation (5). In this case the dynamical system (Y, R, σ) is called base dynamical system (or driving system).
Example 2.2. We consider equation (3). Along with equation (3) consider the family of equations (4), where g ∈ H(f ) := {f τ : τ ∈ R} and f τ is the τ -shift of f with respect to time, i.e., f τ (t, u) := f (t + τ, u) for all (t, u) ∈ R × C r . Suppose that the function f is regular [24], i.e., for all g ∈ H(f ) and u ∈ R n there exists a unique solution ϕ(t, u, g) of equation (4). Denote by Y = H(f ) and (Y, R, σ) a shift dynamical system on Y induced by the Bebutov dynamical system (C(R × C r , R n ), R, σ). Now the family of equations (4) can be written as Below we suppose that equation (5) is regular. Equation (5) is called dissipative (see [8]), if there exists a positive number r such that (6) lim sup t→+∞ ||ϕ(t, u, y)|| < r for all u ∈ C r and y ∈ Y , where || · || is the norm in C r .
In this section we give a simple geometric condition which guarantees existence of a unique almost periodic solution and this solution, generally speaking, is not the unique solution of equation (5) which is bounded on R.
A function f ∈ C(Y ×C r , R n ) is said to be completely continuous if for any bounded subset A ⊂ C r the set f (Y × A) ⊂ R n is bounded.
Let H(f ) be compact. The following statements hold: (i) for any point x ∈ X := C r ×H(f ) there exist a neighborhood U x of the point x and a positive number l x > 0 such that π(l x , U x ) is relatively compact, i.e., the dynamical system (X, R + , π) is locally compact; (ii) if the function f is completely continuous, then for any bounded and positively invariant subset A ⊂ X there exists a positive number t 0 = t 0 (A) such that π(t 0 , A) is a relatively compact subset of X.
Proof. This assertion follows from Lemma 6. We can now state the main results in this section.
Theorem 2.5. Suppose that the following conditions are fulfilled: (i) the function f is completely continuous; (ii) equation (5) is regular and dissipative; (iii) the space Y is compact, and the dynamical system (Y, R, σ) is minimal; where ϕ(t, u i , y) (i = 1, 2) is a solution of equation (5) which is bounded on R.
Then, (i) if the point y is τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent), then equation (40) admits a unique τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent) solution ϕ(t, u y , y) (u y ∈ C r ); (ii) every solution ϕ(t, u, y) is asymptotically τ -periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically almost automorphic, asymptotically recurrent) Proof. Let C r , ϕ, (Y, R, σ) be the cocycle associated to equation (5). Denote by (X, R + , π) the skew-product dynamical system, where X := C r × Y and π := (ϕ, σ) (i.e., π(t, (u, y)) := (ϕ(t, u, y), σ(t, y)) for all x := (u, y) ∈ C r × Y and t ∈ R + ). Consider the non-autonomous dynamical system (X, R + , π), (Y, R, σ), h generated by the cocycle ϕ (respectively, by equation (5)), where h := pr 2 : X → Y . Since Y is compact, it is evident that the dynamical system (Y, R, σ) is compact dissipative and its Levinson center J Y coincides with Y . Now we will show that the skewproduct dynamical system (X, R + , π) is point dissipative. Indeed. Let x := (u, y) ∈ C r × Y = X be an arbitrary point. Notice that the set + x := {π(t, x)| t ∈ R + } is relatively compact. To this end, it is sufficient to show that the set A := pr 1 ( + x ) = {ϕ(t, u, y)| t ∈ R + } is relatively compact in the phase space C r . But the last statement follows from the complete continuity of f , the boundedness of ϕ(t, u, y) on R + , and the Arzelá-Ascoli Theorem. Thus, the ω-limit set ω x of the point x is a nonempty, compact and invariant set of (X, R + , π). Denote by Ω X := {ω x | x ∈ X}. It is easy to see from our assumptions that Ω X is a compact set. Indeed, it is sufficient to note that the set pr 1 (ω x ) = {v ∈ C r | (v, y) ∈ ω x } is a bounded set because, according to the dissipativity of equation (5), we have (8) ||v|| ≤ r for all v ∈ pr 1 (ω x ) and x ∈ X, where r is the positive number appearing in (6). Taking into account (8), the invariance of the set Ω X , and the complete continuity of f , we conclude that the set A = pr 1 (Ω X ) is relatively compact in C r and, consequently, the set Ω X is relatively compact in X. Thus, the dynamical system is point dissipative. Since, thanks to Lemma 2.3, (X, R + , π) is locally dissipative, then by Theorem 1.10 in [8, Ch. 1], it is compactly dissipative. Denote by J X its Levinson center and I y := pr 1 (J X X y ) for all y ∈ Y , where X y := {x ∈ X : h(x) = y}. According to the definition of the set I y ⊆ C r and Theorem 2.24 in [8, Ch. 2, p. 95] (see also Theorem A.1 in Appendix A), u ∈ I y if and only if the solution ϕ(t, u, y) is defined on R and relatively compact (i.e., the set ϕ(R, u, y) ⊆ C r is compact). Thus I y = {u ∈ C r : such that (u, y) ∈ J X }. It is easy to see that condition (7) means that the non-autonomous dynamical system (X, R + , π), (Y, R, σ), h is weak convergent. Now, to finish the proof of the theorem, it is sufficient to apply Theorem 3.5 in [6] (see Theorem A.4 in Appendix A) to the non-autonomous system (X, R + , π), (Y, R, σ), h generated by equation (5).
2. Under the assumptions in Theorem 2.5, there exists a unique almost periodic solution of equation (5), but equation (5) may have more than one solution defined on R and relatively compact.

Convergent
FDEs with finite delay. Let ϕ(·, φ, g) denote the solution of (4) passing through the point φ ∈ C r for t = 0 defined for all t ≥ 0.
Since the solution ϕ(t, u 0 , f ) is bounded on R + , without loss of generality, we can assume that the sequence {ϕ(τ n , u 0 , g)} is convergent and denote by v its limit. (4) is defined and bounded on R. From this fact and inequality (13) it follows that every solution of every equation (4) is bounded on R + . From Corollary 2.4 it follows that every positively semi-trajectory of the skew-product dynamical system (X, R + , π) is relatively compact.
Remark 2.8. Theorem 2.7 remains true if we replace the standard scalar product ·, · on the space R n by an arbitrary scalar product u, is a symmetric and positive defined n × n-matrix.

2.3.
Uniform dissipative FDEs with finite delay. Below we will show that if we replace assumption (10) by a stronger condition, then Theorem 2.7 is true without the requirement that there exists at least one solution which is bounded on R + . Namely, we will establish the following theorem.
Proof. According to (16) we have for all t ∈ R + and, consequently, for all t ∈ R + . From (17) we obtain for all η ∈ C(Y, C r ), g ∈ H(f ) = Y and t ∈ R + . It is clear that, under the conditions of our theorem and thanks to (19), we can define correctly a continuous mapping S t (t ∈ R + ) from C(Y, C r ) into itself and the equality holds for all t, τ ∈ R + , where • is the composition of mappings S t and S τ . Equality (20) means that the family of nonlinear operators {S t } t∈R+ forms a commutative semigroup. Let now γ i ∈ C(Y, C r ) (i = 1, 2). Then, according to inequality (18), we have is the Lipschitz constant of F ) and, consequently, for t > 0 the mapping S t is a contraction. Since the semigroup {S t } t∈R+ is commutative, then it admits a unique fixe point γ, i.e., γ(σ(t, g)) = ϕ(t, γ(g), g) for all g ∈ H(f ) and t ∈ R + . Thus the first statement of our theorem is proved.
The second statement follows from the inequality (18). In fact, we have for all g ∈ H(f ), t ∈ R + and u ∈ C r . Passing to the limit in (22) we obtain the necessary statement. The result is completely proved.
Proof. This statement follows from Theorem 2.9.
Remark 2.11. 1. Actually Theorem 2.9 establishes the convergence of equation (3). 2. Theorem 2.9 remains true if we replace (16) by a more general condition: there are numbers β > 0 and δ ≥ 0 such that More information about different generalizations of this type can be found in the work [11]. Below we will prove this fact which is not based on the ideas used in the proof of Theorem 2.9.
Proof. First, we will show that equation (3) is dissipative. Indeed, denote by w(t) := |φ(t, u, g)| 2 . Then, according to (23), we have . Consider the scalar differential equation on the semi-axis R + . It is easy to check that this equation possesses two fixed points x 0 = 0, x 1 = ( M β ) 2/(1+2β) and the segment [x 0 , x 1 ] is the global attractor for (25). This means, in particular, that for all x ∈ R + , where r 0 := x 1 and by φ(t, x) we denote the unique solution of equation (25) with initial condition φ(0, x) = x (x ∈ R + ). Note that from (24) and (25) it follows that for all t ∈ R + and, consequently, From (27) we obtain where s t ∈ [−r, 0] is some number depending on t. Taking into account (28) and the fact that M β 1/(1+2β) is an absolute constant, we conclude that (3) is dissipative.
Consider the non-autonomous dynamical system (X, R + , π), (Y, R, σ), h generated by equation (3). Note that, owing to our assumptions and the facts established above, the following conditions are fulfilled: 3 the skew-product dynamical system (X, R + , π) associated to equation (3) is locally compact; (iii) by Corollary 2.4 the dynamical system (X, R + , π) is asymptotically compact; (iv) every positive semi-trajectory + x , where x := (u, g) ∈ X g = {(u, g) : u ∈ C r }, is relatively compact in X; (v) the dynamical system (Y, R, σ) if pseudo recurrent. Now to finish the proof of our theorem it is sufficient to apply Corollary 3.12 in [6] (see Theorem A.7 in Appendix A) and Theorem 2.7.
Remark 2.13. Theorem 2.12 remains true if we replace condition (23) by where ζ ∈ K possessing the following properties: This statement can be proved using the same reasoning as that in the proof of Theorem 2.12.

Convergent evolution equations with monotone operators
Let H be a real Hilbert space with inner product ·, · and norm | · | = ·, · , and E be a reflexive Banach space contained in H algebraically and topologically. Furthermore, let E be dense in H, and here H can be identified with a subspace of the dual E of E and ·, · can be extended by continuity to E × E. Recall (see [2,22]) that the operator A is said to be -monotone, if Au 1 − Au 2 , u 1 − u 2 ≥ 0 for all u 1 , u 2 ∈ D(A); -strictly monotone, if for all u 1 , u 2 ∈ D(A) (u 1 = u 2 ); -semi-continuous, if for each u, v ∈ D(A) and w ∈ H the function ϕ : R → R defined by the equality ϕ(t) := A(u + tv), w (for all t ∈ R) is continuous; -uniformly monotone, if there exist positive numbers α and p ≥ 2 such that Note that the family of monotone operators can be partially ordered by including graphics. A monotone operator is called maximal, if it is maximal among the monotone operators.
Let (Y, R, σ) be a dynamical system on the metric space Y . In this subsection we suppose that Y is a compact space. We consider the initial value problem where A : E → E is bounded (generally non-linear), and semi-continuous (see [23]).
A nonlinear "elliptic" operator given by where D is a bounded domain in R n , φ(·) is an increasing function satisfying In the space L p loc (R; B; µ) we define the following family of semi-norms || · || l,p : These semi-norms in (32) define a metrizable topology on L p loc (R; B; µ). The metric given by this topology can be defined, for instance, by  (t, (v, g)) := (ϕ(t, v, g), g t ) and h := pr 2 : X → Y . As it is shown in the work [18], the triplet (X, R + , π), (Y, R, σ), h is a non-autonomous dynamical system.
Applying the general theory developed in [6] (i.e. the results in Appendix A) to the constructed non-autonomous dynamical systems, we obtain the corresponding statements for equation (29). Let us establish some of them.
Theorem 3.2. Suppose that the following conditions are fulfilled: (29) is compact dissipative, i.e., the cocycle ϕ (or equivalently, the skew-product dynamical system generated by equation (29)) generated by equation (29) is compact dissipative; (ii) the space Y is compact, and the dynamical system (Y, R, σ) is minimal; where ϕ(t, u i , y) (i = 1, 2) is solution of equation (29) passing through u i at the initial moment t = 0 which is relatively compact on R.
Then, (29) is convergent, i.e., the cocycle ϕ associated to equation (29) is convergent; (ii) for all y ∈ Y , equation (29) admits a unique solution ϕ(t, x y , y) which is relatively compact on R and uniformly compatible, i.e., M y ⊆ M ϕ(·,xy,y) ; (iii) if the point y is τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent), then (a) equation (29) has a unique τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent) solution; (b) every solution ϕ(t, x, y) is asymptotically τ -periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically almost automorphic, asymptotically recurrent); (c) lim t→∞ |ϕ(t, x, y) − ϕ(t, x y , y)| = 0 for all x ∈ D(A) and y ∈ Y . Remark 3.4. If we suppose that operator A is uniformly monotone, then Theorem 3.3 is also true without the requirement that there exists at least one solution which is relatively compact on R + . Below we will prove this statement.
Then, we recall the following well-known result which will be useful in our proofs.
Theorem 3.5. [1,4,21] Let f : X → X be a ϕ-contraction. Suppose that the mapping ϕ : R + → R + satisfies the following conditions: Then f has a unique fixed point x 0 and lim n→∞ f n (x) = x 0 for all x ∈ X.
Theorem 3.6. Let (Y, R, σ) be pseudo recurrent and operator A be uniformly monotone.
1. If p = 2, then from (35) we have |ϕ(t, u 1 , y) − ϕ(t, u 2 , y)| ≤ e −αt |u 1 − u 2 | for all t ∈ R + , u 1 .u 2 ∈ D(A) and y ∈ Y . To finish the proof in this case it is necessary to use the same reasoning as in the proof of Theorem 2.9.
Let C(Y, D(A)) be the Banach space of all continuous ν : Y → D(A) with the sup-norm. Now we define for all t ∈ R + a mapping S t from C(Y, D(A)) into itself by following rule (S t ν)(y) := ϕ(t, ν(y), σ(−t, y)) for all y ∈ Y . It easy to check that the family of maps {S t } t≥0 forms a semigroup with respect to composition (more exactly S t S τ = S t+τ for all t, τ ∈ R + ). Notice that from (37) and the fact that ω(t, ·) is increasing we have for all t ∈ R + and ν 1 , ν 2 ∈ C(Y, D(A)).

Semi-linear parabolic equations
Let H be a separable Hilbert space with inner product ·, · , and associated norm | · | := ·, · 1/2 , and A be a self-adjoint operator with domain D(A).
One can define an operator f (A) for a wide class of functions f defined on the positive semi-axis as follows: In particular, we can define operators A α for all α ∈ R. For α = −β < 0 this operator is bounded. The space D(A −β ) can be regarded as the completion of the space H with respect to the norm | · | β := |A −β · |.
The following statements hold [12, Ch. II]: (ii) For any β ∈ R, the operator A β can be defined on every space D(A α ) as a bounded operator mapping D(A α ) into D(A α−β ) such that (iii) For all α ∈ R, the space F := D(A α ) is a separable Hilbert space with the inner product ·, · α := A α ·, A α · and the norm | · | α := |A α · |. (iv) The operator A with the domain F 1+α is a positive operator with discrete spectrum in each space F α . (v) The embedding of the space F α into F β for α > β is continuous, i.e., F α ⊂ F β and there exists a positive constant C = C(α, β) such that | · | β ≤ C| · | α . (vi) F α is dense in F β for any α > β. (vii) Let α 1 > α 2 , then the space F α1 is compactly embedded into F α2 , i.e., every sequence bounded in F α1 is relatively compact in F α2 .
as t → τ for every x ∈ F β and β ∈ R; d. For any β ∈ R the exponential operator e −tA defines a dissipative compact dynamical system (F β , e −tA ); e.
Let (Y, ρ) be a compact complete metric space and (Y, R, σ) be a dynamical system on Y . Consider an evolutionary differential equation in the separable Hilbert space H, where A is a linear (generally speaking unbounded) positive operator with discrete spectrum, and F is a non-linear continuous mapping acting from Y × F θ into H, 0 ≤ θ < 1, possessing the property for all u 1 , u 2 ∈ B θ (0, r) := {u ∈ F θ : |u| θ ≤ r}. Here L(r) denotes the Lipschitz constant of F on the set B θ (0, r).
A function u : [0, a) → F θ is said to be a weak solution (in F θ ) of equation (44) passing through the point x ∈ F θ at the initial moment t = 0 (notation ϕ(t, x, y)) if u ∈ C([0, T ], F θ ) and satisfies the integral equation  Proof. Let x 0 ∈ F θ , r > 0, δ > 0 and T > 0. We consider the space C x0,r,δ,T of all continuous functions ψ : which is a complete metric space.
Our lemma is completely proved now.
Equation (44) (equivalently, the cocycle ϕ generated by equation (44)) is said to be dissipative if there exists a positive number R 0 such that for all r > 0 there exists a positive number l = l(r) such that for all t ≥ l(r), ||x|| θ ≤ r and y ∈ Y .  Finally, we can establish the next result. Then, (i) if the point y is τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent), then equation (44) admits a unique τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent) solution ϕ(t, x y , y) (x y ∈ F θ ); A dynamical system is a triplet (X, T, π), where π : T × X → X is a continuous mapping satisfying the following conditions: π(s, π(t, x)) = π(s + t, x) (∀t, τ ∈ T and x ∈ X).
If T = R (R + ) or Z (Z + ), then the dynamical system (X, T, π) is called a group (semi-group). When T = R + or R the dynamical system (X, T, π) is called a flow, but if T ⊆ Z, then (X, T, π) is called a cascade (discrete flow ).
The function π(·, x) : T → X is called a motion passing through the point x at the moment t = 0 and the set Σ x := π(T, x) is called a trajectory of this motion. − compact dissipative if the equality (54) takes place uniformly w.r.t. x on the compact subsets of X; − locally compact if for any point p ∈ X there exist δ p > 0 and l p > 0 such that the set π(l p , B(p, δ p )) is relatively compact, where B(p, δ) := {x ∈ X | ρ(x, p) < δ}.
It can be shown [8,Ch.I] that the set J defined by equality (55) does not depend on the choice of the attractor K, but is characterized only by the properties of the dynamical system (X, T, π) itself. The set J is called the Levinson center of the compact dissipative dynamical system (X, T, π).
Non-autonomous dynamical systems (cocycles) play a very important role in the study of non-autonomous evolutionary differential equations. Under appropriate assumptions every non-autonomous differential equation generates some cocycle (non-autonomous dynamical system).
The family {I y | y ∈ Y } (I y ⊂ W ) of nonempty compact subsets W is called (see, for example, [8]) a compact pullback attractor (uniform pullback attractor ) of a cocycle ϕ, if the following conditions hold: (i) the set I := {I y | y ∈ Y } is relatively compact; (ii) the family {I y | y ∈ Y } is invariant with respect to the cocycle ϕ, i.e. ϕ(t, I y , y) = I σ(t,y) for all t ∈ T + and y ∈ Y ; (iii) for all y ∈ Y (uniformly in y ∈ Y ) and K ∈ C(W ) lim t→+∞ β(ϕ(t, K, σ(−t, y)), I y ) = 0, where β(A, B) := sup{ρ(a, B) : a ∈ A} is the Hausdorff semi-distance, and C(W ) denotes the compact subsets of W .
Below in this Section we suppose that T 2 = S.
The family {I y | y ∈ Y }(I y ⊂ W ) of nonempty compact subsets is called a compact global attractor of the cocycle ϕ, if the following conditions are fulfilled: (i) the set I := {I y | y ∈ Y } is relatively compact; (ii) the family {I y | y ∈ Y } is invariant with respect to the cocycle ϕ; Then, we have the following result. A point x ∈ X is called [10] asymptotically τ -periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically recurrent, asymptotically pseudo recurrent), if there exists a τ -periodic (respectively, quasi periodic, Bohr almost periodic, recurrent, pseudo recurrent) point p ∈ X such that lim t→+∞ ρ(π(t, x), π(t, p)) = 0.
The following result holds.
A non-autonomous dynamical system (X, T 1 , π), (Y, T 2 , σ), h is said to be weak convergent [6], if the following conditions hold: (i) the dynamical systems (X, T 1 , π) and (Y, T 2 , σ) are compact dissipative with Levinson centers J X and J Y respectively; (ii) it follows that lim t→+∞ ρ(π(t, x 1 ), π(t, x 2 )) = 0, for all x 1 , x 2 ∈ J X with h(x 1 ) = h(x 2 ). Remark A.3. It is clear that every convergent non-autonomous dynamical system is weak convergent. The inverse statement, generally speaking, is not true. The paper [6] contains an example confirming this statement.
Then, the following statements hold: (i) there exists a unique compact minimal set M ⊆ J such that (a) the section M X y of the set M consists of a single point m y for all y ∈ Y ; (b) lim t→+∞ ρ(π(t, x), m σ(t,h(x)) ) = 0 holds for all x ∈ X; Denote by L x := {{t n } ∈ M x : t n → +∞}, where M x := {{t n } ⊆ T : such that the sequence {π(t n , x)} is convergent}. Recall [10] that the point x ∈ X is called comparable with y ∈ Y by the character of recurrence in infinity if L x ⊆ L y . (i) (X, T 1 , π) and (Y, T 2 , σ) are two dynamical systems; (ii) the point y ∈ Y is asymptotically stationary (respectively, asymptotically τ -periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent); (iii) the point x is comparable with y ∈ Y by the character of recurrence in infinity.
Let (X, h, Y ) be a bundle [20]. The subset M ⊆ X is said to be conditionally relatively compact, if the pre-image h −1 (Y ) M of every relatively compact subset Y ⊆ Y is a relatively compact subset of X, in particular, M y := h −1 (y) M is relatively compact for every y. The set M is called conditionally compact if it is closed and conditionally relatively compact.
(ii) The NDS (X, T, π), (Y, S, σ), h is positively uniformly stable on J; (iii) every point y ∈ Y is Poisson stable. Then, (i) all motions on J can be uniquely continued to the left and define on J a two-sided dynamical system (J, S, π), i.e., the semi-group dynamical system (X, T, π) generates on J a two-sided dynamical system (J, S, π); (ii) for every y ∈ Y , there are two sequences {t 1 n } → +∞ and {t 2 n } → −∞ such that π(t i n , x) → x (i = 1, 2) as n → ∞ for all x ∈ J y .
Recall that the dynamical system (X, T 1 , π) is called asymptotically compact if for every positively invariant bounded subset M ⊆ X there exists a nonempty compact subset K ⊆ X such that lim t→+∞ β(π(t, M ), K) = 0.