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

The aim of this paper is to describe the structure of global attractors for non-autonomous dynamical systems with recurrent coefficients (with both continuous and discrete time). We consider a special class of this type of systems (the so–called weak convergent systems). It is shown that, for weak convergent systems, the answer to Seifert’s question (Does an almost periodic dissipative equation possess an almost periodic solution?) is affirmative, although, in general, even for scalar equations, the response is negative. We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our paper to the study of 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.


Introduction
Denote by R n the n-dimensional real Euclidian space with the norm | · |, and by C(R × R n , R n ) the space of all continuous functions f : R × R n → R n equipped with the compact-open topology.
Consider the differential equation where f ∈ C(R × R n , R n ). Assume that the right-hand side of (1) satisfies hypotheses ensuring the 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 .
Equation (1) (respectively, the function f ) is called regular, if for every x 0 ∈ R n and g ∈ H(f ) := {f τ : τ ∈ R} (where by bar we denote the closure in the space C(R × R n , R n ) and f τ (t, x) := f (t + τ, x) for all (t, x) ∈ R × R n ), the equation possesses a unique solution ϕ(t, x, g) passing through the point x 0 at the initial moment t = 0, and defined on R + := {t ∈ R| t ≥ 0}.
Theorem 1.1. [9,ChII] Suppose that f ∈ C(R × R n , R n ) and H(f ) is a compact subset of C(R × R n , R n ). Then, the following statements are equivalent: (1) is uniformly dissipative; (ii) there exists a positive number R 0 such that for all (x, g) ∈ R n × H(f ).
At light of Theorem 1.1, it is said that equation (1) is dissipative (in fact, the family of equations (2) is collectively dissipative, but we use this shorter terminology) if (3) holds.
Problem (G. Seifert [20]): 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 [20] and Zhikov [32] established that, in general, even when equation (1) is scalar, the answer to Seifert's question is negative.
Related to this result, there are the following interesting questions: a) To extract some classes of dissipative differential equations for which the response to Seifert's problem is positive; b) To indicate the additional (it is desirable "optimal") conditions which, jointly with dissipativity, guarantee the existence of at least one almost periodic solution of equation (1).
Below we include a short survey on results concerning the questions a) and b). a) For the following classes of dissipative equations of type (1), the response to Seifert's question is affirmative: linear equations [9,ChII], quasi-linear equations (weak non-linear perturbations of linear equations) [7,9]; holomorphic equations [5,6,8,9]. b) Zubov [35] established that equation (1) admits a unique almost periodic solution if it is convergent, i.e., it admits a unique solution which is bounded on R and also uniformly globally asymptotically stable. This result was generalized for equations (1) with recurrent coefficients by Cheban [9,ChII] and with pseudo recurrent coefficients by Caraballo and Cheban [4].
The main result for ODEs (Theorem 4.2 and its generalizations) that we prove in this paper is the following: we show 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)  We present our results in the framework of general non-autonomous dynamical systems (cocycles) and we apply our abstract theory to several classes of differential equations.
The paper is organized as follows.
In Section 2, we collect some notions (global attractor, minimal set, point/compact dissipativity, non-autonomous dynamical systems with convergence, quasi periodicity, Levitan/Bohr almost periodicity, almost automorphy, recurrence, pseudo recurrence, Poisson stability, etc) and facts from the theory of dynamical systems which will be necessary in this paper.
Section 3 is devoted to the study of a special class of non-autonomous dynamical systems (NDS): the so-called NDS with weak convergence. We give a generalization of the notion of convergent NDS. On the one hand, this type of NDS is very close to NDS with convergence (because they conserve some properties of convergent systems) and larger than that of convergent systems. On the other hand, we analyze the class of compact dissipative NDS with nontrivial Levinson center. The main results of our paper are proved in this section, namely Theorem 3.5 and Theorem 3.8 (see also Corollary 3.9 and Corollary 3.10) which provide sufficient conditions for the existence of a unique minimal set in the Levinson center which is homeomorphic to the base dynamical system (driving system). This means, in particular, that if the base dynamical system is a compact minimal set consisting of recurrent (respectively, almost automorphic, Bohr almost periodic, quasi periodic, periodic, stationary) points, then, under the conditions of Theorem 3.5, the Levinson center of a non-autonomous dynamical system contains a unique minimal set which consists of recurrent (respectively, almost automorphic, Bohr almost periodic, quasi periodic, periodic, stationary) points.
In Section 4, we exhibit some applications of our abstract results to different classes of differential equations. Namely, almost periodic and asymptotically almost periodic solutions (Subsection 4.1), uniformly compatible (by the character of recurrence with the right hand side) solutions of strict dissipative equations (Subsection 4.2).

Nonautonomous Dynamical Systems with Convergence
Let us start by recalling some concepts and notations about the theory of nonautonomous dynamical systems which will be necessary for our analysis.
2.1. Compact Global Attractors of Dynamical Systems. Let (X, ρ) be a metric space, R (Z) be the group of real (integer) numbers, R + (Z + ) be the semigroup of nonnegative real (integer) numbers, S be one of the two sets R or Z and T ⊆ S (S + ⊆ T) be a sub-semigroup of the additive group S.
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, s ∈ T and x ∈ X). If T = R (R + ) or Z (Z + ), the dynamical system (X, T, π) is called a group (semigroup). 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 the motion passing through the point x at the initial moment t = 0, and the set Σ x := π(T, x) is called a trajectory of this motion.
A nonempty set M ⊆ X is called positively invariant (negatively invariant, invariant) with respect to the dynamical system (X, T, π) or, simply, positively invariant The set W s (Λ) (W u (Λ)), defined by the equality is called the stable manifold (unstable manifold ) of the set Λ ⊆ X. -asymptotically stable if it is orbitally stable and attracting; -globally asymptotically stable if it is asymptotically stable and W s (M ) = X.
The dynamical system (X, T, π) is called: − point dissipative if there exists a nonempty compact subset K ⊆ X such that, for every x ∈ X, − compact dissipative if the equality (4) takes place uniformly w.r.t. x on the compact subsets of X; − locally complete (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.
Let (X, T, π) be compact dissipative, and K be a compact set attracting every compact subset of X. Let us set It can be shown [9, Ch.I] that the set J defined by equality (5) 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 a Levinson center of the compact dissipative dynamical system (X, T, π).
Some properties of this set can be found in [9,21].
Some sufficient conditions and criteria ensuring the convergence of a dynamical system can be found in [9, Ch.II].
Thus, a non-autonomous dynamical system (X, T 1 , π), (Y, T 2 , σ), h is convergent, if the systems (X, T 1 , π) and (Y, T 2 , σ) are compact dissipative with Levinson centers J X and J Y respectively, and J X has "trivial" sections, i.e., J X X y consists of a single point for all y ∈ J Y . In this case, the Levinson center J X of the dynamical system (X, T 1 , π) is a copy (a homeomorphic image) of the Levinson center J Y of the dynamical system (Y, T 2 , σ). Thus, the dynamics on J X is the same as on J Y .
Remark 2.1. 1. We note that convergent systems are, in some sense, the simplest dissipative dynamical systems. If Y is compact and invariant, T 2 = S, (X, T 1 , π), (Y, T 2 , σ), h is a convergent non-autonomous dynamical system, and J X is the Levinson center of (X, T 1 , π), then (J X , T 2 , π) and (Y, T 2 , σ) are homeomorphic. Although the Levinson center of a convergent system can be completely described, it may be sufficiently complicated.
2. The concept of convergent system of differential equations is well developed (see, for example, B. P. Demidovich [17,18,19], Pavlov et al. [24], V. A. Pliss [25,26], V. I. Zubov [34], and many others). The non-autonomous system of differential equations is called convergent, if it admits a unique solution defined and bounded on R, which is uniformly globally asymptotically stable. It is possible to show that the non-autonomous dynamical system generated by the convergent equation (6) is convergent. However, the concept of convergent non-autonomous dynamical system is much more general (see [9,12] and the bibliography therein).
A point x ∈ X is called almost recurrent (respectively, Bohr almost periodic), if for any ε > 0, there exists a positive number l such that, in any segment of length l, there is an ε−shift (respectively, an ε−almost period) of the point x ∈ X.
If the point x ∈ X is almost recurrent, and the set H(x) := {π(t, x) | t ∈ T} is compact, then x is called recurrent, where the bar denotes the closure in X. A point x ∈ X is called Poisson stable in the positive direction if there exists a sequence {t n } ∈ N x such that t n → +∞ as n → ∞.
Remark 2.2. Every recurrent point is pseudo recurrent, but there exist pseudo recurrent points which are not recurrent (see [28,30]).
An m-dimensional torus is denoted by is an irrational winding of the torus T m and ω ∈ T m .
A point x ∈ X of the dynamical system (X, T, π) is called Levitan almost periodic (see [2,23]) if there exists a dynamical system (Y, T, σ), and a Bohr almost periodic point y ∈ Y such that N y ⊆ N x .
Thus, we can prove the following result.
Proof. The first statement follows directly from the corresponding definition. Let x ∈ X y 0 be an arbitrary point. We will show that lim Indeed, if we suppose that it is not true, then there are ε 0 > 0 and t n → +∞ ({t n } ⊆ T 1 ) such that Since the dynamical system (X, T 1 , π) is compact dissipative, we may suppose that the sequences {π(t n , x)} and {π(t n , x 0 )} are convergent. Denote by p := lim n→∞ π(t n , x), and p 0 := lim n→∞ π(t n , x 0 ). Then, p, p 0 ∈ X q ⊆ J X , where q := lim n→∞ σ(t n , y 0 ). Since the non-autonomous dynamical system (X, is convergent, then p = p 0 . On the other hand, passing to the limit in (7), we obtain ρ(p, p 0 ) ≥ ε 0 > 0. This contradiction proves our statement.

Non-Autonomous Dynamical Systems with Weak Convergence
In this section we will study a class of non-autonomous dynamical systems which is very close to convergent systems, but possessing a non-trivial global attractor. This means that this class of non-autonomous systems will conserve almost all properties of convergent systems, but will have a "non-trivial" global attractor J X , i.e., there exists at least one point y ∈ J Y such that the set J X X y contains more than one point.
A non-autonomous dynamical system (X, T 1 , π), (Y, T 2 , σ), h is said to be weak convergent, 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 Remark 3.1. It is clear that every convergent non-autonomous dynamical system is weak convergent. The opposite statement is not true in general.
Indeed, the last statement can be confirmed by the following example. Let (X, T, π) be an autonomous dynamical system with compact global attractor J, which possesses a unique stationary attracting point p (i.e., lim |t|→+∞ ρ(π(t, x), p) = 0 for all x ∈ J) and J = {p}). For example, consider the dynamical system (X, R, π) on the space X = R 2 , generated by following system of differential equations The phase plane of this dynamical system (8) is described in Figure 1. For more We can now prove the following result which will be crucial in the proof of one of our main results (namely, Theorem 3.5).
, h be a non-autonomous dynamical system, and assume that the following conditions hold: Then, there exists a unique compact minimal set M ⊆ X such that: for all x ∈ X.
Proof. Since the positive semi-trajectory Σ + x0 of x 0 is relatively compact, the ω-limit set ω x 0 of the point x 0 is nonempty, compact, invariant and contains at least one minimal subset M ⊆ ω x 0 . We will prove that the dynamical system (X, T 1 , π) has at most one minimal set. Indeed, if we suppose that M 1 and M 2 are two different minimal sets of (X, T 1 , π), then M 1 M 2 = ∅ and, in particular, M 1y M 2y = ∅ for all y ∈ Y , where M iy := h −1 (y) M i (i = 1, 2). Let x i ∈ M iy and t n → +∞ such that σ(t n , y) → y and π(t n , x i ) →x i ∈ M iy (i = 1, 2) as n → ∞; (10) lim n→∞ ρ(π(t n , x 1 ), π(t n , x 2 )) = 0.
It is now easy to see that there exists such a sequence. From the equality (10) we havex 1 =x 2 ∈ M 1y M 2y . This is a contradiction and, therefore, M is the unique compact minimal set of the dynamical system (X, T 1 , π).
Let y ∈ Y be an arbitrary point, then it is recurrent. By Lemma 6.5.19 in [12, Ch.VI, p.226], there exists a unique recurrent point m y ∈ M y such that the equality (9) holds for all x ∈ M y . Now, we will prove that M y = {m y }. Indeed, if x ∈ M y then, there exists a sequence t n → +∞ such that π(t n , m y ) → x because M is minimal. On the other hand, σ(t n , y) → y and, since the point m y is uniformly compatible by the character of recurrence with the point y, then π(t n , m y ) → m y . Thus, we have x = m y and, consequently, M y = {m y } for all y ∈ Y .
To finish the proof it is sufficient to note that lim t→+∞ ρ(π(t, x), m σ(t,h(x)) ) = 0, for all x ∈ X.
Corollary 3.3. Under the assumptions in Lemma 3.2, the dynamical system (X, T 1 , π) is point dissipative and Ω X := ∪{ω x | x ∈ X} is a compact minimal set.
Below we give an example of point dissipative, but not compact dissipative, dynamical system with weak convergence.
Example 3.4. Let ϕ ∈ C(R, R) be a function possessing the following properties: the function ϕ is monotone increasing from 0 to 1 and it is decreasing from 1 to 2; 6. xϕ(x −1 ) → 0 as x → +∞.
A function ϕ with properties 1. − 6. can be constructed as follows. Let Then, the function ϕ(t) := ϕ 0 (t − 1) is as desired. We set X : where θ is the function from C(R, R) identically equal to 0. It is possible to show that the set X is closed in C(R, R), and it is invariant with respect to shifts. Thus, on the set X is induced a dynamical system (on C(R, R) is defined the dynamical system of translations or Bebutov's dynamical system), which we denote by (X, R, σ). We will indicate some properties of this system: (i) for every function ψ ∈ X the set {σ(t, ψ) : t ∈ R} is relatively compact and ω ψ = α ψ = {θ}, where α ψ denotes the α-limit associated to ψ; (ii) the dynamical system (X, R, σ) is pointwise dissipative, and Ω X = {θ}; (iii) W u (Ω X ) = X and, consequently, the dynamical system (X, R, σ) is not compact dissipative because the set X, evidently, is not compact; (iv) the dynamical system (X, R, σ) does not admit a maximal compact invariant set.
The necessary example is therefore constructed.
The subset A ⊆ X is said to be chain transitive (see [16,22]) if for any a, b ∈ A, and any ε > 0 and L > 0, there are finite sequences We can now establish and prove the main results of our paper.
Proof. According to Lemma 3.2, there exists a unique minimal set M ⊂ X for the dynamical system (X, T 1 , π) with properties (b) and (c). To finish the first statement of our theorem it is sufficient to show that M ⊆ ∂J. Let x ∈ ∂J, then M ⊆ ω x ⊆ ∂J because J is the maximal compact invariant set of (X, T 1 , π), and the set ∂J is also compact and invariant.
Then, we can prove the following theorems concerning the structure of compact positively invariant sets. Theorem 3.6. Let (X, T 1 , π), (Y, T 2 , σ), h be a non-autonomous compact dissipative dynamical system, M = ∅ be a compact positively invariant set. Suppose that the following conditions are fulfilled: Then, M is orbitally stable.
Since M is compact, then we may suppose that the sequence {x n } is convergent. Let x 0 := lim n→+∞ x n , with x y n ∈ M y n , ρ(x n , M ) = ρ(x n , x y n ) and y 0 = h(x 0 ). Then, x 0 = lim n→+∞ x yn and x 0 ∈ M y0 . Let q n = h(x n ), and note that as n → +∞, because q n → y 0 and x qn → x 0 . Taking into account (15), and the asymptotic stability of the set M , we have (16) ρ(π(t n , x n ), π(t n , x qn )) → 0.
But the equalities (14) and (16) are contradictory. Hence, the set M is orbitally stable in (X, T 1 , π).  (i) the dynamical systems (X, T 1 , π) and (Y, T 2 , σ) are point dissipative; (ii) Ω Y is a compact minimal set; (iii) lim t→+∞ ρ(π(t, x 1 ), π(t, x 2 )) = 0, Proof. Let N := Ω Y . Under our assumptions, N is a compact minimal set of the dynamical system (Y, T, σ). Consider the non-autonomous dynamical system (X, T 1 ,π), (N, T 2 ,σ),h , whereX := h −1 (N ), (X, T 1 ,π) (respectively, (N, T 2 ,σ)) is the restriction of (X, T 1 , π) (respectively, (Y, T 2 , σ)) onX (respectively, on N ) andh := h X . According to Lemma 3.2, the dynamical system (X, T 1 ,π) contains a unique compact minimal set M such that h(M ) = N , and for all y ∈ N , the set M y =h −1 (y) consists of a single point. It is clear that the set M is also minimal with respect to the dynamical system (X, T 1 , π). Repeating the arguments in the proof of Lemma 3.2, we can prove that M is the unique compact minimal set of (X, T 1 , π). Let now x ∈ X be an arbitrary point. Then, its ω-limit set ω x is a non-empty, compact and invariant set.
Note that ω x contains at least one compact minimal set. Taking into consideration that M is the unique compact minimal set in (X, T 1 ,π), we then have M ⊆ ω x and, consequently, ω x X y = {m y } for all y ∈ N . Thus, we conclude that ω x = M and, consequently, Ω X = M .
Denote by L x := {{t n } ∈ M x : t n → +∞}. Recall (see [11]) that the point x ∈ X is called comparable with y ∈ Y by the character of recurrence in infinity if L x ⊆ L y . Corollary 3.9. Let (X, T 1 , π), (Y, T 2 , σ), h be a non-autonomous dynamical system such that the following conditions hold: (i) the dynamical systems (X, T 1 , π) and (Y, T 2 , σ) are point dissipative; Proof. The first and second statements follow from Theorem 3.8. To complete the proof we need to establish the third statement. Let x ∈ X, y := h(x) ∈ H + (y 0 ) := {σ(t, y 0 ) : t ∈ T + } and {t n } ∈ L y . We will show that {t n } ∈ L x . Indeed, the sequence {π(t n , x)} is relatively compact because (X, T, π) is point dissipative. Let p be a limit point for the sequence {π(t n , x)}, then there exists a subsequence {t n k } ⊆ {t n } such that p = lim k→∞ π(t n k , x). Denote by q = lim k→∞ σ(t n , y), then h(p) = q. It is clear that q ∈ Ω Y = N and p ∈ L X . Since L X X q contains at most one point, we conclude that p is the unique limit point of the sequence {π(t n , x)} and, consequently, it is convergent.
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, The next result will be crucial for our applications (particularly, Theorem 4.6).
Proof. This statement directly follows from Theorem 3.11 and Lemma 2.4.
Remark 3.13. 1. Note that in [4] we established an analogous result to Theorem 3.11 but under a stronger assumption. Namely, instead of Condition 5. of Theorem 3.11, we used in [4] the following one: for all (x 1 , x 2 ) ∈ X×X \ ∆ X and t > 0.
2. It is clear that (17) implies Condition 5. of Theorem 3.11. The converse is not true. Below we give the corresponding counterexample. It is easy to check that the norm of the operator A is equal to 1 and, consequently, ||A n (φ 1 −φ 2 )|| ≤ ||φ 1 −φ 2 ||. Thus, the dynamical system (X, Z + , π) is V -monotone, if we take the function V : X × X → R + defined by V (φ 1 , φ 2 ) := ||φ 1 − φ 2 ||. In this way, condition (17) does not hold for the dynamical system (X, Z + , π) constructed above. On the other hand, it is also easy to check that the norm of the operator A n is equal to 1 n! . Thus, for all The necessary example is therefore constructed.

Applications
Let X and Y be two complete metric spaces. Denote by C(X, Y ) the space of all continuous functions f : X → Y equipped with the compact-open topology.

4.1.
Almost periodic solutions of almost periodic dissipative systems. Let (Y, R, σ) be a dynamical system on the metric space Y . In this subsection we suppose that Y is a compact space. Consider the differential equation The function f ∈ C(Y × R n , R n ) (respectively, equation (18)) is said to be regular (see [27]) if for all u ∈ R n and y ∈ Y , equation (18) admits a unique solution ϕ(t, u, y) passing through the point u ∈ R n at the initial moment t = 0, and defined on R + .
Thus, the triplet R n , ϕ, (Y, R, σ) is a cocycle (non-autonomous dynamical system) which is associated to (generated by) equation (18). In this case, the dynamical system (Y, R, σ) is called the base dynamical system (or driving system).
Example 4.1. Let us consider the equation where f ∈ C(R × R n , R n ). Along with equation (19), consider the family of equations where g ∈ H(f ) := {f τ : τ ∈ R} and f τ is the τ -shift of f with respect to the time variable t, i.e., f τ (t, u) := f (t+τ, u) for all (t, u) ∈ R×R n . Suppose that the function f is regular (see [27]), i.e., for all g ∈ H(f ) and u ∈ R n there exists a unique solution ϕ(t, u, g) of equation (20). Denote by Y = H(f ) and (Y, R, σ) the shift dynamical system on Y induced by the Bebutov dynamical system (C(R×R n , R n ), R, σ). Now, the family of equations (20) can be written as if we define F ∈ C(Y × R n , R n ) by the equality F (g, u) := g(0, u), for all g ∈ H(f ), and u ∈ R n .
In this section we suppose that equation (18) is regular. Equation (18) is called dissipative (see [9]), if there exists a positive number r such that lim sup t→+∞ |ϕ(t, u, y)| < r for all u ∈ R n and y ∈ Y , where | · | is a norm in R n .
It is well known (see [20,32]) that, a dissipative equation with almost periodic coefficients (Y is an almost periodic minimal set), does not have, in general, an almost periodic solution. For certain classes of dissipative equations of the form (18), in the works [5]- [8] one can find sufficient conditions for the existence of at least one almost periodic solution. In this subsection we give a simple geometric condition which guarantees existence of a unique almost periodic solution, and this solution, in general, is not the unique solution of equation (18) which is bounded on R.
We can now establish the following interesting result.
Theorem 4.2. Suppose that the following conditions are fulfilled: (18) is regular and 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 the solution of equation (18) passing through u i at the initial moment t = 0, which is bounded on R.
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 . By Theorem 2.23 in [9], the skew-product dynamical system (X, R + , π) is compact dissipative. Denote by J X its Levinson center and by I y := pr 1 (J X X y ) for all y ∈ Y , where X y := {x ∈ X : h(x) = y}, and pr 1 is the projection function with respect to the first variable, i.e. pr 1 (α, y) = α. According to the definition of the set I y ⊆ R n , and by Theorem 2.24 in [9], u ∈ I y if and only if the solution ϕ(t, u, y) is defined on R and bounded (i.e., the set ϕ(R, u, y) ⊆ R n is compact). Thus, I y = {u ∈ R n | (u, y) ∈ J X }. It is easy to see the condition (21) means that the non-autonomous dynamical system (X, R + , π), (Y, R, σ), h is weak convergent. To finish the proof, it is sufficient to apply Lemma 6.5.19 in [12, Ch.VI, p.226] and Corollary 3.10 for the non-autonomous system (X, R + , π), (Y, R, σ), h generated by equation (18).  (18), but equation (18) has, generally speaking, more than one solution defined and bounded on R. Below, we will give an example which confirms this statement.
Example 4.4. Consider the following almost periodic system of two differential equations It is easy to check that the almost periodic function ϕ : R → R 2 defined by the equality ϕ(t) := (sin t, sin √ 2t) is a solution of system (22). Let now x := u − sin t and y := v − sin √ 2t. Then, the system (22) reduces to (8). Thus, every solution φ of the system (22) possesses the form φ = ϕ + ψ, where ψ is some solution of the system (8). Since (8) is weak convergent and admits more than one solution which is bounded on R, the system (22) possesses the same property.

4.2.
Uniform compatible solutions of strict dissipative equations. In this section we consider equation (18) when the driving system (Y, R, σ) is pseudo recurrent, and the function f ∈ C(Y × R n , R n ) is strict dissipative with respect to its second variable x ∈ R n , i.e., for all x 1 , x 2 ∈ R n (x 1 = x 2 ) and y ∈ Y .
Recall (see [28,29,30]) that the point x ∈ X is called comparable (respectively, uniformly comparable) by the character of recurrence with the point y ∈ Y if N y ⊆ N x (respectively, M y ⊆ M x ).
Let us now recall a result which plays an important role in the proof of our main result in this subsection.
Theorem 4.5. (See [28,30]) Let (X, T, π) and (Y, T, σ) be two dynamical systems, x ∈ X and y ∈ Y . Then, the following statements hold: (i) If x is comparable by the character of recurrence with y, and y is τ -periodic (respectively, Levitan almost periodic, almost recurrent, Poisson stable), then so is the point x. (ii) If x is uniformly comparable by the character of recurrence with y, and y is τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent), then so is the point x.
Theorem 4.6. Let (Y, R, σ) be pseudo recurrent, f ∈ C(Y × R n , R n ) be strict dissipative with respect to the variable x, and assume that there exists at least one solution ϕ(t, x 0 , y) of equation (18) which is bounded on R + .
Then, (i) equation (18) is convergent, i.e., the cocycle ϕ associated to equation (18) 2. If we replace condition (23) by a stronger condition, then Theorem 4.6 is also true without the requirement that there exists at least one solution which is bounded on R + . Namely, if there exists a function ζ ∈ K such that for all u 1 , u 2 ∈ R n and y ∈ Y , where ζ possesses some additional properties (see, for example, [15]).
3. It is easy to see that Theorem 4.6 remains true also for equation (18) in an arbitrary Hilbert space H, if we suppose that the cocycle ϕ, generated by equation (18), is asymptotically compact (i.e., the corresponding skew-product dynamical system is asymptotically compact), and we replace the condition about the existence of at least one solution which is bounded on R + , by the existence of a relatively compact solution ϕ 0 on R + (this means that ϕ(R + ) is a relatively compact subset from H).