I. U. Bronshtein's Conjecture for Monotone Nonautonomous Dynamical Systems

In this paper we study the problem of almost periodicity of solutions for dissipative differential equations (Bronshtein's conjecture). We give a positive answer to this conjecture for monotone almost periodic systems of differential/difference equations.


Introduction
I. U. Bronshtein's conjecture [4,ChIV,p.273]. If an equation (1) x with right hand side (Bohr) almost periodic in t satisfies the conditions of uniform positive stability and positive dissipativity, then it has at least one (Bohr) almost periodic solution.
2. Even for scalar equations (d = 1) as was shown by A. M. Fink and P. O. Frederickson [15] (see also [13,ChXII]), dissipation (without uniform positive stability) does not imply the existence of almost periodic solutions.
The aim of this paper is studying the problem of existence of Levitan/Bohr almost periodic (respectively, almost automorphic, recurrent and Poisson stable) solutions for dissipative differential equation (1), when the second right hand side is monotone with respect to spacial variable. The existence at least one quasi periodic (respectively, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent, Levitan almost periodic, almost recurrent, Poisson stable) solution of (1) is proved under the condition that every solution of equation (1) is positively uniformly Lyapunov stable.
The paper is organized as follows.
In Section 2 we collected some notions and facts from the theory of dynamical systems (the both autonomous and nonautonomous) which we use in this paper: Poisson stable motions and functions, cocycles, skew-product dynamical systems, monotone non-autonomous dynamical systems, Ellis semigroup.
Section 3 is dedicated to the study the global attractors of cocycles, when the phase space of driving system is noncompact.the structure of the ω-limit set of noncompact semitrajectory for autonomous and nonautonomous dynamical systems.
In Section 4 we formulate I. U. Bronshtein's conjecture for general nonautonomous dynamical systems. The positive answer for monotone nonautonomous dynamical systems is given (Theorem 4.2, Corollary 4.3 and Remark 4.4).
Section 5 is dedicated to the applications of our general results for differential (Theorems 5.15 and 5.16) and difference (Theorems 5.21 and 5.22) equations.

Some general properties of autonomous and nonautonomous dynamical systems
In this section we collect some notions and facts from the autonomous and nonautonomous dynamical systems [8] (see also, [10, Ch.IX]) which we will use below.

2.2.
Some general facts about nonautonomous dynamical systems. In this subsection we give some general facts about nonautonomous dynamical systems without proofs. The more details and the proofs the readers can find in [8] (see also [10,Ch.IX]).
Definition 2.5. Let (X, h, Y ) be a fiber space, i.e., X and Y be two metric spaces and h : X → Y be a homomorphism from X into Y . The subset M ⊆ X is said to be conditionally precompact [8], [10,Ch.IX The set M is called conditionally compact if it is closed and conditionally precompact.
Denote by Φ x the family of all entire trajectories of (X, T + , π) passing through the point x ∈ X at the initial moment t = 0 and Φ := {Φ x : x ∈ X}.
Theorem 2. 16. Let x be comparable with y ∈ Y . If the point y ∈ Y is stationary (respectively, τ -periodic, Levitan almost periodic, almost recurrent, Poisson stable), then the point x ∈ X is so.
Denote by M x := {{t n } ⊂ R : such that {π(t n , x)} converges }.   Theorem 2.22. Let X and Y be two complete metric spaces, the point x be uniformly comparable with y ∈ Y by the character of recurrence. If the point y ∈ Y is recurrent (respectively, almost periodic, almost automorphic, uniformly Poisson stable), then so is the point x ∈ X.
2.5. Monotone Nonautonomous Dynamical Systems. Let R d + := {x ∈ R d : such that x i ≥ 0 (x := (x 1 , . . . , x n )) for any i = 1, 2, . . . , d} be the cone of nonnegative vectors of R d . By R d + on the space R d is defined a partial order.
Remark 2.28. 1. Every recurrent (respectively, uniformly Poisson stable) point is pseudo recurrent. The inverse statement, generally speaking, is not true.
2. If x 0 ∈ X is a pseudo recurrent point, then p ∈ ω p for any p ∈ H(x 0 ).
3. If x 0 is a Lagrange stable point and p ∈ ω p for any p ∈ H(x 0 ), then the point x 0 is pseudo recurrent.

Definition 2.29.
A point x ∈ X is said to be strongly Poisson stable if p ∈ ω p for any p ∈ H(x).
Remark 2.30. Every pseudo recurrent point is strongly Poisson stable. The inverse statement, generally speaking, is not true.

Global Attractors of Cocycles
Let W be a complete metric space.   (ii) U (t, y)ω y (M ) ⊆ ω σ(t,y) (M ) for all y ∈ Y and t ∈ T + , where U (t, y) := ϕ(t, ·, y); (iii) for any point w ∈ ω y (M ) the motion ϕ(t, w, y) is defined on S; (iv) if there exits a nonempty compact K ⊂ W such that Theorem 3.5. [10, ChII] Let W, ϕ, (Y, T, σ) be compactly dissipative and K be the nonempty compact subset of W appearing in the equality (2), then:  Since M ∈ C(W ), then for arbitrary ε > 0 and y ∈ Y there exists a positive number L(ε, y) such that U (t, σ(−t, y))M ⊆ B(I y , ε) for any t ≥ L(ε, y). Note that I ′ y = U (t, σ(−t, y))I ′ σ(t,y) ⊆ U (t, σ(−t, y))M ⊆ B(I y , ε). Since ε is an arbitrary positive number we obtain I ′ y ⊆ I y for any y ∈ Y .
Definition 3.7. Let W, ϕ, (Y, T, σ) be compactly dissipative, K be the nonempty compact subset of W appearing in the equality (2) and I y := ω y (K) for any y ∈ Y . The family of compact subsets {I y | y ∈ Y } is said to be a Levinson center (compact global attractor) of nonautonomous (cocycle) dynamical system W, ϕ, (Y, T, σ) .
where J y := I y × {y}, is invariant with respect to skew-product dynamical system (X, T + , π) (X := W × Y and π := (ϕ, σ)); (ii) if {I y | y ∈ Y } is relatively compact, then the family {I y | y ∈ Y } is upper semicontinuous if and only if the set J is closed in X.
Proof. The first statement is evident.
Suppose that the set J ⊆ X is closed. If we suppose that the family {I y | y ∈ Y } is not upper semicontinuous, then there are ε 0 > 0, y 0 ∈ Y and sequences {y n } ⊂ Y and {u n } ⊂ W such that y n → y 0 as n → ∞, u n ∈ I yn and (3) ρ(u n , I y0 ) ≥ ε 0 .
Since {I y | y ∈ Y } is relatively compact, then without loss of generality we can suppose that the sequence {u n } is convergent. Denote by u 0 := lim n→∞ u n and passing into limit as n → ∞ in inequality (3) we obtain u 0 / ∈ I y0 . On the other hand we have (u n , y n ) ∈ J yn ⊆ J for any n ∈ N and since the set J is closed and (u n , y n ) → (u 0 , y 0 ) as n → ∞, then (u 0 , y 0 ) ∈ J and, consequently, u o ∈ I y0 . The obtained contradiction proves our statement.
Let now the family {I y } upper semicontinuous and (ū,ȳ) ∈ J. Then there exists a sequence {(u n , y n ) ∈ J} such that (u n , y n ) → (ū,ȳ). Since u n ∈ I yn and {I y | y ∈ Y } is upper semicontinuous, then u 0 ∈ I y0 and, consequently, (u 0 , y 0 ) ∈ J y0 ⊆ J. Thus the set J is closed. is a dynamical system on C(T × W, X) which is called a shift dynamical system (dynamical system of translations or Bebutov's dynamical system).
In the quality of the property (A) there can stand stability in the sense of Lagrange (st. L), uniform stability (un. st. L + ) in the sense of Lyapunov, periodicity, almost periodicity, asymptotical almost periodicity and so on.
Then for any y ∈ Y there exists at least one point w y ∈ I y such that the motion ϕ(t, u y , y) is defined on entire axis T and it is Bohr almost periodic.
One of the main goal of this paper is a positive answer to I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems (Corollary 4.3).
Let R d , ϕ, (Y, T, σ) be a cocycle over (Y, T, σ) with fiber R d . (i) if y 0 ∈ ω y0 , then there exists a point a y0 ∈ I y0 such that the full trajectory γ y0 with x 0 := γ y0 (0) = (a y0 , y 0 ) is comparable by character of recurrence with y 0 ; (ii) if y 0 is strongly Poisson stable, then the exists a point a y0 ∈ I y0 such that the full trajectory γ y0 with x 0 := γ y0 (0) = (a y0 , y 0 ) is strongly comparable by character of recurrence with y 0 .
The second statement of Theorem can be proved using the same argument as in the proof of the first statement but instead of Theorem 2.25 we need to apply Theorem 2.31.  be the corresponding skew-product dynamical system, where X := R d × Y and π := (ϕ, σ).
Definition 5.1. The cocycle R d , ϕ, (Y, T, σ) is said to be dissipative if for any y ∈ Y there is a positive number r y such that lim sup t→+∞ |ϕ(t, u, y)| < r y for any y ∈ Y and u ∈ R d , i.e., for all u ∈ R d and y ∈ Y there exists a positive number L(u, y) such that |ϕ(t, u, y)| < r y for any t ≥ L(u, y).
Definition 5.2. The cocycle E, ϕ, (, T, σ) is said to be uniformly dissipative if there exists a positive number r (r is not depend upon y ∈ Y ) such that for any R > 0 there is a positive number L(R) such that |ϕ(t, u, y)| < r for all y ∈ Y and |u| ≤ R and t ≥ L(R).
There is a positive number r 1 such that for all u ∈ R d and y ∈ Y there exists τ = τ (u, y) > 0 for which |ϕ(τ, u, y)| < r 1 .

3.
There is a positive number r 2 such that lim inf t→+∞ |ϕ(t, u, y)| < r 2 for all u ∈ R d and y ∈ Y . 4. There exists a positive number R 0 and for all R > 0 there is l(R) > 0 such that |ϕ(t, u, y)| ≤ R 0 for all t ≥ l(R), u ∈ R d , |u| ≤ R and y ∈ Y .
Note that every condition 1.-4. that figures in Theorem 5.4 is equivalent to the dissipativity of the non-autonomous dynamical system (X, T + , π), (Y, T, σ), h associated by the cocycle R d , ϕ, (Y, T, σ) over (Y, T, σ) with the fiber R d .

5.2.
Ordinary Differential Equations. We will give below an example of a skewproduct dynamical system which plays an important role in the study of nonautonomous differential equations.
Condition (A1). The function f ∈ C(R × R d , R d ) is said to be regular if for every equation (5) the conditions of existence, uniqueness and extendability on R + are fulfilled.
Definition 5.6. Recall that the equation (4) is called dissipative [12], [18], [32], [33], if for all t 0 ∈ R and x 0 ∈ E n there exists a unique solution x(t; x 0 , t 0 ) of the equation (4) passing through the point (x 0 , t 0 ) and if there exists a number R > 0 such that lim t→+∞ sup |x(t; x 0 , t 0 )| < R for all x 0 ∈ R d and t 0 ∈ R. In other words, for every solution x(t; x 0 , t 0 ) there is an instant t 1 = t 0 + l(t 0 , x 0 ), such that |x(t; x 0 , t 0 )| < R for any t ≥ t 1 . If for any r > 0 the number l(t 0 , x 0 ) can be chosen independent on t 0 and x 0 with |x 0 | ≤ r, then the equation (4) is called uniformly dissipative [12].
Below we will establish the relation between the dissipativity of the equation (4) and the dissipativity of the non-autonomous dynamical system generated by the equation (4). (4) is uniformly dissipative, then the cocycle ϕ generated by equation (4) is also uniformly dissipative.
Lemma 5.8. [10, ChIII] Let f ∈ C(R × E n , E n ) be regular. If H(f ) is compact, then equation (4) is uniformly dissipative if and if there is a positive number r such that lim sup t→+∞ |ϕ(t, x 0 , g)| < r (x 0 ∈ R d , g ∈ H(f )) .

Thus, for the equation (4) (f is regular and H(f ) is compact) we established that it is uniformly dissipative if and only if the non-autonomous dynamical system generated by this equation is dissipative.
Condition (A2). Equation (4) is monotone. This means that the cocycle R n , ϕ, (H(f ), R, σ) (or shortly ϕ) generated by (4) is monotone, i.e., if u, v ∈ R d and u ≤ v then ϕ(t, u, g) ≤ ϕ(t, v, g) for all t ≥ 0 and g ∈ H(f ).
Let K be a closed cone in R d . The dual cone to K is the closed cone K * in the dual space R d * of linear functions on R d , defined by where ·, · is the scalar product in R d .
Recall [30], [31,ChV] that the function f ∈ C(R × R d , R d ) is said to be quasimonotone if for any (t, u), (t, v) ∈ R × R d and φ ∈ R d t, v)).
Lemma 5.9. Let f ∈ C(R × R d , R d ) be a regular and quasimonotone function, then the following statements hold: for any t ≥ 0 and g ∈ H(f ); (iv) equation (4) is monotone.
Since f is quasimonote, then we will have and passing into limit in (8) as k → ∞ we obtain that g is quasimonotone too.
Finally, the third statement follows from the first and second statements. Lemma is completely proved. (4) is said to be: -uniformly Lyapunov stable in the positive direction, if for arbitrary ε > 0 there exists δ = δ(ε) > 0 such that where by bar is denoted the closure in R d and ϕ(R + , u 0 , f ) := {ϕ(t, u 0 , f ) : t ∈ R + }.
Remark 5.13. If x 0 := (u 0 , y 0 ) ∈ X := W × Y and α y0 (respectively, γ y0 ) is a point from X defined in Lemma 2.24 then we denote by α u0 (respectively, γ u0 ) a point from W such that α y0 = (α u0 , y 0 ) (respectively, γ y0 = (γ u0 , y 0 )). Definition 5.14. A function f is said to be Poisson stable (respectively, strongly Poisson stable) in t ∈ T uniformly with respect to u on every compact subset of Theorem 5.15. Suppose that the following assumptions are fulfilled: uniformly with respect to u on every compact subset from R d ; -equation (4) is uniformly dissipative; -each solution ϕ(t, u 0 , f ) of equation (4) is positively uniformly Lyapunov stable.
Then under conditions (A1) − (A2) the following statement hold: 1. equation (4) has at least one solution ϕ(t, γ u0 , f ) defined and bounded on R which is compatible and belongs to Levinson center of (4).

2.
if the function f ∈ C(R × R n , R n ) is stationary (respectively, τ -periodic, Levitan almost periodic, almost recurrent, almost automorphic, Poisson stable) in t ∈ R uniformly with respect to u on every compact subset from R n , then ϕ(t, γ u0 , f ) is also stationary (respectively, τ -periodic, Levitan almost periodic, almost recurrent, almost automorphic, Poisson stable).
Proof -the function f ∈ C(R×R d , R d ) is strongly Poisson stale in t ∈ R uniformly with respect to u on every compact subset from R d ; -equation (4) is uniformly dissipative; -each solution ϕ(t, u 0 , g) of every equation (5) is positively uniformly Lyapunov stable.
Then under conditions (A1) − (A2) the following statements hold: Proof. This statement can be proved similarly as Theorem 5.15 using Theorems 4.2 (the second statement) and Corollary 4.3. Theorem is proved.

Difference Equations.
Example 5.17. Consider the equation Along with equation (9) we will consider H-class of (9), i.e., the family of equation where H(f ) := {f τ | τ ∈ Z} and by bar is denoted the closure in the space Denote by ϕ(n, v, g) the solution of equation (10) with initial condition ϕ(0, v, g) = v. From the general properties of difference equations it follows that: (ii) ϕ(n + m, v, g) = ϕ(n, ϕ(m, v, g), σ(m, g)) for all n, m ∈ Z + and (v, g) ∈ R d × H(f ); (iii) the mapping ϕ is continuous.
Suppose that the following conditions hold: (i) u 1 , u 2 ∈ R d and u 1 ≤ u 2 ; (ii) the function f is monotone non-decreasing with respect to variable u ∈ R d , i.e., f (t, u 1 ) ≤ f (t, u 2 ) for any t ∈ Z.
for any t ∈ Z and k ∈ N. Passing in limit as k → ∞ in (11) we obtain g(t, v 1 ) ≤ g(t, v 2 ) for any t ∈ Z.
Theorem 5.21. Suppose that the following assumptions are fulfilled: -the function f ∈ C(Z × R d , R d ) is positively Poisson stable in t ∈ Z uniformly with respect to u on every compact subset from R n ; -equation (9) is uniformly dissipative; -each solution ϕ(t, u 0 , f ) of equation (9) is positively uniformly Lyapunov stable.
Then under condition (D) the following statement hold: 1. equation (9) has at least one solution ϕ(t, γ u0 , f ) defined and bounded on Z which is compatible and belongs to Levinson center of (9). 2. if the function f ∈ C(Z × R d , R d ) is stationary (respectively, τ -periodic, Levitan almost periodic, almost recurrent, almost automorphic, Poisson stable) in t ∈ Z uniformly with respect to u on every compact subset from R d , then ϕ(t, γ u0 , f ) is also stationary (respectively, τ -periodic, Levitan almost periodic, almost recurrent, almost automorphic, Poisson stable).
Proof. Let f ∈ C(Z × R d , R d ) and (C(Z × R d , R d ), Z, σ) be the shift dynamical system no C(Z × R d , R d ). Denote by Y := H(f ) and (Y, Z, σ) the shift dynamical system on H(f ) induced by (C(Z × R d , R d ), Z, σ). Consider the cocycle R n , ϕ, (Y, Z, σ) generated by equation (9). Now to finish the proof of Theorem it is sufficient to apply Theorems 4.2 (the first statement) and Corollary 4.3. Theorem is proved. -the function f ∈ C(Z×R d , R d ) is strongly Poisson stale in t ∈ Z uniformly with respect to u on every compact subset from R d ; -equation (9) is uniformly dissipative; -each solution ϕ(t, u 0 , g) of every equation (10) is positively uniformly Lyapunov stable.
Then under condition (D) the following statements hold: