Levitan/Bohr Almost Periodic and Almost Automorphic Solutions of Second-Order Monotone Differential Equations

LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF SECOND-ORDER MONOTONE DIFFERENTIAL EQUATIONS TOM´ AS CARABALLO AND DAVID CHEBAN Abstract. The aim of this paper is to prove the existence of Levitan/Bohr almost periodic, almost automorphic, recurrent and Poisson stable solutions of the second order differential equation (1) x′′ =f(σ(t, y), x, x′),(y∈Y) where Yis a complete metric space and (Y, R, σ) is a dynamical system (also called a driving system). When the function fin (1) is increasing with respect to its second variable, the existence of 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 (1) admits at least one solution ϕsuch that ϕand ϕ′are bounded on the real axis. 1. Introduction The aim of this paper is to analyze the existence of Levitan/Bohr almost periodic, almost automorphic, recurrent and Poisson stable solutions of the second order differential equation (2) x′′ =f(σ(t, y), x, x′),(y∈Y) where Yis a complete metric space, and (Y, R, σ) is a (driving) dynamical system. The existence of Bohr almost periodic solutions of equation (3) x′′ =f(t, x, x′) with Bohr almost periodic right hand-side fwith respect to time, uniformly with respect to the variables x, x′on every compact subset in R2, was studied by C. Corduneanu in [15] (see also [1]), where it was established that, if ∂f(t,x,u) ∂x ≥k > 0 for all (t, x, u)∈R3,equation (3) admits a unique Bohr almost periodic solution. When the function f(t, x, u) is only increasing (in the large sense), the same problem was studied by Z. Opial in [21], where the following result was established. Theorem 1.1. (Z. Opial [21]) Suppose that the following conditions are fulfilled: Date: October 16, 2010. 1991 Mathematics Subject Classification. primary:34C11, 34C30, 34C35, 34D45, 37C55, 37C60, 37C65, 37C70, 37C75. Key words and phrases. Non-autonomous dynamical systems; skew-product systems; cocycles; quasi-periodic, Bohr/Levitan almost periodic, almost automorphic, pseudo-recurrent solutions, monotone second order equation. 1 2 TOM´ AS CARABALLO AND DAVID CHEBAN (i) f∈C(R3,R)and is increasing in the large sense with respect to the variable x, i.e., the inequality x1≤x2implies f(t, x1, u)≤f(t, x2, u)for all u, t ∈R; (ii) for all r > 0, there exists a number L(r)>0such that |f(t, x1, u1)− f(t, x2, u2)| ≤ L(r)(|x1−x2|+|u1−u2|)for all |xi|,|ui| ≤ r(i= 1,2) and t∈R. Then, the following statements hold: (i) If equation (3) admits a solution usuch that uand its first derivative u′are bounded on R, then this equation admits at least one Bohr almost periodic solution. (ii) If u(t)and v(t)are two Bohr almost periodic solutions of equation (3), then there exists a constant c∈Rsuch that u(t)−v(t) = cfor all t∈R. (iii) If the function fis strictly increasing with respect to the second variable x∈R, then equation (3) admits at most one Bohr almost periodic solution. Some generalization of Theorem 1.1 when (3) is a vectorial equation (i.e., f∈ C(Y×Rn×Rn,Rn) (n≥2) are established in [16] and [13]. In [12], P. Cieutat studied the existence of bounded and Bohr almost periodic solutions of the following Li´enard equation (4) x′′ +f(x)x′+g(x) = p(t), where p:R→Ris a Bohr almost periodic function, f(x)≥0 and gis a strictly decreasing function. Namely, it was proved in [12] that every solution, which is bounded on R+, is asymptotically Bohr almost periodic, and there exists a unique Bohr almost periodic solution of equation (4). A typical model for such equation (4) is x′′ +cx′+ 1/xα=p(t),(x∈(0,+∞)), where c≥0, α > 0 and pis Bohr almost periodic. Recently, the existence of almost automorphic solutions of equation (4) with almost automorphic forcing term pwas studied by Cieutat et al. in [14], where they proved the asymptotically almost automorphy of every solution which is bounded on R+, and the existence of a unique almost automorphic solution of equation (4). In the periodic case (i.e. when pis periodic), the dynamics of equation (5) was intensively studied by P. Mart´ınez-Amores and P. J. Torres [20] and J. Campos and P. J. Torres [4]. Desheng Li and Jinqiao Duan [19] analyzed the structure of the set of bounded solutions for equation (2). In particular, they proved the existence of a unique periodic (respectively, quasi-periodic, Bohr almost periodic) solution of equation (2) if the point y∈Yis also periodic (respectively, quasi-periodic, Bohr almost periodic), and the function fis strictly increasing with respect to its second variable. Namely, the following theorem was proved in [19]. Theorem 1.2. [19] Suppose that the following conditions are fulfilled: (i) (H, ρ)is a compact complete metric space and (H, R, θ)is a minimal dynamical system on H; LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 3 (ii) f:H×I×R→Ris a continuous map, where I:= (a, b)⊆R; (iii) For any compact subset V⊂(a, b)×R, there exists an L > 0such that |f(h, x, p)−f(h, y, q)| ≤ L(|x−y|+|p−q|)for all (x, p),(y, q)∈Vand h∈H; (iv) f(h, x, p)is strictly increasing in x; (v) For any compact interval I⊂Ithere exists c0>0such that |f(h, x, p)| ≤ c0(1 + |p|2)for all (h, x, p)∈H×I×R. Then: (i) There exists a continuous map Γ : H7→ Rsuch that for each h∈H, γh(t) := Γ(θ(t, h)) is the unique solution of equation (5) x′′ =f(θ(t, h), x, x′) which is bounded on R; (ii) For each h∈H, there exists a continuous decreasing function Φhdefined on a maximal nonempty open interval D(Φh)⊂I, such that for any x∈D(Φh),x(t) := ψh(t, x, Φh(x)) is the unique solution of (5), which is bounded on R+, that satisfies x(0) = x, where ψ(t, x, x′)denotes the unique solution of equation (5) passing through the point (x, x′)∈I×Rat the initial moment t= 0; (iii) For any compact interval D⊂D(Φh) lim t→+∞(|ψh(t, x, Φh(x)) −γh(t)|+|ψ′ h(t, x, Φh(x)) −γ′ h(t)|) = 0 uniformly with respect to x∈D. We note that, in all of the previously cited works (with the exception of [21]), an assumption of strict monotony is imposed. In the present paper, we consider equation (2) when the function fis increasing with respect to its second variable in the large sense. All of our results will be formulated and proved for this case which includes, of course, the strictly increasing one. The paper is organized as follows. In Section 2, we collect some notions (quasi periodicity, Levitan/Bohr almost periodicity, almost automorphy, recurrence, pseudo recurrence, Poisson stability) facts and constructions (Bebutov dynamical systems, skew-product dynamical systems, cocycles etc) from the theory of dynamical systems which will be necessary in this paper. Section 3 is dedicated to the study of a special class of non-autonomous dynamical systems (NDS): the so-called NDS with convergence. The main result in this section is Theorem 3.9 which provides sufficient conditions for the convergence of a NDS. An application of Theorem 3.9 to study the dynamics of the scalar one-dimensional equation x′=f(σ(t, y), x) (y∈Y) with pseudo recurrent base (Y, R, σ) (driving system) is carried out in Section 4. The main result of this section is Theorem 4.2. Levitan almost periodic and almost automorphic solutions of a second order equation of the form x′′ =f(σ(t, y), x, x′) and with increasing f(in the large sense) 4 TOM´ AS CARABALLO AND DAVID CHEBAN are analyzed in Section 5. The main results of this section are Theorem 5.4 and Corollary 5.5. Section 6 is devoted to the existence of quasi-periodic, Bohr almost periodic and recurrent solutions (in the sense of Birkhoff) of the equation x′′ =f(σ(t, y), x, x′) with increasing f(in the large sense). The main results proved in this section are Theorem 6.1 and Corollary 6.2. Finally, in Section 7, we discuss some generalizations of our main results (theorems 5.4 and 6.1). One of this type of results is established in Theorem 7.1 (see also corollaries 7.2 and 7.3). 2. Bohr/Levitan Almost Periodic and Almost Automorphic Motions of Dynamical Systems We recall now some notions, facts and constructions from the theory of dynamical systems. Although we could refer the readers to other publications for these preliminaries (see, for instance, Caraballo and Cheban [5, 6]), in order to keep our paper as much self-contained as possible, we prefer to include the results here. 2.1. Recurrent, Bohr Almost Periodic and Almost Automorphic Motions. Let (X, ρ) be a complete metric space, Sbe one of the two sets Ror Z, and T⊆S(S+⊆T) be a sub-semigroup of the additive group S, where S+:= {s∈S:s≥0}. Let (X, T, π) be a dynamical system on X, i.e., let π:T×X→Xbe a continuous function such that π(0, x) = xfor all x∈X, and π(t1+t2, x) = π(t2, π(t1, x)), for all x∈X, and t1, t2∈T. Given ε > 0,a number τ∈Tis called an ε−shift (respectively, an ε−almost period) of x, if ρ(π(τ, x), x)< ε (respectively, ρ(π(τ+t, x), π(t, x)) < ε for all t∈T). A point x∈Xis called almost recurrent (respectively, Bohr almost periodic), if for any ε > 0 there exists a positive number lsuch 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∈Xis almost recurrent and the set H(x) := {π(t, x)|t∈T}is compact, then xis called recurrent, where the bar denotes the closure in X. Denote by Nx:= {{tn} ⊂ T: such that {π(tn, x)} → xand {tn} → ∞} and Mx:= {{tn} ⊂ T: such that {π(tn, x)}is convergent and {tn} → ∞}. A point x∈Xis called Poisson stable in the positive direction if there exists a sequence {tn} ∈ Nxsuch that tn→+∞as n→ ∞. Let (X, T, π) be a two-sided dynamical system (i.e., T=S). A point x∈Xis called Poisson stable in the negative direction if there exists a sequence {tn} ∈ Nx such that tn→ −∞ as n→ ∞. The point x∈Xis called Poisson stable if it is Poisson stable in both directions. LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 5 A dynamical system (X, T, π) is said to be (i) transitive, if there exists a point x0∈Xsuch that H(x0) = X, where H(x0) := {π(t, x0) : t∈T}; (ii) pseudo recurrent if Xis compact, the dynamical system (X, T, π) is transitive, and every point x∈Xis Poisson stable. A point x∈Xis called [26, 28] pseudo recurrent if the dynamical system (H(x),T, π) is pseudo recurrent. Remark 2.1. Every recurrent point is pseudo recurrent, but there exist pseudo recurrent points which are not recurrent [26, 28]. An m-dimensional torus is denoted by Tm:= Rm/2πZ.Let (Tm,T, σ) be an irrational winding of Tm, i.e., σ(t, ν) := (ν1t, ν2t,...,νmt) for all t∈Sand ν∈ T m. A point x∈Xis called quasi-periodic with the frequency ν:= (ν1, ν2,...,νm)∈ Tm, if there exists a continuous function Φ : Tm→Xsuch that π(t, x) := Φ(σ(t, ω)) for all t∈T,where (Tm,T, σ) is an irrational winding of the torus Tmand ω∈ T m. A point x∈Xof the dynamical system (X, T, π) is called Levitan almost periodic [18], if there exists a dynamical system (Y, T, σ) and a Bohr almost periodic point y∈Ysuch that Ny⊆Nx. Remark 2.2. Let xi∈Xi(i= 1,2,...,m) be a Levitan almost periodic point of the dynamical system (Xi,T, πi).Then the point x:= (x1, x2,...,xm)∈X:= X1×X2×... ×Xmis also Levitan almost periodic in the product dynamical system (X, T, π),where π:T×X→Xis defined by the equality π(t, x) := (π1(t, x1), π2(t, x2), . . . , πm(t, xm)) for all t∈Tand x:= (x1, x2,...,xm)∈X. A point x∈Xis called stable in the sense of Lagrange (st.L) (respectively, stable in the sense of Lagrange in the positive direction (st.L+)), if its trajectory {π(t, x) : t∈T}(respectively, its positive semi-trajectory {π(t, x) : t∈T+}) is relatively compact, where T+; = {t∈T:t≥0}. A point x∈Xis called almost automorphic [18, 24] in the dynamical system (X, T, π),if the following conditions hold: (i) xis st.L; (ii) there exists a dynamical system (Y, T, σ),a homomorphism hfrom (X, T, π) onto (Y, T, σ), and a point y∈Ywhich is almost periodic, in the sense of Bohr, such that h−1(y) = {x}. Remark 2.3. Notice the following well-known facts. 1. Every almost automorphic point is Levitan almost periodic. 2. A Levitan almost periodic point is almost automorphic if and only if is stable in the sense of Lagrange. 2.2. Shift Dynamical Systems, Levitan/Bohr Almost Periodic and Almost Automorphic Functions. Below we recall a general method of construction of dynamical systems on spaces of continuous functions. In this way, we will 6 TOM´ AS CARABALLO AND DAVID CHEBAN obtain many well-known dynamical systems on some functional spaces (see, for example, [2, 23, 26]). Let (X, T, π) be a dynamical system on X, Y a complete pseudo metric space, and P a family of pseudo metrics on Y. We denote by C(X, Y ) the family of all continuous functions f:X→Yequipped with the compact-open topology. This topology is given by the following family of pseudo metrics {dp K}(p∈ P, K ∈ C(X)),where dp K(f, g) := sup x∈K p(f(x), g(x)) and C(X) denotes the family of all compact subsets of X. For all τ∈Twe define a mapping στ:C(X, Y )→C(X, Y ) by the following equality: (στf)(x) := f(π(τ, x)), x ∈X. We note that the family of mappings {στ:τ∈T}possesses the next properties: a. σ0=idC(X,Y ); b. στ1◦στ2=στ1+τ2, for all τ1, τ2∈T; c. στis continuous for all τ∈T. Lemma 2.4. [7] The mapping σ:T×C(X, Y )→C(X, Y ),defined by the equality σ(τ, f) := στf(f∈C(X, Y ), τ ∈T),is continuous, and the triple (C(X, Y ),T, σ) is a dynamical system on C(X, Y ). Consider now some examples of dynamical systems of the form (C(X, Y ),T, σ), which are useful in the applications. Example 2.5. Let X=T, and denote by (X, T, π) a dynamical system on T, where π(t, x) := x+t. The dynamical system (C(T, Y ),T, σ) is called Bebutov’s dynamical system [2, 23, 26] (dynamical system of translations, or shifts dynamical system). It is said that the function ϕ∈C(T, Y )possesses a property (A), if the motion σ(·, ϕ) : T→C(T, Y ), generated by this function, possesses this property in the Bebutov dynamical system (C(T, Y ),T, σ). As property (A) we can take periodicity, quasi-periodicity, Bohr/Levitan almost periodicity, almost automorphy, recurrence, pseudo recurrence, Poisson stability, etc. Example 2.6. Let X:= T×W, where Wis a metric space, and let (X, T, π) denote a dynamical system on Xdefined in the following way: π(t, (s, w)) := (s+t, w). Using the general method proposed above, we can define on C(T×W, Y ) a dynamical system of translations (C(T×W, Y ),T, σ). The function f∈C(T×W, Y ) is called Bohr/Levitan almost periodic (quasiperiodic, recurrent, almost automorphic, etc) with respect to t∈T, uniformly in won every compact from W, if the motion σ(·, f) is Bohr/Levitan almost periodic (quasi-periodic, recurrent, almost automorphic, etc.) in the dynamical system (C(T×W, Y ),T, σ). Remark 2.7. Recall that for a compact metric space W, the topology on C(T× W, Y )is metrizable. For example, the equality d(f, g) := ∞ X k=1 1 2k dk(f, g) 1 + dk(f, g) LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 7 defines a complete metric on the space C(T×W, X)which is compatible with the compact-open topology on C(T×W, X), where dk(f, g) := max |t|≤k, x∈Wρ(f(t, x), g(t, x)). The space C(T×W, Y )is topologically isomorphic to C(T, C(W, Y )) (see [26]), and also the shifts dynamical systems (C(T×W, Y ),T, σ)and (C(T, C(W, Y )),T, σ)are dynamically isomorphic. 2.3. Cocycles, Skew-Product Dynamical Systems and Non-Autonomous Dynamical Systems. Let T1⊆T2be two sub-semigroups of the group S(S+⊆ T1). A triplet h(X, T1, π),(Y, T2, σ), hi, where his a homomorphism from (X, T1, π) onto (Y, T2, σ) (i.e., his continuous and h(π(t, x)) = σ(t, h(x)) for all t∈T1and x∈X), is called a non-autonomous dynamical system. Let (Y, T2, σ) be a dynamical system, Wa complete metric space, and ϕa continuous mapping from T1×W×Yinto W, possessing the following properties: a. ϕ(0, u, y) = u(u∈W, y ∈Y); b. ϕ(t+τ, u, y) = ϕ(τ, ϕ(t, u, y), σ(t, y)) (t, τ ∈T1, u ∈W, y ∈Y). Then, the triplet hW, ϕ, (Y, T2, σ)i(or shortly ϕ) is called [23] a cocycle on (Y, T2, σ) with fiber W. Let X:= W×Yand let us define a mapping π:X×T1→Xas follows: π((u, y), t) := (ϕ(t, u, y), σ(t, y)) (i.e., π= (ϕ, σ)). Then, it is easy to see that (X, T1, π) is a dynamical system on X, which is called a skew-product dynamical system [23] and h=pr2:X→Yis a homomorphism from (X, T1, π) onto (Y, T2, σ) and, hence, h(X, T1, π),(Y, T2, σ), hiis a non-autonomous dynamical system. Thus, if we have a cocycle hW, ϕ, (Y, T2, σ)ion the dynamical system (Y, T2, σ) with fiber W, then it generates a non-autonomous dynamical system h(X, T1, π), (Y, T2, σ), hi(X:= W×Y), called a non-autonomous dynamical system generated by the cocycle hW, ϕ, (Y, T2, σ)ion (Y, T2, σ). 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 a cocycle (a non-autonomous dynamical system). Below we give some examples of this type. Example 2.8. Consider the system of differential equations (6) u′=F(y, u) y′=G(y), where Y⊆Em(for example, Y=Tmis an m–torus), G∈C(Y, En) and F∈ C(Y×En, En). Suppose that, for the system (6), the conditions ensuring existence, uniqueness and extendability of solutions to R+are fulfilled. Denote by (Y, R+, σ) a dynamical system on Ygenerated by the second equation of the system (6) and by ϕ(t, u, y) we denote the solution of the equation u′=F(σ(t, y), u) passing through the point u∈Enat t= 0. Then, the mapping ϕ:R+×En×Y→ Ensatisfies conditions a. and b. from the definition of cocycle and, consequently, 8 TOM´ AS CARABALLO AND DAVID CHEBAN system (6) generates a non-autonomous dynamical system h(X, R+, π),(Y, R+, σ), hi (where X:= En×Y,π:= (ϕ, σ) and h:= pr2:X→Y). Example 2.9. Let (Y, R, σ) be a dynamical system on the metric space Y. We consider the equation (7) u′=F(σ(y, t), u) (y∈Y), where F∈C(Y×Rn,Rn). Suppose again that, for equation (7), the conditions for the existence, uniqueness and extendability of solutions to R+are fulfilled. The non-autonomous dynamical system h(X, R+, π),(Y, R, σ), hi(respectively, the cocycle hE, ϕ, (Y, R, σ)i), where X:= Rn×Y,π:= (ϕ, σ), ϕ(·, x, y) is the solution of (7) passing through the point xat time t= 0, and h:= pr2:X→Yis generated by equation (7). Example 2.10. We consider the equation (8) u′=f(t, u), where f∈C(R×Rn,Rn). Along with equation (8), consider the family of equations (9) u′=g(t, u), where g∈H(f) := {fτ:τ∈R}and fτis the τ-shift of fwith respect to the time variable t, i.e., fτ(t, u) := f(t+τ, u) for all (t, u)∈R×Rn. Suppose that the function fis regular [23], i.e., for all g∈H(f) and u∈Rnthere exists a unique solution ϕ(t, u, g) of equation (9). Denote by Y=H(f) and (Y, R, σ) a shift dynamical system on Yinduced by the Bebutov dynamical system (C(R×Rn,Rn),R, σ). Now the family of equations (9) can be written as (7) if we take the mapping F∈C(Y×Rn,Rn) defined by F(g, u) := g(0, u), for all g∈H(f) and u∈Rn. A solution ϕ(t, u, y) of equation (7) is called [26, 28] compatible (respectively, uniformly compatible) by the character of recurrence if Ny⊆Nϕ(respectively, My⊆ Mϕ), where Nϕ(respectively, Mϕ) is the set of all sequences {tn} ⊂ Rsuch that {ϕ(t+tn, u, y}converges to ϕ(t, u, y) (respectively, {ϕ(t+tn, u, y}converges) in the space C(T,Rn). Remark 2.11. The sequence {ϕ(t+tn, u, y)}converges to the function ψin the space C(T,Rn)if and only if {ϕ(tn, u, y)}converges to ψ(0). Theorem 2.12. [26, 28] The following statements hold: 1. Let y∈Ybe a stationary (respectively, τ-periodic, Levitan almost periodic, almost recurrent, Poisson stable) point. If ϕ(t, u, y)is a compatible solution of equation (7), then so is ϕ(t, u, y). 2. Let y∈Ybe a stationary (respectively, τ-periodic, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent) point. If ϕ(t, u, y)is a uniformly compatible solution of equation (7), then so is ϕ(t, u, y). Example 2.13. Let us consider a second order differential equation (10) x′′ =f(σ(t, y), x, x′),(y∈Y) where f∈C(Y×Rn×Rn,Rn), and state a criterion for the existence of Levitan almost periodic and almost automorphic solutions for this equation. Below we will suppose that the function fis regular, i.e., for all y∈Yand x, x′∈Rnthe equation LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 9 (10) admits a unique solution ϕ(t, x, x′, y) defined on R+with the initial conditions ϕ(0, x, x′, y) = xand ϕ′(0, x, x′, y) = x′. As it is well-known, we can reduce equation (10) to the following equivalent system (11) u′=v v′=f(σ(t, y), u, v), (y∈Y) or to the equation z′=F(σ(t, y), z) on the product space Rn×Rn,where z:= (u, v) and F∈C(Y×Rn×Rn,Rn×Rn) is the function defined by the equality F(y, z) := (v, f(y, u, v)) for all y∈Yand z:= (u, v)∈Rn×Rn. Theorem 2.14. [26] Let ϕ∈C(R,Rn)be a continuously differentiable function. If its derivative ϕ′∈C(R,Rn)is uniformly continuous on R, then ϕ′is uniformly comparable by the character of recurrence with ϕ, i.e., Mϕ⊆Mϕ′. We can now prove the following result. Lemma 2.15. Suppose that the following conditions hold: (i) Yis compact; (ii) f∈C(Y×Rn×Rn,Rn)is regular; (iii) ϕ(t, x0, x′ 0, y)is a solution of equation (10) defined and bounded on Rand such that its derivative ϕ′(t, x0, x′ 0, y)is also bounded on R. Then, the following two statements are equivalent: a. The solution ϕ(t, x0, x′ 0, y)of equation (10) is compatible (respectively, uniformly compatible) by the character of recurrence with the right-hand side; b. The solution (ϕ(t, x0, x′ 0, y), ϕ′(t, x0, x′ 0, y)) of equation (11) is compatible (respectively, uniformly compatible) by the character of recurrence with the right-hand side. Proof. The implication b.=⇒a. is evident. Thus, to prove the lemma, it is sufficient to establish the converse implication. Let ϕ(t, x0, x′ 0, y) be a solution of equation (10) such that ϕ(t, x0, x′ 0, y) and ϕ′(t, x0, x′ 0, y) are defined and bounded on R. Then Ny⊆Nϕ(respectively, My⊆Mϕ). We need to show that the inclusion Ny⊆Nϕ′(respectively, My⊆Mϕ′) also holds. Indeed, let {tn} ∈ Ny (respectively, {tn} ∈ My), then the sequence {σ(tn, y)}converges to y(respectively, the sequence {σ(tn, y)}converges to some point ˜y∈Y). Consequently, the functional sequence {f(σ(t+tn, y), u, v)}converges to f(σ(t, y), u, v) (respectively, to f(σ(t, ˜y), u, v)) uniformly with respect to ton every compact subset from R and u, v ∈Q:= ϕ(R, x0, x′ 0, y)×ϕ′(R, x0, x′ 0, y). Since Ny⊆Nϕ(respectively, My⊆Mϕ), the sequence {ϕ(tn, x0, x′ 0, y)}converges to x0(respectively, to some point ˜x0∈R). Since the function f∈C(Y×Rn×Rn,Rn) is regular, then the functional sequence {ϕ(t+tn, x0, x′ 0, y)}converges to the function ϕ(t, x0, x′ 0, y) (respectively, to ϕ(t, ˜x0,˜x0′,˜y)) uniformly with respect to ton every compact subset from R. Note that, under the conditions of Lemma, the second derivative ϕ′′(t, x0, x′ 0, y) of the function ϕ(t, x0, x′ 0, y) is bounded on Rand, consequently, the first derivative ϕ′(t, x0, x′ 0, y) is uniformly continuous in t∈R. Thus, according 16 TOM´ AS CARABALLO AND DAVID CHEBAN If (16) admits a solution which is bounded on R+, then it possesses a unique stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent) solution which is globally uniformly asymptotically stable. Proof. This statement follows directly from Theorem 4.2 and Remark 3.1 (item 2 (ii)).  Remark 4.4. 1. The analog of Theorem 4.2 (as well as Corollary 4.3) holds if we replace the condition “fis strictly decreasing” by “fis strictly increasing”. This case can be reduced to the considered one by the time substitution t→ −t. 2. Note that Theorem 4.2 and Corollary 4.3 remain true also for a vectorial equation (i.e., for systems of equations). Indeed, to this end, we assume that f∈C(Y× Rn,Rn), and replace the condition “fis strictly decreasing” by the condition hf(y, u1)−f(y, u2), u1−u2i<0 for all y∈Yand u1, u2∈Rn(u16=u2), where h,iis the scalar product on the space Rn. 3. If the function f∈C(Y×R,R)is continuously differentiable with respect to x∈Rand (18) ∂f ∂x(y, x)≤ −k < 0 for all y∈Yand x∈R, then Theorem 4.2 and Corollary 4.3 also hold without the requirement that equation (16) admits at least one solution which is bounded on R+. Owing to condition (18), it follows (19) hf(y, u1)−f(y, u2), u1−u2i ≤ −k|u1−u2|2 for all y∈Yand u1, u2∈R. But condition (19) guarantees (see [11]) that equation (16) is convergent. 4. We plan to study in more details the multi-dimensional case in one of our next publications. 5. Levitan almost periodic and almost automorphic solutions of second order differential equations In this section we consider a scalar differential equation of the type (10), i.e., n= 1. In the sequel, we suppose that the function f∈C(Y×R2,R) is regular and increasing (in the large sense) with respect to the variable x, i.e., if u1≤u2then f(y, u1, v)≤f(y, u2, v) for all y∈Yand v∈R. Lemma 5.1. [22] Let u(t), v(t)be two solutions of equation (10) defined on R. Then, only one of the following three cases is possible: (i) The function u(t)−v(t)is monotone on the real axis R; (ii) u(t)−v(t)is positive on R, and there exists a number t0∈Rsuch that this function is non-decreasing on the interval (t0,+∞), and non-increasing on (−∞, t0); LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 17 (iii) The function u(t)−v(t)is negative on R, and there exists a number t0∈R such that it is non-increasing on the interval (t0,+∞), and non-decreasing on (−∞, t0). Let ϕ∈C(R,R), and denote by aϕ:= inf{ϕ(t)|t∈R}and bϕ:= sup{ϕ(t)|t∈R}. Remark 5.2. Notice that the following facts take place: 1. aϕ≤bϕfor all ϕ∈C(R,R). 2. The inequalities (20) aϕ≤aψ≤bψ≤bϕ hold for all ψ∈H(ϕ). 3. If the function ϕis recurrent, then aϕ=aψand bψ=bϕ for all ψ∈H(ϕ). Theorem 5.3. [25, 29] Let f∈C(R×Rn,Rn)be Poisson stable with respect to the time variable t∈R. If the equation x′=f(t, x) admits a solution ϕwhich is bounded on R, then it admits at least a Poisson stable solution ψ∈H(ϕ). Let us now establish our first main result in this section. Theorem 5.4. Suppose that f∈C(Y×R2,R)is regular and increasing (in the large sense) with respect to the variable x∈R, and assume that the point y∈Yis Poisson stable. Then, the following statements hold: (i) If (10) admits a solution φsuch that φand φ′are bounded on R, then it has at least one compatible (by the character of recurrence with the right-hand side) solution; (ii) If u(t)and v(t)are two compatible solutions of equation (10), then u(t)− v(t) = cfor all t∈R, where c∈Ris some constant; (iii) If the function fis strictly increasing with respect to the variable x∈R, then equation (10) admits at most one compatible solution which is bounded on R. Proof. Let φ∈C(R,R) be a solution of equation (10) such that φand φ′are bounded on R. To prove the first statement, on account of Lemma 2.15, it is sufficient to show that the function φis comparable with yby the character of recurrence, i.e., the functional sequence {φ(t+tn)}converges to φ(t) uniformly on every compact subset from R, for every sequence {tn} ∈ Ny. Consider the motion σ(t, φ) in the shift dynamical system (Bebutov’s system) (C(R,R),R, σ). According to Theorem 5.3, the set H(φ) := {σ(τ, φ)|τ∈R}contains at least one Poisson stable solution ϕ∈H(φ) of equation (10) (in fact, the function ϕand the point yare jointly Poisson stable) . We will prove that the solution ϕis compatible. To this end, we will show that equation (10) possesses at most one 18 TOM´ AS CARABALLO AND DAVID CHEBAN solution from H(ϕ)⊆H(φ). Indeed, if ψ∈H(ϕ) is a solution of equation (10) and r(t) := ψ(t)−ϕ(t) for all t∈R, then, by Lemma 5.1, there exist the limits lim t→+∞r(t) = c+,lim t→−∞ r(t) = c− and |c+|+|c−|>0. Suppose, for example, that c+>0. Then, by the joint Poisson stability of the point yand the solution ϕ, there exists a sequence {tn} ∈ Ny∩Nϕsuch that tn→+∞as n→ ∞. Without loss of generality, we can suppose that the sequence {ψ(t+tn)} is convergent in the space C(R,R). Let ¯ ψbe its limit, i.e., ¯ ψ(t) = lim t→+∞ψ(t+tn). Then, (21) ¯ ψ(t) = ϕ(t) + c+for all t∈R. From (20) and the fact that ¯ ψ∈H(ψ)⊆H(ϕ), we have (22) aϕ≤aψ≤a¯ ψ≤b¯ ψ≤bψ≤bϕ. On the other hand, from (21) we have b¯ ψ=bϕ+c+. From the last equality and (22) we obtain c+≤0.This contradiction proves our statement. The other cases can be treated similarly. Let now u(t) and v(t) be two compatible solutions of equation (10). Then, thanks to Lemma 5.1, there exists a number t0∈Rsuch that the function r(t) := u(t)−v(t) is monotone on one of the two intervals: (−∞, t0) or (t0,+∞). Consider, for example, the case when r(t) is monotone on the interval (−∞, t0). Since the solutions uand vare compatible, and the point yis Poisson stable, the function r(t) is Poisson stable too. In particular, it is Poisson stable in the negative direction. On the other hand, this function is monotone on the interval (−∞, t0) and, consequently, is a constant. Thus u(t)−v(t) = cfor all t∈R, where c∈Ris some constant. Finally, we prove the third statement of our theorem. Suppose that the function fis strictly increasing with respect to the variable x∈R. If we suppose that equation (10) admits two different solutions uand vwhich are bounded on R, then the function (23) r(t) := u(t)−v(t) (t∈R) possesses the limits c±:= lim t→±∞ r(t) and |c−|+|c+|>0. Suppose, for example, that c+>0. Then, we take a sequence {tn} ∈ Nysuch that tn→+∞and the functional sequences {u(t+tn)}and {v(t+tn)}are convergent (since the functions uand vare solutions of (10) which are bounded on R. Denote by ¯u(respectively, ¯v) the limit of the sequence {u(t+tn)}(respectively, {v(t+tn)})). From equality (23) we have ¯u(t) := ¯v(t) + c+for allt∈R and, consequently, we obtain f(σ(t, y),¯v(t),¯v′(t)) = f(σ(t, y),¯v(t) + c+,¯v′(t)) for all t∈R. The last identity contradicts the strict monotony of the function fwith respect to the variable x. This contradiction completes the proof.  As a consequence of Theorem 5.4 and Theorem 2.12, we have the second main result in this section. LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 19 Corollary 5.5. Suppose that f∈C(Y×R2,R)is regular and monotone increasing (in the large sense) with respect to the variable x∈R, and the point y∈Yis stationary (respectively, τ–periodic, Levitan almost periodic, almost automorphic, almost recurrent, Poisson stable). Then, the following statements hold: (i) If (10) admits a solution φsuch that φand φ′are bounded on R, then it has at least one stationary (respectively, τ–periodic, Levitan almost periodic, almost automorphic, almost recurrent, Poisson stable) solution; (ii) If u(t)and v(t)are two stationary (respectively, τ–periodic, Levitan almost periodic, almost automorphic, almost recurrent, Poisson stable) solutions of equation (10), then u(t)−v(t) = cfor all t∈R, where c∈Ris some constant; (iii) If the function fis strictly increasing with respect to the variable x∈R, then equation (10) admits at most one stationary solution (respectively, τ– periodic, Levitan almost periodic, almost automorphic, almost recurrent, Poisson stable) which is bounded on R. 6. Quasi-periodic, Bohr almost periodic, almost automorphic and recurrent solutions In this section we analyze problem (10) in the scalar case, i.e., n= 1, and suppose that Yis compact and (Y, R, σ) is a minimal dynamical system, i.e., Ydoes not contain a proper compact invariant subset. Our main result below ensures that compatibility is now uniform. Theorem 6.1. Suppose that f∈C(Y×R2,R)is regular and monotone increasing (in the large sense) with respect to the variable x∈R. Then, the following statements hold: (i) If (10) admits a solution ϕsuch that ϕand ϕ′are bounded on Rthen it has at least one uniformly compatible (by the character of recurrence with the right-hand side) solution; (ii) If u(t)and v(t)are two uniformly compatible solutions of equation (10), then u(t)−v(t) = cfor all t∈R, where c∈Ris some constant; (iii) If the function fis strictly increasing with respect to the variable x∈R, then equation (10) admits at most one uniformly compatible solution. Proof. Let ϕ∈C(R,R) be a solution such that ϕand varφ′are bounded on R. To prove the first statement, taking into account Lemma 2.15, it is sufficient to show that ϕis uniformly comparable with yby the character of recurrence, i.e., the functional sequence {ϕ(t+tn)}is convergent uniformly on every compact subset from R, for every sequence {tn} ∈ My. Denote by X:= C(R,R)×Y and (X, R, π) the product dynamical system, i.e., π(τ, (ϕ, y)) := (ϕτ, σ(τ, y)) for all (ϕ, y)∈C(R,R)×Yand τ∈R, where ϕτis a τ–shift of the function ϕ (ϕτ(t) := ϕ(t+τ) for all t∈R). Consider the motion π(t, (ϕ, y)) in the product dynamical system (X, R, π). Under the conditions of our theorem, this motion is stable in the sense of Lagrange, i.e., the set H(ϕ, y) := {π(τ, (ϕ, y))|τ∈R} is compact. According to Birkhoff’s theorem, the set H(ϕ, y) contains at least 20 TOM´ AS CARABALLO AND DAVID CHEBAN one minimal set M ⊆ H(ϕ, y). Note that the mapping h:= pr2:M 7→ Y is an homomorphism of the dynamical system (H(ϕ, y),R, π) onto (Y, R, σ) and, consequently, My:= {(ψ, y) : (ψ, y)∈H(ϕ, y)}is a nonempty compact subset of H(ϕ, y). Now we will show that the set Myconsists of a single point for every y∈Y. Indeed, if we assume the opposite, then there exists a point y0∈Ysuch that My0contains at least two different points (vi, y0) (i= 1,2 and v16=v2). According to Theorem 5.4, without loss of generality we may suppose, for example, that v1 is comparable by the character of recurrence with the point y0, i.e., Ny0⊆Nv1. On the other hand, by Lemma 5.1, there exist the limits lim t→±∞ r(t) = c±and |c−|+|c+|>0, where r(t) := v2(t)−v1(t) for all t∈R. Suppose, for example, that c−>0. Then, taking into account the fact that the point (v1, y0) is negatively Poisson stable, we have a sequence {tn} ∈ Nv1∩Ny0such that tn→ −∞ as n→ ∞. We can suppose that the sequence {v2(t+tn)}is convergent. Denote by ¯v2its limit. Then, we have ¯v2(t) = v1(t) + c−for all t∈Rand, consequently, we have (24) a¯v2=av1+c−. But the functions v1,¯v2∈H(v1), and the function v1is recurrent and, consequently, we have (25) a¯v2=av1=av2. From (24) and (25) it follows that c−= 0. This contradiction proves our statement. The other cases can be considered in a similar way. Thus, we have established that the set Myconsists of a single point for all y∈Y. Let φbe a solution of equation (10) such that {(φ, y)}=My. Now, it is easy to show that the solution φis uniformly compatible. Indeed, let {tn} ∈ My. Then, the sequence {σ(tn, y)} converges. Denote by ˜yits limit. We will show that the functional sequence {φ(t+ tn)}is also convergent in the space C(R,R). If it is not true, then there exist at least two points of accumulation ψi(i= 1,2 and ψ16=ψ2) for this sequence. On the other hand, it is easy to see that (ψi,˜y)∈ M˜y(i= 1,2). The last inclusion contradicts the fact that every subsets My⊆ M consists of a single point for all y∈Y. This contradiction proves the first statement of our theorem. The second and third statements follow from Theorem 5.4.  Corollary 6.2. Suppose that f∈C(Y×R2,R)is regular and monotone increasing (in the large sense) with respect to the variable x∈R, and the point y∈Yis stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent). Then, the following statements hold: (i) If (10) admits a solution ϕsuch that ϕand ϕ′are bounded on R, then it has at least one stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solution; (ii) If u(t)and v(t)are two stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solutions of equation (10), then u(t)− v(t) = cfor all t∈R, where c∈Ris some constant; (iii) If the function fis strictly increasing with respect to the variable x∈R, then equation (10) admits at most one stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solution. Proof. These statements follow from Theorem 6.1 and Theorem 2.12.  LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 21 Remark 6.3. In the particular case in which Yis a Bohr almost periodic minimal set, then Corollary 6.2 coincides with the result proved by Opial in [22]. 7. Some generalizations Let now I:= (a, b),where a, b ∈[−∞,+∞]. For example, I=R, I = (0,+∞), I= (a, b) and a, b ∈R, etc. Consider equation (10) when f∈C(Y×I×R,R). For example, for the equation x′′ +cx′+1 xα=f(σ(t, y)) we have f(y, x, x′) := −cx′−1/xα+f(y) and I= (0,+∞), where α > 0. In this context, we will now highlight an extended meaning to the concept of boundedness on R. To this respect, a solution ϕ∈C(R,R) of equation (10) is said to be bounded on R(respectively, on R+) if Q:= ϕ(R) is a compact subset from I, i.e., if there exist two real numbers αand βsuch that a < α ≤ϕ(t)≤β < b for all t∈R (respectively, t∈R+). All of our results about our second order equation (10) (especially, theorems 5.4, 6.1 and Corollaries 5.5 and 6.2) remain true also when f∈C(Y×I×R,R). We will formulate for example the following statements. Theorem 7.1. Suppose that f∈C(Y×I×R,R)is regular and increasing (in the large sense) with respect to the variable x∈I. Then, the following statements hold: (i) If (10) admits a solution ϕsuch that ϕand ϕ′are bounded on R, then it has at least one uniformly compatible (by the character of recurrence with the right-hand side) solution; (ii) If u(t)and v(t)are two uniformly compatible solutions of equation (10), then u(t)−v(t) = cfor all t∈R, where c∈Ris some constant; (iii) If the function fis strictly increasing with respect to the variable x∈I, then equation (10) admits at most one uniformly compatible solution. Proof. We omit the proof because is completely similar to the proof of Theorem 6.1.  Corollary 7.2. Suppose that the function f∈C(Y×I×R,R)is regular and increasing (in the large sense) with respect to the variable x∈I, and the point y∈Yis stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent). Then, the following statements hold: (i) If (10) admits a solution ϕsuch that ϕand ϕ′are bounded on R, then it has at least one stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solution; (ii) If u(t)and v(t)are two stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solutions of equation (10), then u(t)− v(t) = cfor all t∈R, where c∈Ris some constant; (iii) If the function fis strictly increasing with respect to the variable x∈I, then equation (10) admits at most one stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solution. 22 TOM´ AS CARABALLO AND DAVID CHEBAN Proof. This result follows from Theorem 7.1 and Theorem 2.12.  Corollary 7.3. Suppose that the following conditions are fulfilled: (i) f∈C(Y×I×R,R)and there exists a constant C > 0such that (26) |f(y, x, x′)| ≤ C(1 + |x′|2) for all (y, x, x′)∈Y×I×R; (ii) The function fis regular and monotone increasing (in the large sense) with respect to the variable x∈R; (iii) The point y∈Yis stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent). Then, the following statements hold: (i) If (10) admits a solution which is bounded on R, then it has at least one stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solution; (ii) If u(t)and v(t)are two stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solutions of equation (10), then u(t)− v(t) = cfor all t∈R, where c∈Ris some constant; (iii) If the function fis strictly increasing with respect to the variable x∈I, then equation (10) admits at most one stationary (respectively, τ–periodic, quasi-periodic, Bohr almost periodic, recurrent) solution. Proof. These statements follow from Corollary 7.2. To this end, it is sufficient to note that under condition (26), if ϕ∈C(R,R) is a solution of equation (10) which is bounded on R, then its derivative ϕ′is also bounded on R(see Lemma 2.1 [19] and also Lemma 5.1 from [17, Ch.XII]).  Remark 7.4. 1. Corollary 7.3 (item (iii)) improves and generalizes some of the results from [4, 12, 14, 19] when the function fis strictly increasing with respect to the second variable. 2. We plan to study in more detail this case (fis strictly increasing with respect to second variable) in one of our future publication. Acknowledgements. We would like to thank the referee for interesting suggestions which allowed us to improve the presentation of this paper. This paper was written while the second author was visiting the University of Sevilla (February–September 2010) under the Programa de Movilidad de Profesores Universitarios y Extranjeros (Ministerio de Educaci´on, Spain) grant SAB2009-0078. He would like to thank people of this university for their very kind hospitality. He also gratefully acknowledges the financial support of the Ministerio de Educaci´on (Spain). The first author is partially supported by grant MTM2008-00088 (Ministerio de Ciencia e Innovaci´on, Spain) and Proyecto de Excelencia P07-FQM02468 (Junta de Andaluc´ıa, Spain). LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 23 References [1] Belova M. M., Bounded solutions of nonlinear second order differential equations. Mat. Sbornik, 56 (98) (1962), 469–503.(in Russian) [2] Bronsteyn I. U., Extensions of Minimal Transformation Group. Noordhoff, 1979. [3] Bhatia N. P. and Szeg¨o G. P., Stability Theory of Dynamical Systems. Lecture Notes in Mathematics. Springer, Berlin–Heidelberg–New York, 1970. [4] Campos J. and Torres P. J., On the structure of the set of bounded solutions on a periodic Li´enard equation. Proc. Amer. Math. Soc., 127 (5) (1999), 1453–1462. [5] Caraballo T., and Cheban D. N., Almost periodic and almost automorphic solutions of linear differential/difference equations without Favards separation condition. I, J. Differential Equations 246 (2009), 108128 [6] Caraballo T., and Cheban D. N., Almost periodic and almost automorphic solutions of linear differential/difference equations without Favards separation condition. II, J. Differential Equations 246 (2009), 11641186 [7] Cheban D. N., Global Attractors of Non-autonomous Dissipative Dynamical systems. Interdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, 528pp. [8] Cheban D. N., Levitan Almost Periodic and Almost Automorphic Solutions of V-monotone Differential Equations. J.Dynamics and Differential Equations, 20 (2008), No.3, 669–697. [9] Cheban D.N. Asymptotically Almost Periodic Solutions of Differential Equations. Hindawi Publishing Corporation, New York – Cairo, 2009, 203pp. [10] Cheban D. N., Global Attractors of Set-Valued Dynamical and Control Systems. Nova Science Publishers, New York, 2010, 302pp.(to appear, 2nd quarter). [11] Cheban D. N. and Schmalfuss B., Invariant Manifolds, Global Attractors, Almost Automrphic and Almost Periodic Solutions of Non-Autonomous Differential Equations. J. Math. Anal. Appl., 340 (2008), no.1, 374–393. [12] Cieutat P., On the structure of the set bounded solutions on an almost periodic Li´enard equation. Nonlinear Analysis, 58 (2004), no. 7-8, 885–898. [13] Cieutat P., Maximum principle and existence of almost-periodic solutions of second-order differential systems. Differential and Integral Equations, 17 (2004), no. 7-8, 921–942. [14] Cieutat P., Fatajou S. and N’Guerekata G. M., Bounded and almost automorphic solutions of Lineard equation with a singular nonlinearity. EJQTDE, (2008), No. 21, 1–15. [15] Corduneanu C., Solut¸ii aproape periodice ale equat¸iilor different¸iale neliniare de ordinul al dilea. Comun. Acad. Rep. Rom. V, no.5 (1955), pp.793–797. [16] A. M. Fink, Almost automorphic and almost periodic solutions which minimize functionals. Tˆohoku Math. J., 20 (1968), pp.323–332. [17] Hartman Ph., Ordinary Differential Equations. J. W. Wiley and Sons, New York; London; Sydney: 1964. [18] Levitan B. M. and Zhikov V. V., Almost Periodic Functions and Differential Equations. Cambridge Univ. Press. London, 1982. [19] Li D. and Duan J., Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations. J. Differential Equations, 246 (2009), no. 5, 1754–1773. [20] Mart´ınez-Amores P. and Torres P. J., Dynamics of periodic differential equation with a singular nonlinearity of attractive type. J. Math. Anal. Appl., 202 (1996), 1027–1039. [21] Opial Z., Sur les solutions presque-periodiques des equations differentielles du premier et du second ordre. Annales Polonici Mathematici, VII (1959), 51–61. [22] Opial Z., Sut l’existence des solutions p´eriodique de l’´equation diff´erentielle du second ordre. Bulletin de l’Academy Polonaise des Sciences, Serie des Sci. Math., Astr. et Phys., Vol. VII (1959), No.2, 71–75. [23] Sell G. R., Lectures on Topological Dynamics and Differential Equations, vol.2 of Van Nostrand Reinhold math. studies. Van Nostrand–Reinbold, London, 1971. [24] Shen W. and Yi Y., Almost automorphic and almost periodic dynamics in skew-product semiflows. Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93 pp. [25] Shcherbakov B. A., Poisson stable solutions of differential equations, and topological dynamics. Differencial’nye Uravnenija, 5 (1969), 2144–2155. [26] Shcherbakov B. A., Topologic Dynamics and Poisson Stability of Solutions of Differential Equations. S¸tiint¸a, Chi¸sin˘au, 1972. 24 TOM´ AS CARABALLO AND DAVID CHEBAN [27] Shcherbakov B. A., The compatible recurrence of the bounded solutions of first order differential equations. Differencial’nye Uravnenija, 10 (1974), 270–275. [28] Shcherbakov B. A., Poisson Stability of Motions of Dynamical Systems and Solutions f Differential Equations. S¸tiint¸a, Chi¸sin˘au, 1985. [29] Shcherbakov B. A. and Fal’ko N. S., The minimality of sets and the Poisson stability of motions in homomorphic dynamical systems. Differencial’nye Uravnenija, 13 (1977), no. 6, 1091–1097. [30] Taro U., Sur les courbes d´efinies par les ´equations diff´erentielles dans l’espace ´a mdimensions. Annales Scientifiques de l’E. N. S. 3-eme serie, tome 70, no. 4 (1953), 287-360. [31] Zhikov V. V., Monotonicity in the Theory of Nonlinear Almost Periodical Operationel Equations. Matematicheskii Sbornik, 90 (132), (1972) No.2, 214–228. E-mail address, T. Caraballo: [email protected] E-mail address, D. Cheban: [email protected] (T. Caraballo) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. Correos 1160, 41080-Sevilla (Spain) (D. Cheban) State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chis¸in˘ au, Moldova