Almost Periodic and Asymptotically Almost Periodic Solutions of Liénard Equations

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume 00, Number 0, Xxxx XXXX pp. 000–000 ALMOST PERIODIC AND ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS TOM´ AS CARABALLO AND DAVID CHEBAN Abstract. The aim of this paper is to study the almost periodic and asymptotically almost periodic solutions on (0,+∞) of the Li´enard equation x′′ +f(x)x′+g(x) = F(t), where F:T→R(T=R+or R) is an almost periodic or asymptotically almost periodic function and g: (a, b)→Ris a strictly decreasing function. We study also this problem for the vectorial Li´enard equation. We analyze this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers [3, 7] to prove the existence of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of Li´enard equations (both scalar and vectorial). 1. Introduction. In this paper we study the existence of almost periodic and asymptotically almost periodic solutions of the Li´enard equation x′′ +f(x)x′+g(x) = F(t),(1) where F:T→R(T=R+:= [0,+∞) or R:= (−∞,+∞)) is a continuous or locally integrable function and f, g : (a, b)→R(−∞ ≤ a < b ≤+∞) are locally Lipschitz continuous functions. We assume that the following conditions are fulfilled: (i) gis strictly decreasing; (ii) f(x)≥0 for all x∈(a, b); (iii) Fis almost periodic (respectively, almost automorphic, recurrent, pseudo recurrent) or asymptotically almost periodic (respectively, asymptotically recurrent, asymptotically pseudo recurrent). The typical equation of type (1) is x′′ +cx′+1 xα=F(t), where c≥0, α > 0 and F:T→Ris an almost periodic or asymptotically almost periodic function. Date: November 19, 2010. 2000 Mathematics Subject Classification. primary: 34C11, 34C15, 34C27, 34C35, 34D05, 34D23, 34D45, 37C55, 37C60, 37C70, 37C75. Key words and phrases. Non-autonomous dynamical systems; skew-product systems; cocycles; global attractor;convergent systems; quasi-periodic, almost periodic, almost automorphic, recurrent solutions; asymptotically almost periodic solutions; Lienard equation. 1 2 TOM´ AS CARABALLO AND DAVID CHEBAN In the periodic case (i.e., when Fis periodic), the dynamics of equation (1) was intensively studied by P. Mart´ınez-Amores and P. J. Torres [13] and J. Campos and P. J. Torres [2]. For the almost periodic case (i.e. for almost periodic F) these results were generalized by P. Cieutat in [8]. The almost automorphic and asymptotically almost automorphic solutions of equation (1) were studied by P. Cieutat et al. [9], while the existence of pseudo almost periodic solutions of equation (1) was analyzed by El Hadi Ait Dads et al. [9]. Our main result in the present paper states that, when the function Fis τ– periodic (respectively, quasi periodic, almost periodic, almost automorphic, recurrent, pseudo recurrent), if equation (1) admits a solution which is bounded on R+, then it has a unique τ–periodic (respectively, quasi periodic, almost periodic, almost automorphic, recurrent, pseudo recurrent) solution, and every solution of (1), bounded on R+, is asymptotically τ–periodic (respectively, asymptotically quasi periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent). We obtain also an analog of this result when the function Fis asymptotically τ–periodic (respectively, asymptotically quasi periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent). These results are new and contain, as particular cases, some of the results cited above. We present our results in the framework of general non-autonomous dynamical systems (cocycles) and we apply our abstract theory developed in [3, 7] to Li´enard differential equations (both scalar and vectorial). 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. We give here also some results concerning a special class of non-autonomous dynamical system (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. Section 3 is devoted to the existence of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of Li´enard equation (1). In Sections 4 we present some results about Sp– asymptotically almost periodic (asymptotically almost periodic in the sense of Stepanoff) solutions of Li´enard equation (1). Finally, Sections 5 is devoted to study the problem of almost periodicity (respectively, almost automorphy, recurrence, pseudo recurrence) and asymptotically almost periodicity (respectively, asymptotically almost automorphy, asymptotically recurrence, asymptotically pseudo recurrence) of solutions for the vectorial Li´enard equation. ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS 3 2. Nonautonomous Dynamical Systems with Convergence. Let us start by recalling some concepts and notations about the theory of non-autonomous dynamical systems which will be necessary for our analysis. 2.1. Compact Global Attractors of Dynamical Systems. Let (X, ρ) be a metric space, Rbe the group of real numbers, R+be the semi-group of nonnegative real numbers, Tbe one of the two sets Ror R+. Adynamical system is a triplet (X, T, π), where π:T×X→Xis a continuous mapping satisfying the following conditions: π(0, x) = x(∀x∈X) and π(s, π(t, x)) = π(s+t, x) (∀t, τ ∈Tand x∈X). When T=R+(respectively, R), the dynamical system (X, T, π) is called a semi-flow (respectively, flow). The function π(·, x) : T→Xis called a motion passing through the point xat the moment t= 0 and the set Σx:= π(T, x) is called the trajectory of this motion. A nonempty set M⊆Xis called positively invariant (negatively invariant, invariant) with respect to the dynamical system (X, T, π) or, simply, positively invariant (negatively invariant, invariant), if π(t, M)⊆M(M⊆π(t, M), π(t, M) = M) for every t∈T. A closed positively invariant set, which does not contain any own closed positively invariant subset, is called minimal. It is easy to see that every positively invariant minimal set is invariant. The dynamical system (X, T, π) is called: −point dissipative if there exists a nonempty compact subset K⊆Xsuch that for every x∈X lim t→+∞ρ(π(t, x), K) = 0; (2) −compact dissipative if there exists a nonempty compact subset K⊆Xsuch that lim t→+∞ρ(π(t, x), K) = 0 uniformly with respect to xon compact subsets of X. Let (X, T, π) be compact dissipative and Kbe a compact set attracting every compact subset of X. Let us set J:= ω(K) := \ t≥0[ τ≥t π(τ, K).(3) It can be shown [5, Ch.I] that the set Jdefined by equality (3) does not depend on the choice of the attracting set K, but is characterized only by the properties of the dynamical system (X, T, π) itself. The set Jis called the Levinson center of the compact dissipative dynamical system (X, T, π). 2.2. Non-Autonomous Dynamical Systems with Convergence. Recall that given two dynamical systems (X, T1, π) and (Y, T2, σ), a triplet h(X, T1, π),(Y, T2, σ), hi, where his a homomorphism from (X, T1, π) onto (Y, T2, σ), is called a nonautonomous dynamical system. The non-autonomous dynamical system h(X, T1, π), (Y, T2, σ), hiis said to be convergent (see [5]) if the following conditions are fulfilled: (i) the dynamical systems (X, T1, π) and (Y, T2, σ) are compact dissipative; (ii) the set JXTXycontains no more than one point for all y∈JY, where Xy:= h−1(y) := {x∈X|h(x) = y}and JX(respectively, JY) is the Levinson center of the dynamical system (X, T1, π) (respectively, (Y, T2, σ)). 4 TOM´ AS CARABALLO AND DAVID CHEBAN Thus, a non-autonomous dynamical system h(X, T1, π),(Y, T2, σ), hiis convergent, if the systems (X, T1, π) and (Y, T2, σ) are compact dissipative with Levinson centers JXand JYrespectively, and JXpossesses “trivial” sections, i.e., JXTXy consists of a single point for all y∈JY. In this case the Levinson center JXof the dynamical system h(X, T1, π) is a copy (an homeomorphic image) of the Levinson center JYof the dynamical system (Y, T2, σ). Thus, the dynamics on JXis the same as on JY. Remark 2.1. We note that convergent systems are in some sense the simplest dissipative dynamical systems. If Yis compact, invariant, T2=R,h(X, T1, π), (Y, T2, σ), hiis a convergent non-autonomous dynamical system and JXis the Levinson center of (X, T1, π), then (JX,T2, π) and (Y, T2, σ) are homeomorphic. Although the Levinson center of a convergent system can be completely described, it may be sufficiently complicated. Recall [7] that the point x∈Xis called asymptotically τ–periodic (respectively, asymptotically quasi periodic,asymptotically Bohr almost periodic,asymptotically recurrent,asymptotically pseudo recurrent), if there exists a τ-periodic (respectively, quasi periodic, Bohr almost periodic, recurrent, pseudo recurrent) point p∈Xsuch that lim t→+∞ρ(π(t, x), π(t, p)) = 0. 2.3. 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 “nontrivial” global attractor JX, i.e., there exists at least one point y∈JYsuch that the set JXTXycontains more than one point. A non-autonomous dynamical system h(X, T1, π),(Y, T2, σ), hiis said to be weak convergent, if the following conditions hold: (i) the dynamical systems (X, T1, π) and (Y, T2, σ) are compact dissipative with Levinson centers JXand JYrespectively; (ii) it follows that lim t→+∞ρ(π(t, x1), π(t, x2)) = 0, for all x1, x2∈JXwith h(x1) = h(x2). Given x∈X, let us denote by Mx:= {{tn} ⊆ T: such that the sequence {π(tn, x)}converges}and Lx:= {{tn} ∈ Mx:tn→+∞}. Similarly, for any y∈Y, denote by My:= {{tn} ⊆ T: such that the sequence {σ(tn, y)}converges} and Ly:= {{tn} ∈ My:tn→+∞}. Remark 2.2. 1. Recall that the point x∈Xis called [14]-[16] comparable (respectively, uniformly comparable) by the character of recurrence with the point y∈Y, if Ny⊆Nx(respectively, My⊆Mx), where Nx:= {{tn} ∈ Mx,such that {π(tn, x} → x}}. 2. The notions of comparability and uniform comparability of motions by the character of recurrence play a very important role [14]-[16] in the study of stability in the sense of Poisson (in particular, periodicity, quasi-periodicity, almost periodicity, almost automorphy, recurrence, etc) of solutions of differential equations with Poisson stable coefficients. ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS 5 Recall [7] that the point x∈Xis called comparable with y∈Yby the character of recurrence in infinity if Ly⊆Lx. Remark 2.3. Note that the notion of comparability by the character of recurrence in infinity plays an important role [7] in the problem of existence of asymptotically almost periodic solutions of differential equations with asymptotically almost periodic coefficients. The next theorem contains sufficient conditions ensuring asymptotical stationarity (asymptotical periodicity, asymptotical almost periodicity, etc) for points which are comparable by the character of recurrence in infinity. Theorem 2.4. [7] Suppose that the following conditions hold: (i) (X, T1, π)and (Y, T2, σ)are two dynamical systems; (ii) the point y∈Yis asymptotically stationary (respectively, asymptotically τ– periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent); (iii) the point xis comparable with y∈Yby the character of recurrence in infinity. Then, the point xis also asymptotically stationary (respectively, asymptotically τ–periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent). Let (X, T, π) be a dynamical system. Denote by ΩX:= S{ωx|x∈X},where ωxis the ω-limit set of the point x. The following results, which have been proved in [3, 4], ensure the existence of compact minimal sets for point dissipative nonautonomous dynamical systems, as well as the existence of stationary (asymptotically periodic, asymptotically quasi-periodic, asymptotically almost periodic, etc...) points in the space X. Corollary 2.5. [3, 4] Let h(X, T1, π),(Y, T2, σ), hibe a non-autonomous dynamical system such that the following conditions hold: (i) the dynamical systems (X, T1, π)and (Y, T2, σ)are point dissipative; (ii) ΩYis a compact minimal set; (iii) lim t→+∞ρ(π(t, x1), π(t, x2)) = 0 holds for all x1, x2∈Xwith h(x1) = h(x2); (iv) for every y∈ΩY, the set L˜ XT˜ Xycontains at most one point, where L˜ X:= {x∈˜ X:there exists at least one entire motion γ(·) = π(·, x)through the point xsuch that γ(R)⊆˜ Xand γ(R)is relatively compact}, where ˜ X:= h−1(ΩY). Then, there exists a unique compact minimal set M⊆Xsuch that (i) the section MTXyof the set Mconsists of a single point myfor all y∈Y; (ii) ΩX=M; (iii) every point x∈Xis comparable with h(x)by the character of recurrence in infinity. Corollary 2.6. [3, 4] Let h(X, T1, π),(Y, T2, σ), hibe a non-autonomous dynamical system such that the following conditions hold: (i) the dynamical system (X, T, π)is point dissipative; (ii) there exists a point y0∈Ysuch that Y:= H+(y0) := {σ(t, y0) : t∈T+}; 6 TOM´ AS CARABALLO AND DAVID CHEBAN (iii) the point y0is asymptotically stationary (respectively, asymptotically τ–periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent); (iv) lim t→+∞ρ(π(t, x1), π(t, x2)) = 0 holds for all x1, x2∈Xwith h(x1) = h(x2); (v) for every y∈ΩYthe set L˜ XT˜ Xycontains at most one point, where the sets L˜ Xand ˜ Xyare the ones defined in the previous corollary. Then, there exists a unique compact minimal set M⊆Xsuch that (i) the section MTXyof the set Mconsists of a single point myfor all y∈Y; (ii) ΩX=M; (iii) every point x∈Xis asymptotically stationary (respectively, asymptotically τ– periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent). 2.4. Pseudo Recurrent Dynamical Systems with Convergence. A nonautonomous dynamical system h(X, T1, π),(Y, T2, σ), hiis said to be uniformly stable in the positive direction on compact subsets of Xif, for arbitrary ε > 0 and compact subset K⊂X, there is δ=δ(ε, K)>0 such that inequality ρ(x1, x2)< δ (x1, x2∈K, h(x1) = h(x2)) implies that ρ(π(t, x1), π(t, x2)) < ε for t∈T+ 1,where T+ 1:= {t∈T1:t≥0}. Denote by X˙ ×X={(x1, x2)∈X×X|h(x1) = h(x2)}. If there exists a function V:X˙ ×X→R+with the following properties: (i) Vis continuous; (ii) Vis positive defined, i.e., V(x1, x2) = 0 if and only if x1=x2; (iii) V(π(t, x1), π(t, x2)) ≤V(x1, x2) for all (x1, x2)∈X˙ ×Xand t∈T+ 1, then the non-autonomous dynamical system h(X, T1, π),(Y, T2, σ), hiis called (see [5, 6], [12], and [17]) V–monotone. Let (X, h, Y ) be a fiber space, i.e., Xand Ybe two metric spaces and h:X→Y be a homomorphism from Xonto Y. The subset M⊆Xis said to be conditionally relatively compact, if the pre-image h−1(Y′)TMof every relatively compact subset Y′⊆Yis a relatively compact subset of X, in particular, My:= h−1(y)TMis relatively compact for every y. The set Mis called conditionally compact if it is closed and conditionally relatively compact. Example 2.7. Let Kbe a compact space, X:= K×Y,h=pr2:X→Y, then the triplet (X, h, Y )is a fiber space. The space Xis conditionally compact, but not compact. Denote by K:= {a∈C(R+,R+)|a(0) = 0, a is strictly increasing}. Recall that the dynamical system (X, T1, π) is called asymptotically compact if for every positively invariant bounded subset M⊆Xthere exists a nonempty compact subset K⊆Xsuch that lim t→+∞β(π(t, M), K) = 0, where β(A, B) := sup a∈A ρ(a, B) and ρ(a, B) := inf b∈Bρ(a, b). Now, we state two results proved in [3, 4] which provide some sufficient conditions ensuring the convergence character of a non-autonomous dynamical systems, as well ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS 7 as the existence of periodic (respectively, quasi-periodic, almost periodic, etc) points in the fiber of a periodic (respectively, quasi-periodic, almost periodic, etc) point. Theorem 2.8. Let h(X, T, π),(Y, R, σ), hibe a non-autonomous dynamical system satisfying the following conditions: 1. the dynamical system (Y, R, σ)is pseudo recurrent; 2. the dynamical system (X, T, π)is asymptotically compact; 3. there exists a point x0∈Xy0with relatively compact positive semi-trajectory Σ+ x0:= {π(t, x0) : t≥0}; 4. the non-autonomous dynamical system h(X, T, π),(Y, R, σ), hiis V–monotone; 5. for all (x1, x2)∈LX˙ ×LX\∆X(where ∆X:= {(x, x) : x∈X}) there exists a positive number t0=t0(x1, x2)∈Tsuch that V(π(t0, x1), π(t0, x2)) < V(x1, x2); 6. there are functions a, b ∈ K such that Im(a) = Im(b)and a(ρ(x1, x2)) ≤ V(x1, x2)≤b(ρ(x1, x2)) for all (x1, x2)∈X˙ ×X. Then, the following statements take place: (i) the NDS h(X, T1, π),(Y, R, σ), hiis convergent; (ii) JX=ωx0; (iii) h(JX) = Y. Corollary 2.9. Let h(X, T, π),(Y, R, σ), hibe a non-autonomous dynamical system such that: (i) the dynamical system (Y, R, σ)is transitive, i.e., there exists a point y0∈Y such that H(y0) = Y; (ii) the point y0is τ–periodic (respectively, quasi periodic, Bohr almost periodic, recurrent, pseudo recurrent); (iii) the dynamical system (X, T, π)is asymptotically compact; (iv) there exists a point x0∈Xy0with relatively compact positive semi-trajectory Σ+ x0:= {π(t, x0) : t≥0}; (v) the non-autonomous dynamical system h(X,T,π),(Y,R,σ),hiis V–monotone; (vi) for all (x1, x2)∈LX˙ ×LX\∆X(where ∆X:= {(x, x) : x∈X}) there exists a positive number t0=t0(x1, x2)∈Tsuch that V(π(t0, x1), π(t0, x2)) < V(x1, x2); (vii) there are functions a, b ∈ K such that Im(a) = Im(b)and a(ρ(x1, x2)≤ V(x1, x2)≤b(ρ(x1, x2)) for all (x1, x2)∈X˙ ×X. Then, (i) there exists a unique τ–periodic (respectively, quasi periodic, Bohr almost periodic, recurrent, pseudo recurrent) point x0∈Xy0:= {x∈X:h(x) = y0}; (ii) every point x∈Xis asymptotically τ–periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically recurrent, asymptotically pseudo recurrent). 3. Almost periodic and asymptotically almost periodic solutions of Li´enard equations. Consider the following Li´enard equation x′′ +f(x)x′+g(x) = F(t),(4) where F:T→Ris a continuous function and f, g :I→R(I:= (a, b) with −∞ ≤ a < b ≤+∞) are locally Lipschitz continuous functions. We assume that the functions f, g and Fsatisfy the following conditions: 8 TOM´ AS CARABALLO AND DAVID CHEBAN (i) gis strictly decreasing; (ii) f(x)≥0 for all x∈I; (iii) sup t∈T |F(t)|<+∞. As we already pointed out, the typical example for equation (4) is given by x′′ +cx′+1 xα=F(t), where cis a nonnegative constant, α > 0 and F:R→Ris a periodic (respectively, quasi periodic, almost periodic, almost automorphic, recurrent) function. Recall that a function ϕ:T→Iis said to be bounded if Q:= ϕ(T) is a compact subset of I. Remark 3.1. A continuous function ϕ:T→Iis bounded if and only if [mϕ, Mϕ]⊂ I, where mϕ:= inf t∈Tϕ(t) and Mϕ:= sup t∈T ϕ(t). Denote by Cb(T,R) the set of all continuous functions F:T→Rwith norm ||F|| := sup t∈T |F(t)|<+∞. The following results are well known. Lemma 3.2. [8] Let I:= (t0,+∞)with t0=−∞ or t0∈Rand F∈Cb(T,R). If ϕ(t)is a solution of equation (4) which is bounded on R+(respectively, bounded on R), then the derivatives ϕ′(t)and ϕ′′(t)are also bounded on R+(respectively, bounded on R). Denote by ϕ(t, u, v, F ) the unique solution of equation (4) satisfying the initial conditions ϕ(0, u, v, F ) = uand ϕ′(0, u, v, F ) = v. We have the following theorem. Theorem 3.3. [1, 8] The following statements result to be true: (i) if F∈Cb(R+,R), then for any pair of solutions ϕ(t, ui, vi, F )(i= 1,2) of equation (4), which are bounded on R+, we have lim t→+∞(|ϕ(t, u1, v1, F )−ϕ(t, u2, v2, F)|+|ϕ′(t, u1, v1, F )−ϕ′(t, u2, v2, F)|) = 0; (ii) if F∈Cb(R,R), then equation (4) admits at most one solution which is bounded on R. Remark 3.4. Note that Theorem 3.3 remains true if we replace the condition F∈Cb(T,R) (T=R+for item (i) and T=Rfor item (ii)) by F∈Sp(T,R), where Sp(T,R) is the space of all functions ϕ∈Lp loc(T,R) satisfying the condition |ϕ|Sp:= sup t∈T (Rt+1 t|ϕ(s)|pds)1/p <+∞and p≥1. This statement may be proved with a slight modification of the proof of Theorem 3.3. Denote by C(T,R) the set of all continuous functions F:T→Rendowed with the compact-open topology, by Fτthe τ–shift of F(τ∈T), that is, Fτ(t) := F(t+τ) for all t∈T, and (C(T,R),T, σ) the shift dynamical system (Bebutov’s dynamical system), i.e., σ(τ, F ) := Fτfor all τ∈Tand F∈C(T,R). It is said that the function F∈C(T,R)possesses the property (S)(for example, periodicity, almost periodicity, recurrence, asymptotically almost periodicity and so on), if the motion σ(τ, F ),generated by the function Fin the shift dynamical system (C(T,R),T, σ), possesses this property. ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS 9 The solution ϕ(t) of equation (4) is called [7, 14, 15, 16] compatible (respectively, uniform compatible) by the character of recurrence, if the motion σ(t, (ϕ, ϕ′)), generated by by function (ϕ, ϕ′)∈C(R,R)×C(R,R) is comparable (respectively, uniform comparable) by the character of recurrence with the motion σ(τ, F ), i.e., NF⊆N(ϕ,ϕ′)(respectively, MF⊆M(ϕ,ϕ′)), MF:= {{tn}: the sequence {σ(tn, F )} is convergent},LF:= {{tn} ∈ MF: such that tn→+∞as n→ ∞} and ϕ′is the derivative of the function ϕ. Example 3.5. Denote by y=x′, then equation (4) can be reduced to the following equivalent first order system x′=y y′=−g(x)−f(x)y+F(t).(5) Along with system (5), consider its H–class, i.e., the family of systems x′=y y′=−g(x)−f(x)y+G(t),(6) where G∈H(F). Recall that we denote by ϕ(t, u, v, F) the unique solution of equation (4) satisfying the initial conditions ϕ(0, u, v, F ) = uand ϕ′(0, u, v, F ) = vand defined on R+(or on R). Then, (ϕ(t, u, v, F ), ϕ′(t, u, v, F )) is the unique solution of system (5) with the initial data (ϕ(0, u, v, F ), ϕ′(0, u, v, F )) = (u, v)∈R2. Let Y=H(F) := {σ(t, F ) : t∈R}and let (Y, R, σ) be the shift dynamical system on H(F), induced by Bebutov’s dynamical system (C(R,R),R, σ). We set W:= R2×H(F),˜ X:= {((u, v), G)∈W: there exists a unique solution ϕ(t, u, v) of equation (4) through the point (u, v)∈R2at the initial moment t= 0 and defined on R+},π(t, ((u, v), G)) := (ϕ(t, u, v, G), ϕ′(t, u, v, G)) for all t∈R+and ((u, v), G)∈˜ X, where (ϕ(t, u, v, G), ϕ′(t, u, v, G)) is the unique solution of system (6) with initial data (ϕ(0, u, v, G), ϕ′(0, u, v, G)) = (u, v). Let now ϕ(t, u, v, F ) be a solution of equation (4) bounded on R+, and we denote by X=H+((u, v, F )) := {(ϕ(τ, u, v, F ), ϕ′(τ, u, v, F ), Fτ) : τ∈R+},where Fτ:= F(·+τ). From Lemma 3.2 it follows that the set Xis shift invariant, i.e., π(t, X)⊆Xfor all t∈R+and X⊆˜ X. Let h=pr2:X7→ Ybe the second projection of Xonto Y, then the triplet h(X, R+, π),(Y, R, σ), hiis a non-autonomous dynamical system, generated by equation (4) (or system of equations (5)) and the solution ϕ(t, u, v, F ). A solution ϕof equation (4) is called compatible by recurrence in infinity [7], if LF⊆L(ϕ,ϕ′), where ϕ′is the derivative of the function ϕand Lϕ:= {{tn} ∈ Mϕ: tn→+∞}. Theorem 3.6. Suppose that F∈C(R+,R)and Fis asymptotically recurrent. Then, every solution ϕ(t, u, v)of equation (4), which is bounded on R+, is compatible by recurrence in infinity. Proof. Consider the non-autonomous dynamical system h(X, R+, π),(Y, R, σ), hi generated by equation (4) and its solution ϕ(t, u, v, F ) (see Example 3.5). Since Fis asymptotically recurrent, the dynamical system (Y, R, σ) is compact dissipative and its Levinson center JY=ωFis a compact minimal set, where ωFis the ω–limit set of the point F∈C(R,R) in the shift dynamical system (Y, R, σ). By Lemma 3.2, the set X=H+((u, v, F)) ⊆R2×C(R,R) is compact. Let now