ALMOST PERIODIC AND ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS OF LIÉNARD EQUATIONS

The aim of this paper is to study the almost periodic and asymptotically almost periodic solutions on (0, +∞) of the Liénard 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) → R is a strictly decreasing function. We study also this problem for the vectorial Liénard 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 Liénard equations (both scalar and vectorial).


Introduction
In this paper we study the existence of almost periodic and asymptotically almost periodic solutions of the Liénard equation (1) x + f (x)x + g(x) = F (t), We assume that the following conditions are fulfilled: (i) g is strictly decreasing; (ii) f (x) ≥ 0 for all x ∈ (a, b); (iii) F is 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 where c ≥ 0, α > 0 and F : T → R is an almost periodic or asymptotically almost periodic function.
In the periodic case (i.e., when F is periodic), the dynamics of equation (1) was intensively studied by P. Martinez-Amores and P. J. Torres [13] and J. Campos and P. J. Torres [2]. For the almost periodic case (almost periodic F ) these results were generalized by P. Cieutat in [8].
The almost automorphic and asymptotically almost automorphic solutions of equation (1) was studied by P. Cieutat et al. [9].
The problem of existence of pseudo almost periodic solutions of equation (1) was analyzed by El Hadi Ait Dads et al. [9].
Our main result states that, when the function F is 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 F is 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 Liénard 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 Liénard equation (1).
In Sections 4 we present some results about S p -asymptotically almost periodic (asymptotically almost periodic in the sense of Stepanoff) solutions of Liénard 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 Liénard equation.

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 be the group of real numbers, R + be the semi-group of nonnegative real numbers, T be one of the two sets R or R + .
The function π(·, x) : T → X is called a motion passing through the point x at the moment t = 0 and the set Σ x := π(T, x) is called the 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 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: uniformly with respect to x on compact subsets from X.
Let (X, T, π) be compact dissipative and K be a compact set attracting every compact subset from X. Let us set It can be shown [5,Ch.I] that the set J defined 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 J is called the Levinson center of the compact dissipative dynamical system (X, T, π).

2.2.
Non-Autonomous Dynamical Systems with Convergence. Recall (see [5]) that a non-autonomous dynamical system (X,T 1 ,π),(Y,T 2 ,σ),h is said to be convergent if the following conditions are fulfilled: (i) the dynamical systems (X, T 1 , π) and (Y, T 2 , σ) are compact dissipative; (ii) the set J X X y contains no more than one point for all y ∈ J Y , where X y := h −1 (y) := {x|x ∈ X, h(x) = y} and J X (respectively, J Y ) is the Levinson center of dynamical system (X, T 1 , π) (respectively, (Y, T 2 , σ) ).
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 possesses "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 (an 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. We note that convergent systems are in some sense the simplest dissipative dynamical systems. If Y is compact, invariant, T 2 = R, (X, T 1 , π), (Y, T 2 , σ), h is a convergent non-autonomous dynamical system and J is the Levinson center of (X, T 1 , π), then (J, 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.

2.3.
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 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 Theorem 2.2. [7] Suppose that the following conditions hold: (ii) the point y ∈ Y is asymptotically stationary (respectively, asymptotically τ -periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent); (iii) the point x is comparable with y ∈ Y by the character of recurrence in infinity.
Let (X, T, π) be a dynamical system. Denote by Ω X : where ω x is the ω-limit set of the point x.
, h be a non-autonomous dynamical system such that the following conditions hold: Then, there exists a unique compact minimal set M ⊆ X such that by the character of recurrence in infinity.
, h be a non-autonomous dynamical system such that the following conditions hold: (iii) the point y 0 is asymptotically stationary (respectively, asymptotically τperiodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent); Then, there exists a unique compact minimal set M ⊆ X such that (iii) every point x ∈ X is asymptotically stationary (respectively, asymptotically τ -periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent).

Pseudo Recurrent Dynamical Systems with Convergence.
A non-auto- and C(X) denotes the set of all compact subsets from X.
Let (X, h, Y ) be a fiber space, i.e., X and Y be two metric spaces and h : The set M is called conditionally compact if it is closed and conditionally relatively compact.
is a fiber space, the space X is conditionally compact, but not compact.
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 Now, we state two results proved in [3,4].
3. Almost periodic and asymptotically almost periodic solutions of Liénard equation.
Consider the following Liénard equation where F : T → R is 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 F satisfy the following conditions: The typical example for equation (4) is given by where c is a nonnegative constant, α > 0 and F : R → R is a periodic (respectively, quasi periodic, almost periodic, almost automorphic, recurrent) function. The following results are well known. (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). (4) satisfying the initial conditions ϕ(0, u, v, F ) = u and ϕ (0, u, v, F ) = v. We have the following theorem.

Denote by ϕ(t, u, v, F ) the unique solution of equation
, then for any pair of solutions ϕ(t, u i , v i , F ) (i = 1, 2) of equation (4), which are bounded on R + , we have , then equation (4) admits at most one solution which is bounded on 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 F in the shift dynamical system (C(T, R), T, σ), possesses this property. (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., N F ⊆ N (ϕ,ϕ ) (respectively, M F ⊆ M (ϕ,ϕ ) ), M F := {{t n } : the sequence {σ(t n , F )} is convergent}, L F := {{t n } ∈ M F : such that t n → +∞ as n → ∞} and ϕ is the derivative of the function ϕ.

The solution ϕ(t) of equation
Example 3.5. Denote by y = x , then equation (4) can be reduced to the system (5) x = y y = −g(x) − f (x)y + F (t).
Along with system (5), consider its H-class, i.e., the family of systems where G ∈ H(F ).
Theorem 3.6. Suppose that F ∈ C(R + , R) and F is 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 (X, R + , π), (Y, R, σ), h generated by equation (4) and its solution ϕ(t, u, v, F ) (see Example 3.5). Since F is asymptotically recurrent, the dynamical system (Y, R, σ) is compact dissipative and its Levinson center be two solutions of system (6). According to Theorem 3.3 we have On the other hand if G ∈ Ω F (ω-limit set of the function F ), then G ∈ C b (R, R) and by Theorem 3.3 system (6) admits at most one solution which is bounded on R . Now to finish the proof of Theorem it is sufficient to apply Corollary 2.3.
Proof. This statement follows from Theorem 3.6 and Corollary 2.4.

4.2.
Stepanoff asymptotically almost periodic solutions. Let us start by defining the concept of Stepanoff asymptotically almost period solutions.   1) the function ϕ is asymptotically S p almost periodic, i.e., the motion σ(·, ϕ) is asymptotically almost periodic in the dynamical system (L p loc (R + ; B; µ), R, σ); 2) there exist an S p almost periodic function p and a function ω ∈ L p loc (R + ; B; µ) such that p ∈ L p loc (R; B; µ), ϕ = p + ω and lim   (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).
Proof. We omit the proof because it is a slight modification of the proof of Lemma 3.2.
Lemma 4.6. [7,ChI] If F ∈ L p loc (R + ; R) and F is asymptotically recurrent, then sup Theorem 4.7. Suppose that F ∈ L p loc (R + ; R) and F is asymptotically recurrent. Then every bounded on R + solution ϕ(t, u, v) of equation (4) is compatible by recurrence in infinity.
Proof. This statement can be proved using the same arguments as in the proof of Theorem 3.3 (see also Remark 3.4), Theorem 3.6 and using lemmas 4.5 and 4.6.
Proof. The proof follows from Theorem 3.6 and Corollary 2.4. Remark 4.9. We would like to stress that in Corollary 4.8 the function F is asymptotically almost periodic in the sense of Stepanoff, but the solution ϕ(t, u, v, F ) is asymptotically almost periodic in the sence of Fréchet [10,11].
Remark 4.10. When the function F is S p asymptotically almost periodic, a particular case of Corollary 3.7 was established by Ait Dads et al. [1]. Namely, they proved this statement in the case when F (t) = P (t) + Ω(t) for all t ∈ R + , where P ∈ C(R, R) is a Bohr almost periodic function and Ω ∈ C(R + , R) with the following properties: sup t∈R+ |Ω(t)| < +∞ and lim t→+∞ t+1 t |Ω(s)|ds = 0. It is evident, that the function F with the properties listed above is S p (with p = 1) asymptotically almost periodic, but the inverse statement is not true.

Convergence of Forced Vectorial Liénard Equations
Let (Y, R, σ) be a two-sided dynamical system. Consider the following vectorial Liénard equation where f ∈ C(Y, R m ), C : R m → R m is a symmetric and nonsingular linear operator, and ∇F denotes the gradient of the convex function F on R m . If the operator C is positive definite, then we introduce the product space R m × R m R 2m endowed with the inner product associated to the quadratic form Q given by Remark 5.1. Note that the inequality holds for all (u, v) ∈ R 2m , where α := max(1, C −1 ) and β := max(1, where ||C|| is the norm of operator C. Equation (9) can be re-written in the form (10) U (t) + G(σ(t, y), U (t)) = 0, where G ∈ C(Y × R 2m , R 2m ) and the partial function G(y, ·, ·) is strictly monotone for each y ∈ Y with respect to the inner product associated to Q, i.e., for each . Indeed, by letting v(t) := u (t) + ∇F (u(t)); U (t) := (u(t), v(t)), equation (9) reduces to where Φ(u, v) := F (u), J := 0 −I C 0 and F(y) := (0, f (y)).

Lemma 5.2. [8]
Let I be the interval [0, +∞) or the whole real line R. Let f (t) := f (σ(t, y)) (∀t ∈ I) be bounded on I. If u ∈ C 2 (I, R n ) is a solution of equation (9) which is bounded on I, then u and u are also bounded on I.
Theorem 5.3. Suppose that the following conditions are fulfilled: (iii) the equation (9) admits a solution u 0 which is bounded on R + . Then, (i) the equation (9) is convergent, i.e., the non-autonomous dynamical system (cocycle) generated by (9) is convergent; (ii) if the point y 0 ∈ Y is a τ -periodic (quasi periodic, almost periodic in the sense of Bohr, almost automorphic, recurrent, pseudo recurrent) point, then equation (9) has a unique τ -periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent) solution u such that M y 0 ⊆ M u ; (iii) every solution of equation (9), which is bounded on R + , is asymptotically τ -periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent).
Let u 0 be a solution of equation (9) which is bounded on R + , then by Lemma 5.2 U 0 := (u 0 , u 0 ) is the solution of equation (10) which is bounded on R + . To finish the proof, it is sufficient to refer to Theorem 2.6 and Corollary 2.7.
Example 5.4. We consider the equation (11) x + p(x)x + ax = f (σ(t, y)), where p ∈ C(R, R), f ∈ C(Y, R) and a is a positive number. Denote by u = x and v = u + F(u), where F(u) = u 0 p(s)ds, then we obtain the system σ(t, y)).