Structure of the Set of Bounded Solutions and Existence of Pseudo Almost Periodic Solutions of a Vector Liénard Di erential Equation

In applied sciences, somepractical problems concerningmechanics, the engineering techniques elds, economy, control theory, physics, chemistry, biology, medicine, atomic energy, information theory, etc.. are associated with the so called Liénard or modi ed Liénard’s equation. By this time, the qualitative properties of solutions of scalar Liénard ormodi ed Liénard’s equation have been intensively discussed, see the references therein. Liénard equation has been investigated by many authors from various points of view. One of the reason why many mathematicians have been studied this kind of equations is that a broad class of phenomena in sciences and engineering is presented by Liénard’s equation. However, to the best of our knowledge from the literature, the pseudo almost periodicity of solutions for vector Liénard’s equation has not been discussed in the literature up to now. In this work, we study some properties of bounded and pseudo almost periodic solutions of the vector Liénard’s equation.More exactly,we study someof the properties of bounded, asymptotically almost periodic or pseudo almost periodic solutions of the following Liénard system:


Introduction
In applied sciences, some practical problems concerning mechanics, the engineering techniques elds, economy, control theory, physics, chemistry, biology, medicine, atomic energy, information theory, etc.. are associated with the so called Liénard or modi ed Liénard's equation. By this time, the qualitative properties of solutions of scalar Liénard or modi ed Liénard's equation have been intensively discussed, see the references therein.
Liénard equation has been investigated by many authors from various points of view. One of the reason why many mathematicians have been studied this kind of equations is that a broad class of phenomena in sciences and engineering is presented by Liénard's equation.
However, to the best of our knowledge from the literature, the pseudo almost periodicity of solutions for vector Liénard's equation has not been discussed in the literature up to now.
In this work, we study some properties of bounded and pseudo almost periodic solutions of the vector Liénard's equation. More exactly, we study some of the properties of bounded, asymptotically almost periodic or pseudo almost periodic solutions of the following Liénard system: where p : R −→ R N is continuous, bounded, pseudo almost periodic function. F : R N → R, and G : R N → R N .
On the other hand, the Liénard's equation with a periodic forcing term p has also been studied by many authors. It is known that under some assumptions, the Liénard's problem has multiple periodic solutions and sometimes the dynamics of the solutions are chaotic. We will see that under suitable conditions, n− dimensional Liénard's equation has pseudo almost periodic solutions, in the case where p(t) is so.
The space R N is endowed with its usual inner product < x | y >:= N i= x i y i and || . || denotes the associated Euclidean norm. Denote by ∇F(x) (resp. ∇ F(x)) the gradient (resp. the Hessian) of F at a point x.
The model of equation. (1) is where c ≥ , α > and p : R −→ R is an almost periodic function, that appears when the restoring force is a singular nonlinearity which becomes in nite in zero. Martínez-Amores and Torres in [18], then Campos and Torres in [7] described the dynamics of equation. (1) in the periodic case, namely the forcing term p is periodic. Recall that the existence of periodic solutions of equation. (1) without friction term (F = ) is proved by Lazer and Solimini in [17] and by Habets and Sanchez in [15] for some Liénard's equations with singularity, more general than equation. (1). In [7], Campos and Torres proved that the existence of a bounded solution on ( , +∞) implies the existence of a unique periodic solution that attracts all bounded solutions on ( , +∞). Besides they state that the set of initial conditions of bounded solutions on ( , +∞) is the graph of a continuous nondecreasing function. Then Cieutat [9] extended results of the paper [7] to the almost periodic case. In [7], the authors used topological tools, such as free homeomorphisms (c.f. [6]), together with truncation arguments. The homeomorphisms used in [7], are the Poincaré operators of equation (1), therefore these topological tools are not adapted to the almost periodic case. In [9], the methods used are essentially the recurrence property of the almost periodic functions. A well known extension of almost periodicity is the notion of asymptotically almost periodicity historically due to Fréchet (c.f. [14] for details). In [20], Zhang introduced an other extension of the almost periodic functions, the so-called pseudo almost periodic functions. In [21], Zhang investigated the existence of pseudo almost periodic solution for a pseudo almost periodic nonlinear perturbation of a linear autonomous ordinary di erential equations. Ait Dads and Arino [4] introduced the generalized pseudo almost periodic functions and extended Zhang's results. More details on the concept of pseudo almost periodicity can be found in [2,4,21 ]. The question of pseudo almost periodic solutions to some di erential equations has been studied by various authors [3,5,12,16,19,22] and references therein.
Our aim is to extend the main results of the paper [1] to the bounded or pseudo almost periodic solutions in vector case. Notably, we describe the set of initial conditions of the bounded solutions on ( , +∞) and we state some results of the existence of pseudo almost periodic solutions.
The paper is organized as follows: in section , we give some preliminary results with the problem in general case. In section , we study the structure of solutions that are bounded in the future, when the second member p is bounded. The main result of this section (Theorem 16) extends a result in the almost periodic case [Theorem 2.2, [9]] to the bounded case. To establish this last result, we state some results of comparison of bounded solutions. In section , we consider some particular cases, the rst one is when F is quadratic and in the second subsection, we are concerned with the case that G is a linear function. We establish that all solutions that are bounded in the future, are asymptotically almost periodic, when the second member p is asymptotically almost periodic. In fact, we state this result for a larger class of second member p. Then in the section , we study the existence of pseudo almost periodic solutions.

General case
From a convex open subset Ω of R N , a C function F : Ω −→ R and two continuous functions: G : Ω −→ R N and p : R −→ R N , we consider the following forced vector Liénard's equation : We assume that ∇ F and G are locally lipschitzian and that the following hypotheses hold: (H1) −G is strictly monotone on Ω, namely : (3)

Lemma 4.
Under the Hypothesis (H2), let x and x , be two bounded in the future and continuous functions.
Since the two sequences (x (tn)) and (x (tn)) are with values in a compact K, they have respectively an adhesion value x * and x * . From (4) and (6), we deduce that By using Lemma 3, we obtain a contradiction: ∇F(x * ) = ∇F(x * ); then the relation (5) is established.
Proof. There exists a compact K of R N which contains the ranges of x and x . Let ε > . Denoting The function (u, v) −→ <∇F(u)−∇F(v) | u − v> is continuous and nonnegative on the compact Kε (c.f. Lemma 3), so there exists δ > such that We deduce the inclusion of the following ergodic sets : and consequently the ergodicity of implies the ergodicity of t −→|| ∇F(x (t)) − ∇F(x (t)) ||.
Proof. It su ces to apply Lemma 4 and Lemma 6 to S(x) = ∇F(x) and S(x) = G(x).

Lemma 8. Under the hypothesis (H1), if x and x , are two continuous and bounded functions over
Proof. It su ces to apply Lemma 4 and Lemma 6 to S(x) = ∇F(x) and S(x) = G(x).

Lemma 9.
Under the hypothesis (H3), let I = (t , +∞) with t = −∞ or t ∈ R. If x is a solution of equation (3) which is bounded in the future (resp. bounded on R), i.e. x(t) ∈ K for all t > t > −∞ (resp. t ∈ R), then x ′ and x ′′ satisfy sup and Proof. Let t ∈ I. By Taylor's formula, we obtain that From equation (3), we deduce that by integrating by parts, we obtain By (9)-(11), one has || x ′ (t) ||≤ c , for all t ∈ I, where c is de ned by (7). By equation (3), we deduce where c is the constant de ned by (8).
and lim n→+∞ n+ n || p(t) || dt = . (3) is the sum of p = p ap + p e with p ap ∈ AP(R, R N ) and p e ∈ E + ((c, +∞) , R N ). Let x be a solution which is bounded in the future of equation (3). If there exist a real sequence (tn)n and p ap

Lemma 10. Under the hypothesis (H3), assume that the second member p of equation
and lim Then there exists a subsequence of (tn)n such that where x * is a bounded solution over R of the following equation Proof. Since x is a bounded solution in the future, then there exists t ∈ R such that From Lemma (9), we obtain sup and sup Let (τ , +∞), for n ∈ N be large enough (τ + tn ≥ t ). Then the function t −→ x(. + tn) is de ned on (τ , +∞) and from (17), we have From (18) and (19), we deduce By considering (tn) n as a sequence which goes to −∞. From Ascoli's Theorem and diagonalization principle, we can assert the existence of x * ∈ C (R, R N ) and a subsequence of (tn)n such that uniformly on each compact subinterval of R. By (20) and (21), we deduce that x * is bounded on R: It remains to prove that x * is a solution of equation (16). Let (τ , τ ). Since the function t −→ x(t + tn) is a solution of the following equation: on (τ , τ ) and by integrating over (τ , t) for τ ≤ t ≤ τ , we obtain that x is a solution on (τ , τ ) of Moreover, one has From (14), (21)-(24), we deduce that x * satis es Consequently x * is a solution on (τ , τ ) for the equation (16). Since this equality is satis ed on every interval (τ , τ ), then x * is a solution on R of equation (16).

Lemma 11. Let S be a symmetric matrix of order N, with eigenvalues
where Proof. We may assume that λ = , since for <x | y> = , one has We may also assume that R = , since For || x ||= and <x | y> = , one has Due to the fact that S is a symmetric matrix, then there exists an orthonormal basis From (27) and (28), one obtains that for all x and y ∈ R N such that || x ||= and <x | y> = . Putting one has || x ||= and <x | y> = , From (29)-(31), one obtains (26), then (25) holds.

Bounded solutions
In this section, rst we study the structure of solutions that are bounded in the future when the second member p is bounded. Now, we formulate a theorem on the existence of bounded solutions for equation (3).

De nition 12. We say that
Consider the two following hypotheses : for all t ∈ R and x ∈ R N such that || x ||= R.

Theorem 13. Under the hypotheses (H2) and (H3), if hypotheses (C1) or (C2) holds, then equation (3) has at least a bounded solution on R.
Proof. 1) The hypothesis (C2) implies (C1). We choose x = . For || x ||= R, we have and with −G is c-strongly monotone, we deduce that hence for R su ciently large, we obtain The map f is continuous and equation (3) is equivalent to the following system : with the change of the unknown function By using Lemma 11, we deduce that Using (C1), one obtains for all t ∈ R, x and y ∈ R N such that || x ||= R and <x | y> = , Moreover, we have the condition for all t ∈ R, x and y ∈ R N such that || where α and β ∈ ( , +∞), By (33) consequently, we obtain the existence of a solution for the equation (3), that is bounded on R.

Particular cases
. First particular case: when F is quadratic. Now, we consider the following particular case (3) becomes: The hypothesis (H2) becomes (H2-2) B is semi de nite positive i.e. : <Bx | x> ≥ for all x in R N .
for every t where both solutions are de ned and (ii) Equation (35) has at most one bounded solution on R.
For the proof, we need the following Lemma : (35), which is bounded on R, then its derivative x ′ is also bounded on R.

Lemma 15. Let p ∈ BC(R, X). If x is a solution of equation
Proof. In the particular case where X is a nite-dimensional space, the following estimate is given in (Lemma 3.1, [10]), therefore by continuity of ∇F, the derivative x ′ is bounded on R. However in the in nite-dimensional space, this last estimation on the derivative x ′ and the proof given in [10] are also valid, because this proof use essentially Taylor's formula and integration by parts. By hypothesis ∇F is Lipschitzian, then ∇F maps bounded sets into bounded sets, therefore x ′ is bounded on R. Putting The function α is derivable and Since the solutions x and x are di erent, we deduce that α ′ (t) > , for all t ≥ T, then lim t→+∞ α(t) = sup t≥T α(t) < +∞.

Theorem 16. Assume that the forcing term p is almost periodic. (i) If equation (35) has at least one solution that is bounded in the future, then equation (35) has exactly one solution x that is bounded on R. Moreover this solution x and its derivatives x ′ and x ′′ are almost periodic and satisfy mod(x) ⊂ mod(p). (ii) Every y bounded solution in the future is asymptotically almost periodic
Proof. i) Let φ be a solution of equation (35) which is bounded in the future. Then there exist a compact K ⊂ Ω and t ∈ R such that for all t ≥ t , φ(t) ∈ K.
By Lemma 9, there exists c > such that sup By [Theorem 6.2, p. 99, [14]], equation (35) has at least one solution x de ned on R such that then x is bounded on R. By Proposition 14, we have the uniqueness of solution that is bounded on R.
For each p * ∈ H(p)(the hull set of p), we consider the following equation By proposition 14, for each p * ∈ H(p), equation (45) has exactly one solution x * satisfying (46). By using [Theorem 10, p. 170, [14]], we obtain the bounded solution x; so that x and its derivative x ′ are almost periodic and mod(x) ⊂ mod(p). The almost periodicity of the second derivative is a direct consequence of equation (35).
(ii) Since the almost periodic solution x is bounded in the future, by Proposition 14, we obtain (43). (35) is the sum p = p ap + p e , where p ap ∈ AP(R, R N ) and p e ∈ E + ((c, +∞) ; R N ).

(i) If equation (35) has at least one solution that is bounded in the future, then equation
has exactly one solution φ that is bounded on R. Moreover, this solution φ and its derivatives φ ′ and φ ′′ are almost periodic and mod(φ) ⊂ mod(p ap ).

(ii) Every solution x that is bounded in the future of equation (35) is asymptotically almost periodic. i.e.:
Since p ap is almost periodic, there exist a subsequence of (tn)n and p ap * ∈ AP(R, R N ) satisfying (14). By Lemma 10, we deduce the existence of a subsequence of (tn)n and the existence of solution x * bounded on R for the equation satisfying (15). Since φ is an almost periodic solution of equation (47), then φ is a bounded solution on R for the equation (35) with the perturbation p e = . By Lemma 10 and (14), we obtain the existence of a subsequence of (tn)n such that for all t ∈ R, lim where φ * is a bounded solution on R of equation (51). By (15), (50) and (52 ), we deduce that Since x * and φ * are two bounded solutions on R of equation (51), we obtain assertion i) of this theorem, which gives the equality of the last two solutions, which contradicts (53), consequently (49)  Proposition 18. Suppose that p = p ap + p e where p ap ∈ AP(R, R N ) and p e ∈ E(R, R N ). If equation (35) has a bounded solution x on R, then this solution is unique and asymptotically almost periodic at −∞ and at +∞: where φ denotes the almost periodic solution of equation (47) is similar to the one of (48). So (55) is proved.

. Second particular case : G is linear
Now we consider the particular case G(x) = Cx with C ∈ L(R N , R N ). The equation (3) becomes: The hypothesis (H1) becomes (H1-3) C is a de nite negative and symmetric matrix : <Cx | x> < for all x in Putting The function f is bounded and continuously di erentiable on (T , +∞), moreover one has From the convexity of F, we deduce that f ′ (t) ≤ , then The second derivative of f is bounded, then f ′ is uniformly continuous, so with (60), we deduce that We will establish (57). Even if it means to work on the eigenspaces of C, we can assume that C = λI, with λ < . Let λ = −ω , we can assume that λ = − , even if it means to replace h and k by h (t) = h( t ω ) and . Then, one has By the variation constant formula, we obtain that the solutions h which are bounded in the future of equation (63) are given by We can assume that c = , due to the fact that t −→ e −t goes to zero, so is for its derivative when t tends to +∞. By integration by parts, we obtain With the same argument, we deduce that Since f is decreasing, we obtain f (t) = , it follows that f ′ (t) = for all t ∈ R. By using Lemma 3 and (59), one has k(t) = for all t ∈ R and with (58), we deduce that Putting r(t) := || h(t) || . Since C is negative de nite, we have r ′′ (t) ≥ for all t ∈ R, then r is convex and bounded in R, consequently r is constant on R, consequently r ′′ (t) = for all t ∈ R. From hypothesis madded on C and by (64), we deduce that h(t) = , then x (t) = x (t) for all t ∈ R.

Theorem 20. Assume p is almost periodic (i) If equation (56) has at least one solution that is bounded in the future, then (56) has exactly one bounded solution on R. Moreover, this solution x and its derivatives x ′ and x ′′ are almost periodic and mod(x) ⊂ mod(p). (ii) Every bounded solution in the future y is asymptotically almost periodic:
Proof. (i) The proof is similar to the one given in Theorem 16. Let φ be a solution that is bounded in the future of equation (56). There exist a compact K ⊂ Ω and t ∈ R such that for all t ≥ t , φ(t) ∈ K.
By Lemma 9, there exists c > such that sup t≥t || φ ′ (t) ||≤ c . If we denote K := K × B( , c ) the compact subset of Ω × R N , we obtain From [Theorem 6.2, p. 99, [14]] and [Lemma 4.3, p. 104, [4]], we obtain that for each p * ∈ H(p), there exists a solution x * of equation such that By proposition 19, for each p * ∈ H(p), equation (66) has an unique solution x * satisfying (59). By using [Theorem 10, p. 170, [14]], we obtain that the bounded solution x; so that x and its derivative x ′ are almost periodic and mod(x) ⊂ mod(p). The almost periodicity of the second derivative x ′′ follows from (56).
(ii) Let us prove that lim Assume the contrary, then there exist ε > and a numerical sequence (tn)n satisfying lim n→+∞ tn = −∞ and inf Since p is almost periodic, there exist a subsequence of (tn)n and p * ∈ AP(R, R N ) verifying From Lemma 10 we deduce (The ergodic perturbation of p is zero : p e = ), Existence of a subsequence of (tn)n and existence of a solution x * (resp. y * ) bounded on R of equation (66) verifying for all t ∈ R, lim n→+∞ y(t + tn) = y * (t).
Since x * and y * are two bounded solutions on R of equation (66), From the assertion i) of this theorem, we obtain that, the equality of these last solutions, which is a contradiction with (72), consequently (68) is satis ed. Since the function x ′′ − y ′′ is bounded on (t , +∞) (c.f. Lemma 9), then x ′ − y ′ is uniformly continuous on (t , +∞), then (68) implies that lim which ends the proof of ii) .

Theorem 21.
Suppose that p = p ap + p e where p ap ∈ AP(R, R N ) and p e ∈ E + ((c, +∞) , R N ). i) If equation (56) has at least one solution that is bounded in the future, then equation has a unique solution φ that is bounded on R. Moreover, this solution φ and its derivatives φ ′ and φ ′′ are almost periodic and mod(φ) ⊂mod(p ap ). (56) is asymptotically almost periodic: From Lemma 10, we deduce the existence of a solution φ that is bounded on R for the equation (73). By Theorem 20, we deduce the uniqueness of the bounded solution φ on R, and that φ, φ ′ and φ ′′ are almost periodic and the formula of the module.

ii) Every solution x bounded in the future of equation
(ii) We claim that lim Assume the contrary, then there exist ε > and a numerical sequence (tn)n verifying lim n→+∞ tn = +∞ and inf Since p ap is an almost periodic function, then there exist a subsequence of (tn)n and p ap * ∈ AP(R, R N ) verifying (14). From Lemma 10, we deduce the existence of a subsequence of (tn)n and the existence of a x * bounded on R of the following equation satisfying (15). Since φ is an almost periodic solution of equation (73), then φ is a bounded solution on R of equation (56) with the perturbation null: p e = . By Lemma 10 and by (14), there exists a subsequence of (tn)n such that for all t ∈ R, lim where φ * is a bounded solution on R of equation (76). From (15), (??) and (77), one has Since x * and φ * are two bounded solutions on R of equation (76), By assertion i) of this theorem, we obtain x * (t) = φ * (t) for each t ∈ R, which contradicts (78), Therefore (75) is satis ed. Since x ′′ − φ ′′ is bounded on (t , +∞) (c.f. Lemma 9), then x ′ − φ ′ is uniformly continuous on (t , +∞), hence (75) implies which ends the proof of (ii).

Proposition 22.
Assume that p = p ap + p e where p ap ∈ AP(R, R N ) and p e ∈ E(R; R N ). If equation (56) has a solution x that is bounded on R, then this solution is unique and is asymptotically almost periodic at −∞ and at +∞: lim where φ denotes the almost periodic solution of equation (73).
Proof. The uniqueness of the bounded solution of equation 56 results from proposition 19. By using Theorem 21, we obtain the existence of the almost periodic solution of equation (73) verifying (74). The proof of is similar to the proof of (74). Then the equality (79) is achieved. Now, we state a result of existence and uniqueness of the µ−pseudo almost periodic solution.

µ− Pseudo almost periodic functions
In this section, we de ne new concepts, the µ−ergodic functions and the µ− pseudo almost periodic functions, then we give some properties of these functions. The notion of µ− pseudo almost periodicity is a generalization of the pseudo almost periodicity introduced by Zhang [20]; it is also a generalization of weighted pseudo almost periodicity given by Diagana [12]. For more details on this functions we refer the reader to [13]. Throughout this section E is a Banach space and BC(R, E) denotes the Banach space of bounded continuous functions from R to E, equipped with the supremum norm f ∞ = sup t∈R f (t) . We denote by B the Lebesgue σ− eld of R and by M the set of all positive measures µ on B satisfying µ(R) = +∞ and µ ([a, b]) < +∞ for all a, b ∈ R (a ≤ b).

De nition 23. Let µ ∈ M. A bounded continuous function f
We denote the space of all such functions by E(R, E, µ).

De nition 24. Let µ ∈ M.
A bounded continuous function f : R → E is said to be µ− pseudo almost periodic if f is written in the form: where g ∈ AP(R, E) and ϕ ∈ E(R, E, µ).
We denote the space of all such functions by PAP(R, E, µ), then we have the following inclusions Proposition 25. [13] Let µ ∈ M. Then E(R, E, µ, . ∞ ) is a Banach space.
From the de nition of PAP(R, E, µ), we easily deduce the following result: Proposition 26. [13] Let µ ∈ M. Then PAP(R, E, µ) is a vector space.
Next result is a characterization of µ− ergodic functions.
Consider the following hypotheses: (D1) There exist x ∈ Ω and R > such that B(x , R) ⊂ Ω and for all t ∈ R and x ∈ R N such that || x ||= R.
(D2) Ω = R N , −G is c-strongly monotone on R N and c > δ .
Corollary 33. If we assume that hypothesis (D1) or (D2) is satis ed, then equation (35) has a unique bounded solution on R.
Proof. It follows from Theorem 13 applied with F(x) = <Bx | x> and the proposition 14. Q.E.D.
Theorem 34. Assume p is µ−pseudo almost periodic. If hypothesis (D1) or (D2) is satis ed, then equation (35) has a unique solution x which is bounded on R and µ−pseudo almost periodic and veri es mod(x) ⊂ mod(p). Furthermore, if we denote by p ap (resp. y) the almost periodic component of p (resp. x), then y is the almost periodic solution of equation (47).

Remark 35.
In the almost periodic case (p ∈ AP(R, R N )), the unique bounded solution is almost periodic and veri es the modulus formula.
Proof. From Corollary 33, we obtain the existence and uniqueness of the solution x that is bounded on R for the equation (35). Firstly we state the existence and uniqueness of the almost periodic y for the equation (47) and mod(y) ⊂ mod(p ap ). Since if p veri es (D1) or (D2), then so is for p ap . In fact for (D2), it is easily to see that since D ) does not depend of the second member p.
Using the hypothesis (H ) and (86), we obtain that ≤ <G(y(t)) − G(x(t)) | h(t)> ≤|| h ||∞|| p e (t) || +α ′ (t), where || . ||∞ denotes the sup norm on R. By integrating (87)  Since p e is ergodic and α is bounded on R, by taking the limit as T → +∞, we deduce that with the function t −→ <G(x(t)) − G(y(t)) | h(t)> is ergodic. We deduce from Lemma 8 that h = x − y is ergodic, then x is pseudo almost periodic and y is the almost periodic component of x: y = x ap .