Faculté des Sciences

AbstractWe study some properties of bounded and C (1) -almost automor-phic solutions of the following Li´enard equation:x 00 +f(x)x 0 +g(x) = p(t),where p : R −→ R is an almost automorphic function, f, g : (a,b) −→R are continuous functions and g is strictly decreasing.AMS classiﬁcation: 34C11, 34C27, 34D05.Key words: Almost automorphic solutions, bounded solutions, Li´e-nard equations. 1 Introduction In this paper, we study some properties of bounded or C (1) -almost automor-phic solutions of the following Li´enard equation:x 00 +f(x)x 0 +g(x) = p(t), (1.1)where p: R −→ R is an almost automorphic function and f,g: (a,b) → R,(−∞ ≤ a 0 and p: R −→ R is an almost automorphic function, thatappears when the restoring force is a singular nonlinearity which becomesinﬁnite atzero. Mart´inez-Amores andTorres in [13], then Campos and Torresin [5] describe the dynamics of Equation (1.1) in the periodic case, namelythe forcing term pis periodic. Recall that the existence of periodic solutionsof Equation (1.1) without friction term (f = 0) is proved by Lazer andSolimini in [12] and by Habets and Sanchez in [11] for some Li´enard equationsEJQTDE, 2008 No. 21, p. 2


Introduction
In this paper, we study some properties of bounded or C (1) -almost automorphic solutions of the following Liénard equation: where p : R −→ R is an almost automorphic function and f, g : (a, b) → R, (−∞ ≤ a < b ≤ +∞) are continuous functions.The following assumptions will be used in proving the main results: (H1) f and g : (a, b) −→ R are locally Lipschitz continuous.
The model of Equation (1.1) is where c ≥ 0, α > 0 and p : R −→ R is an almost automorphic function, that appears when the restoring force is a singular nonlinearity which becomes infinite at zero.Martínez-Amores and Torres in [13], then Campos and Torres in [5] describe the dynamics of Equation (1.1) in the periodic case, namely the forcing term p is periodic.Recall that the existence of periodic solutions of Equation (1.1) without friction term (f = 0) is proved by Lazer and Solimini in [12] and by Habets and Sanchez in [11] for some Liénard equations EJQTDE, 2008 No. 21, p. 2 with singularities, more general than Equation (1.1).In [5], Campos and Torres prove that the existence of a bounded solution on (0, +∞) implies the existence of a unique periodic solution that attracts all bounded solutions on (0, +∞).Moreover, they proved that the set of initial conditions of bounded solutions on (0, +∞) is the graph of a continuous nondecreasing function.
Then Cieutat extends these results to the almost periodic case in [6].In [5], Campos and Torres use topological tools, such as free homeomorphisms (c.f.[4]), together with truncation arguments.The homeomorphisms used in [5], are the Poincaré operators of Equation (1.1), therefore these topological tools are not adapted to the almost periodic case.In [6], the method used is essentially the recurrence property of the almost periodic functions.This last property says that once a value is taken by φ(t) at some point t ∈ R, it will be "almost" taken arbitrarily far in the future and in the past.Later, Ait Dads et al. [1] in the bounded case, namely the forcing term p is continuous and bounded, prove the uniqueness of the bounded solutions on (−∞, +∞) and describe the set of initial conditions of bounded solutions on (0, +∞).
Then they establish a result of existence and uniqueness of the pseudo almost periodic solution.
The notion of almost automorphic is a generalization of almost periodicity.It has been introduced in the literature by Bochner in relation to some aspect of differential geometry [2,3] and more recently, this notion was developed by N'Guérékata (see for instance [14,15]).
Our aim is to extend some results of [5,6] to the almost automorphic case, namely to prove that the existence of a bounded solution on (0, +∞) implies the existence of a unique almost automorphic solution that attracts all bounded solutions on (0, +∞).Then we state and prove a result on the existence of almost automorphic solutions.
Let us first fix some notations and definitions.We say that a function u ∈ C(R) (continuous) is almost automorphic if for any sequence of real numbers (t n ) n , there exists a subsequence of ( If we denote by AA(R) the space of all almost automorphic R-valued functions, then it turns out to be a Banach space under the sup-norm.
Because of pointwise convergence, the function v ∈ L ∞ (R) (the space of essentially bounded measurable functions in R), but not necessarily continuous.It is also clear from the definition above that almost periodic functions (in the sense of Bochner [2,10]) are almost automorphic.If we denote AP (R), the space of all almost periodic R-valued functions, we have AP (R) ⊂ AA(R).
A function u ∈ C(R) is said to be C (n) -almost automorphic if it is almost automorphic up to its nth derivative.We denote the space of all such functions by AA (n) (R) (see [8]).
If the limit in (1.3) is uniform on any compact subset K ⊂ R, we say that u is compact almost automorphic.If we denote AA c (R), the space of compact almost automorphic R-valued functions and BC(R) the space of bounded and continuous R-valued functions, we have Similarly AA (n) c (R) will denote the space of all C (n) -compact almost automorphic functions.For more details on almost automorphic functions, we refer to [14,15].
The bounded solutions considered in this paper, are the solutions such that their range is relatively compact in the domain (a, b) of Equation (1.1).More precisely, for a bounded solution x, we impose the existence of a compact set such that ∀t ∈ R, In the almost periodic case, this type of conditions was assumed by Corduneanu in [7,Chapter 4] and by Yoshizawa in [18,Chapter 3].Without these conditions, the tools of the study of almost automorphic solutions of differential equations are often unusable.For these reasons, we say that a function We also say that a function Remark that if x is a periodic solution of Equation (1.1), then x is bounded on R (in the sense of above definition), but an almost periodic solution, therefore an almost automorphic solution, is not necessarily bounded on R (of course sup t∈R | x(t) |< +∞), because there exists an almost periodic solution x such that inf t∈R x(t) = a (if a ∈ R).For example, we consider x(t) := cos(t) − cos(2πt) + 2. Since x(t) > 0 for all t ∈ R, then x is an almost periodic solution of Equation (1.1) where a := 0, b := +∞, f (x) := 0, g(x) := −x and p(t) := ((2π The paper is organized as follows: we announce the main results (Theorem 2.1) in Section 2 and we give its proof in Section 3. Section 4 is devoted to an example.

Main Result
Theorem 2.1.Assume that hypotheses (H1)-(H3) hold, and let p ∈ AA(R).In addition, assume that Equation (1.1) has at least one solution that is bounded in the future.Then the following statements hold true: i) Equation (1.1) has exactly one solution φ that is bounded on R. Moreover φ ∈ AA (1)  c (R). ii) Every solution x bounded in the future of Equation (1.1) is asymptotically almost automorphic, in the sense that: The proof of Theorem 2.1 will be given in Section 3.
Remark.For the proof of Theorem 2.1, we use a result on the structure of solutions that are bounded in the future and on the uniqueness of the EJQTDE, 2008 No. 21, p. 5 bounded solution on R when the second member p is bounded and continuous (c.f.Proposition 3.1).This last proposition is established in [1].Firstly, for the proof of Theorem 2.1, we state the existence of a solution that is bounded in the future implies the existence of a bounded solution on the whole real line.This result is well-known when the second member p is almost periodic (for instance [9,10]).In the almost automorphic case, this result is stated when p is compact almost automorphic.For example, Fink has established similar result [9, Lemma 2], which is valid even for the following differential system in R n : x (t) = F (t, x(t)).We cannot use [9, Lemma 2] because we do not assume that p is compact almost automorphic, but only almost automorphic.Secondly, we prove that the unique bounded solution is compact almost periodic.Since we assume that p is only almost automorphic, we cannot use [9, Corollary 1].
Corollary 2.2.Assume that hypotheses (H1)-(H3) hold.In addition suppose that p ∈ AA(R).If inf t∈R p(t) and sup t∈R p(t) are in the range of g: g(a, b), then Equation (1.1) has a unique bounded solution x on R which is compact almost automorphic.Moreover this solution is asymptotically almost automorphic and its derivative is also compact almost automorphic.
Remark.In the particular case of Equation (1.2), one has the existence and uniqueness of compact almost automorphic solution, when the second member p satisfies 0 < inf t∈R p(t) ≤ sup t∈R p(t) < +∞ and p is almost automorphic.
Proof of Corollary 2.2.We use Theorem 2.1.It suffices to prove the existence of a solution of Equation (1.1) that is bounded on R. For that we adapt a result of Opial [16,Théorème 2].In the particular case where p(t) = p 0 for each t ∈ R, i.e. inf t∈R p(t) = sup t∈R p(t), there exists x 0 ∈ (a, b) such that g(x 0 ) = p 0 , therefore x(t) = x 0 for each t ∈ R, is a solution that is bounded on the R. Now we assume that inf t∈R p(t) < sup t∈R p(t).By hypothesis on the range of g and by (H2), there exist A and B ∈ R such that g(A) = sup t∈R p(t) and g(B) = inf t∈R p(t) and a < A < B < b.Let f and g be extensions of f /[A,B] and g /[A,B] .The extension f is defined by f : R −→ R with EJQTDE, 2008 No. 21, p. 6 In a similar way we define g.Obviously f and g are continuous.Now set ii) V and T are nonnegative and continuous functions on By using [16, Théorème 2], we can assert that the equation x = F (t, x, x ) admits at least a solution x satisfying A ≤ x(t) ≤ B for each t ∈ R, therefore x is a solution of Equation (1.1) that is bounded on R.This ends the proof.

Proof of Theorem 2.1
The object of this section is to prove Theorem 2.1.For the reader's convenience, we recall the following results.Proposition 3.1.(Ait Dads, Lhachimi and Cieutat [1]).Assume that hypotheses (H1)-(H3) hold.We also suppose that p ∈ BC(R).Then we get: i) Any pair of distinct solutions of Equation (1.1) x 1 and x 2 bounded in the future, satisfy for every t where both solutions are defined and ii) Equation (1.1) has at most one bounded solution on R.

EJQTDE, 2008
No. 21, p. 7 Remark.Relation (3.1) implies that t −→| x 1 (t) − x 2 (t) | is strictly decreasing and any two distinct solutions bounded in the future have no common point.Lemma 3.2.(Cieutat [6]).Assume that p ∈ BC(R), f and g ∈ C(a, b).Let I = (t 0 , +∞) with t 0 = −∞ or t 0 ∈ R. If x is a solution of Equation (1.1) which is bounded in the future (respectively bounded on R), i.e. a < A ≤ x(t) ≤ B < b for all t > t 0 (respectively t ∈ R), then the derivatives x and x are bounded in the future (respectively bounded on R), i.e.Let p ∈ L ∞ (R).We say that x is a weak solution on R of Equation (1.1), if x ∈ C 1 (R) (of class C 1 ) and satisfies for each s and t ∈ R such that s ≤ t.
Obviously a classical solution is a weak solution and in the particular case where p is continuous, the notion of weak solution and classical solution are equivalent.EJQTDE, 2008 No. 21, p. 8 Lemma 3.3.Let e ∈ L ∞ (R) and f , g ∈ C(R).We assume that u is a weak solution bounded on R of such that u ∈ L ∞ (R) and u is k-Lipschitzian on R for some constant k.If there exist a numerical sequence then there exists a subsequence of (t n ) n denoted (t n ) n such that as n → +∞, (3.9) uniformly on each compact subset of R, where v is a weak solution bounded on ) Proof.Since u is a bounded on R, there exist A and B ∈ R such that for each t ∈ R a < A ≤ u(t) ≤ B < b.
If we denote by u n (t) := u(t + t n ), (3.12) then u n ∈ C 1 (R) and satisfies, for each t ∈ R and n ∈ N and thus we obtain for each s, t ∈ R and n ∈ N. From (3.13) and (3.15), we deduce that for each t ∈ R, {u n (t); n ∈ N} is a bounded subset of R and the sequence (u n ) n is equicontinuous.By help of Arzela Ascoli's theorem [17, p. 312], we can EJQTDE, 2008 No. 21, p. 9 assert that {u n ; n ∈ N} is a relatively compact subset of C(R) endowed with the topology of compact convergence.From the sequence (t n ) n , we can extract a subsequence (t n ) n such that there exists v ∈ C(R) and (3.9) holds.Moreover since u is k-Lipschitzian on R, then one has for each s, t ∈ R and n ∈ N. Using (3.14), (3.16) and applying Arzela Ascoli's theorem, we deduce that there exist w ∈ C(R) and a subsequence of (t n ) n (which we denote by the same) such that as n → +∞ uniformly on each compact subset of R. With (3.9), we deduce that w = v , consequently (3.10) holds.By assumptions, u ∈ C 1 (R), u ∈ L ∞ (R) and u is k-Lipschitzian, then the convergence (3.9) and (3.10) and relations (3.13), (3.14) and (3.16) It remains to prove that v is a weak solution of Equation (3.11).Since u is a weak solution of Equation (3.7), then for each s ≤ t, we have To check that φ and its derivative φ are compact almost automorphic, we have to prove that if (t n ) n is any sequence of real numbers, then one can pick up a subsequence of (t n ) n such that ii) It is straightforward from Proposition 3.1.
.17) Moreover, we have | e(σ + t n ) |≤ sup t∈R | e(t) |< +∞ for each σ ∈ [s, t] and by n ) dσ = t s p * (σ) dσ, (3.33) thus with (3.30)-(3.33),we deduce that x * is a weak solution on R of x * + f (x * )x * + g(x * ) = p * (t).* is bounded on R and x * ∈ L ∞ (R) and x * is Lipschitzian.Applying Lemma 3.3, u = x * , e = p * and the sequence (−t n ) n (c.f.(3.21)), we obtain the existence of a weak solution φ of Equation (1.1) that is bounded on R. Since p is a continuous function, then φ is a classical solution on R of Equation (1.1).The uniqueness of the bounded solution of Equation (1.1) follows from Proposition 3.1.
In fact by assumption, we can choose a subsequence of (t n ) n such that (3.20) and(3.21)hold.By applying Lemma 3.3 with u = φ, e = p and the sequence (t n ) n we obtain (3.35) and (3.36)where φ * is a weak solution on R of Equation (3.34), which satisfies all hypotheses of Lemma 3.3.Applying again Lemma 3.3 to u = φ * , e = p * and the sequence (−t n ) n , we obtain that (t − t n ) = ψ (t) (3.40) (for a subsequence) where ψ is a weak solution on R of Equation (1.1).Since p is continuous, then ψ is a classical solution on R of Equation (1.1).By uniqueness of the solution of Equation (1.1) that is bounded on R, we deduce that ψ = φ, therefore (3.35)-(3.38)are fulfilled, thus φ and φ are compact almost automorphic. *