New criteria for the existence of periodic and almost periodic solutions for some evolution equations

In this work we give a new criteria for the existence of periodic and almost periodic solutions for some dieren tial equation in a Banach space. The linear part is nondensely dened and satises the Hille-Yosida condition. We prove the existence of periodic and almost periodic solutions with condition that is more general than the known exponential dichotomy. We apply the new criteria for the existence of periodic and almost periodic solutions for some partial functional dieren tial equation whose linear part


Introduction
In this work, we are concerned with the existence of periodic and almost periodic solutions for the following differential equation: where A : D(A) ⊂ E → E is a nondensely defined linear operator on a Banach space E and f : R → E is continuous, p-periodic or almost periodic (f is not identically zero).For every t ≥ 0, B(t) is a bounded linear operator on E. Throughout this work, we suppose that A is a Hille-Yosida operator which means that there exist M 0 ≥ 1 and ω 0 ∈ R such that (ω 0 , +∞) ⊂ ρ(A) and |R (λ, A) n | ≤ M 0 (λ − ω 0 ) n , for n ∈ N and λ > ω 0 , (2) where ρ(A) is the resolvent set of A and R (λ, A) = (λ − A) −1 .
Differential equations with nondense domain have many applications in partial differential equations.About this topic we refer to [14] where the authors studied the well-posedness of equation ( 1) with B = 0 and A is a Hille-Yosida operator.The existence of periodic and almost periodic solutions for partial functional differential equations has been extensively studied in literature, for the reader we refer to [5], [9], [10], [11], [12] and references therein.In [8], the authors established the existence of periodic and almost periodic solutions of equation ( 1) and they applied their results for the following partial functional differential equation: where C is the space of continuous functions from [−r, 0] into E endowed with the uniform norm topology, for every t ≥ 0, the history function x t ∈ C is defined by K(t) is a bounded linear operator from C to E and t → K(t)ϕ is continuous, for every ϕ, p-periodic in t and h : R → E is continuous and p-periodic.
The famous Massera's Theorem [9] on two dimensional periodic ordinary differential equations explains the relationship between the boundedness of solutions and periodic solutions.In this work we use Massera's approach [9], we give sufficient conditions such that the equivalence between the existence of a p−periodic solution and the existence of a bounded solution holds.Note that Massera's approach holds for equation (1) if A generates a compact semigroup on E. Since in this case the Poincaré map is compact for p > r.In [6], the authors proved the existence of a periodic solution for nonlinear partial functional EJQTDE, 2004 No. 6, p. 2 differential equations that are bounded and ultimate bounded, using Horn's fixed point theorem they proved that the Poincaré map has a fixed point which gives a periodic solution.Recently in [10], the authors obtained a new spectral criteria for the existence of bounded solutions for the following difference equation where (y n ) n∈Z ∈ l ∞ (E) is given and P is a bounded linear operator on E and they applied the criteria to show the existence of periodic and almost periodic solutions for some partial functional differential equations with infinite delay and some differential equations in Banach spaces.
The results obtained in this paper (together with the idea) are intimately relates to those in [10].Several results in [10] are extended to the equation ( 1) whose linear part is nondensely defined.
The organization of this work is as follows: in section 2, we recall some preliminary results that will be used later.In section 3, we establish a new criteria for the existence of p-periodic and almost periodic solutions of equation ( 1).Finally we propose an application to equation (3).

Preliminary results
In the following we assume that (H 1 ) A is a Hille-Yosida operator.
(H 2 ) For every t ≥ 0, the operator B(t) is a bounded linear operator on E, p-periodic in t and t → B(t)x is continuous, for every x in E. Let us introduce some notions which will be used in this work.
Proposition 2.1 [14] For every s ∈ R and x 0 ∈ D(A), equation (1) has a unique integral solution for t ≥ s.
Then A 0 generates a strongly continuous semigroup (T 0 (t)) t≥0 on D(A).
Proposition 2.2 [8] Assume that (H 1 ) and (H 2 ) hold.Then (U B (t, s)) t≥s is an evolution family: Moreover the integral solution of equation ( 1) is given by Definition 2.2 (U B (t, s)) t≥s has an exponential dichotomy on D(A) with constant β > 1 and L ≥ 1, if there exists a bounded strongly continuous family of projection (P (t)) t∈R on D(A) such that for t ≥ s one has For the sequel, C b (R, E) denotes the space of bounded continuous functions on R with values in E.
Theorem 2.2 [8] Assume that (H 1 ) and (H 2 ) hold.Then the following propositions are equivalent: i) U B (t, s) t≥s has an exponential dichotomy, ii) for any f in C b (R, E), equation (1) has a unique integral solution in C b (R, E).
If we suppose that B(t + p) = B(t), for all t in R, then the evolution family U B (t, s) t≥s is p-periodic: Recall that a function v ∈ C b (R, E), is said to be almost periodic if the set {v τ : τ ∈ R} is relatively compact in C b (R, E), where v τ is defined by iii) sp(αu) ⊂ sp(u).iv) If u is uniformly continuous, sp(u) is countable and E doesn't contain a copy of c 0 , then u is almost periodic.
The spectrum σ(u) of a bounded continuous function u is defined by: σ(u) = e isp(u) .Then a criteria for the existence of a p−periodic solution of equation ( 1) is obtained in [8].
Then equation ( 1) has at most one solution in C b (R, E).Moreover if f is almost periodic, then equation ( 1) has a unique almost periodic solution.
Remark:U B (p, 0) is called the monodromy operator.Condition ( 9) is more general than the exponential dichotomy condition.Indeed, if the evolution family U B (t, s) t≥s has an exponential dichotomy, then where ρ(U B (p, 0)) denotes the resolvent set of U B (p, 0).Moreover, it's wellknown that f is p-periodic if and only if sp(f ) ⊂ 2π p Z. Consequently if 1 ∈ ρ(U B (p, 0)), then equation ( 1) has a unique p-periodic solution.In the following, we give an extension of Theorem 2.3 and we prove the existence of a p-periodic solution of equation ( 1) if ( 1) has a bounded solution on the whole line and 1 is isolated in σ(U B (p, 0).

Periodic solutions
where g is defined by Periodicity of f and U B (t, s) t≥s imply that g is p−periodic.Let (x n ) n∈Z and (g n ) n∈Z be defined by x n = u(np) and g n = g(np), for n ∈ Z.
EJQTDE, 2004 No. 6, p. 6 Then periodic integral solutions of equation ( 1) correspond to constant solutions of the following discrete equation Since f is not identically zero, then g n+1 = g n = 0 and σ((g n ) n∈Z ) = {1} .By Theorem 2.4, we deduce that equation ( 11) has a bounded solution (x n ) n∈Z such that σ((x n ) n∈Z ) = σ((g n ) n∈Z ) = {1} .By Lemma (2.1), we conclude that x n+1 = x n for every n ∈ Z and by uniqueness of solutions with initial data we get that the integral solution of equation ( 1) starting from x 0 is p−periodic.Proof.Arguing as above, we get that equation ( 11) has a bounded solution (x n ) n∈Z such that σ((x n ) n∈Z ) = σ((g n ) n∈Z ) = {−1} and by Lemma (2.1), we obtain that x n+1 = −x n , which gives an anti p−periodic integral solution of equation (1).

Almost periodic solutions
Let g be defined by g(t) = lim If σ(g(n) n∈Z ) is countable and E doesn't contain a copy of c 0 , then equation ( 1) has an almost periodic integral solution if and only if it has a bounded integral solution on the whole line.
We start by the following fundamental Lemma which plays an important role in the proof of Theorem 3.3, its proof is similar to the one given in [10].

Lemma 3 . 1
Assume that U B (t, s) t≥s is a 1−periodic evolution family.Let w be a solution on the whole line ofw(t) = U B (t, s)w(s) + lim λ→∞ t s U B (t, τ )B λ θ(τ )dτ , t ≥ s, Remark 3.1 Condition 10 means that if 1 is in σ Γ (U B (p, 0)), then 1 is an isolated point in σ Γ (U B (p, 0)).Assume that (H 1 ), (H 2 ) hold and f is anti-periodic which means that f (t + p) = −f (t), for all t ∈ R. If σ Γ (U B (p, 0))\ {−1} is closed, then equation (1) has an anti p−periodic integral solution if and only if it has a bounded integral solution on the whole line.