Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 6, 1-12; http://www.math.u-szeged.hu/ejqtde/

New Criteria for the Existence of Periodic and Almost Periodic Solutions for Some Evolution Equations in Banach

Spaces

Khalil EZZINBI^{1}^{∗}, ^{†}and Mustapha JAZAR^{2}^{‡},

1Université Cadi Ayyad Faculté des Sciences Semlalia Département de Mathématiques B.P. 2390, Marrakech, Morocco e-mail: ezzinbi@ucam.ac.ma

2Université Libanaise, Faculté des Sciences Département de Mathématiques B.P. 155-012, Beyrouth, Liban e-mail: mjazar@ul.edu.lb

Abstract

In this work we give a new criteria for the existence of periodic and almost periodic solutions for some differential equation in a Banach space. The linear part is nondensely defined and satisfies 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 differential equation whose linear part is nondensely defined. Key Words: Hille-Yosida operator, integral solution, evolution family, discrete equation, monodromy operator, partial functional differential equations. AMS (MOS)1991 Subject classification: 34C25-34C27-34C28

∗The first author is supported from TWAS under grant project 00-412 RG/MATHS/AF/AC.

†To whom all correspondence should be sent.

‡The second author is supported by a grant from the Lebanese University and the Lebanese National Council for Scientific Research.

1 Introduction

In this work, we are concerned with the existence of periodic and almost periodic solutions for the following differential equation:

\{ \begin{matrix}\frac{d}{dt}x(t) = (A + B(t)) x(t) + f(t), for t ≥ s \\ x(s) = x_{0} \end{matrix}  (1)

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}| ≤ \frac{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:

\{ \begin{matrix}\frac{d}{dt}x(t) = Ax(t) + K(t)x_{t} + h(t), for t ≥ s, \\ x_{s} = φ ∈ C = C ([−r, 0] ; E) , \end{matrix}  (3)

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

x_{t}(θ) = x(t + θ), for θ ∈ [−r, 0] .  (4)

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 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

x_{n+1} = P x_{n} + y_{n+1}, n ∈ ℤ.  (5)

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).

2 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.

Definition 2.1 A continuous function x : [s, ∞) → E is said to be an integral solution of equation (1) if

i) ∫_{s}^{t} x(τ)dτ ∈ D(A), for t ≥ s ii) x(t) = x(s) + A ∫_{s}^{t} x(τ)dτ + ∫_{s}^{t} B(τ)x(τ)dτ + ∫_{s}^{t} f(τ)dτ, for t ≥ s.

Proposition 2.1 [14] For every s ∈ ℝ and x_{0} ∈ \overline{D(A)}, equation (1) has a unique integral solution for t ≥ s.

Theorem 2.1 (Theorem 4.1.2 in [13])Let A_{0} be the part of A in \overline{D(A)} which is defined by

\{ \begin{matrix}D(A_{0}) = \{x ∈ D(A) : Ax ∈ \overline{D(A)}\} \\ A_{0}x = Ax. \end{matrix}  (6)

Then A_{0} generates a strongly continuous semigroup (T_{0}(t))_{t≥0} on \overline{D(A)}.

By [8], the integral solution x of equation (1) is given by,

x(t) = T_{0}(t − s)x_{0} + lim_{λ→∞} ∫_{s}^{t} T_{0}(t − τ)B_{λ} (B(τ)x(τ) + f(τ)) dτ.  (7)

where B_{λ} = λR (λ, A) . Define U_{B}(t, s)_{t≥s} by

U_{B}(t, s)x_{0} = x(t, s, x_{0}), for x_{0} ∈ \overline{D(A)},

where x(., ., x_{0}) is the integral solution of the equation (1) with f = 0.

Proposition 2.2 [8] Assume that (H_{1}) and (H_{2}) hold. Then (U_{B}(t, s))_{t≥s} is an evolution family:

i) U_{B}(t, t) = I, for every t ∈ ℝ, ii) U_{B}(t, s)U_{B}(s, r) = U_{B}(t, r), for t ≥ s ≥ r, iii) for all x_{0} ∈ \overline{D(A)}, (t, s) → U_{B}(t, s)x_{0} is continuous. Moreover the integral solution of equation (1) is given by

u(t) = U_{B}(t, s)u(s) + lim_{λ→∞} ∫_{s}^{t} U_{B}(t, τ)B_{λ}f(τ)dτ, t ≥ s.  (8)

Definition 2.2 (U_{B}(t, s))_{t≥s} has an exponential dichotomy on \overline{D(A)} with constant β > 1 and L ≥ 1, if there exists a bounded strongly continuous family of projection (P (t))_{t∈ℝ} on \overline{D(A)} such that for t ≥ s one has

i) P (t)U_{B}(t, s) = U_{B}(t, s)P (s). ii) the map U_{B}(t, s) : (Id − P (s))\overline{D(A)} → (Id − P (t))\overline{D(A)} is invertible. iii) |U_{B}(t, s)z| ≤ Le^{−β(t−s)} |z| , for z ∈ P (s)\overline{D(A)}. iv) |U_{B}(t, s)^{−1}z| ≤ Le^{−β(t−s)} |z| , for z ∈ (Id − P (t))\overline{D(A)}.

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:

U_{B}(t + p, s + p) = U_{B}(t, s), for t ≥ s.

Definition 2.3 The Carleman spectrum sp(u) of a function u in C_{b}(ℝ, E), is consisting of ξ ∈ ℝ such that the Fourier-Carleman transform

û(λ) = \{ \begin{matrix}∫_{0}^{∞} e^{−λt}u(t)dt, Reλ > 0 \\ − ∫_{0}^{∞} e^{λt}u(−t)dt, Reλ < 0 \end{matrix}

has no holomorphic extension to a neighborhood of iξ.

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

v_{τ} (s) = v(τ + s), for s ∈ ℝ.

Proposition 2.3 [7] Let u ∈ C_{b}(ℝ, E). Then the following statements hold true:

i) sp(u) is closed. ii) If u^{n} is a sequence in C_{b}(ℝ, E) converging to u uniformly and sp(u^{n}) ⊂ Λ, for any n ≥ 0, then sp(u) ⊂ \overline{Λ}. 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) = \overline{e^{isp(u)}}. Then a criteria for the existence of a p−periodic solution of equation (1) is obtained in [8].

Theorem 2.3 (Theorem 3.7 and Corollary 3.8 in [8]) Let f be in C_{b}(ℝ, E) such that

σ(U_{B}(p, 0) ∩ \overline{\{e^{iηp} : η ∈ sp(f)\}} = ∅.  (9)

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

Γ = {z ∈ ℂ : |z| = 1} ⊂ ρ(U_{B}(p, 0)),

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) ⊂ \frac{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).

Define l^{∞}(E) = {(α_{n})_{n∈ℤ} ⊂ E : sup_{n∈ℤ} |α_{n}| < ∞} . For any α = (α_{n})_{n∈ℤ} ∈ l^{∞}(E)

and any p ∈ ℤ, we define S(p)α by

S(p)α = (α_{n+p})_{n∈ℤ}.

Definition 2.4 [10] Let (α_{n})_{n∈ℤ} be a sequence in l^{∞}(E). Then the subset of all λ on the unit circle Γ = {z ∈ ℂ : |z| = 1} at which

α̂(λ) = \{ \begin{matrix}∑_{n=0}^{∞} λ^{n−1}S(n)α, |λ| > 1 \\ − ∑_{n=1}^{∞} λ^{n−1}S(−n)α, |λ| < 1 \end{matrix}

has non holomorphic extension, is said to be the spectrum σ(α) of the sequence (α_{n})_{n∈ℤ}.

Let α ∈ l^{∞}(E). Then α is said to be almost periodic if {(α_{n+k})_{n∈ℤ} : k ∈ ℤ} is relatively compact in l^{∞}(E).

Lemma 2.1 (Corollary 2.5 in [10]) Let α be an element of l^{∞}(E). Then α_{n} = α_{n+1} ≠ 0, for all n ∈ ℤ if and only if σ(α) = {1}. Similarly, α_{n} = −α_{n+1} ≠ 0 for all n ∈ ℤ if and only if σ(α) = {−1} .

Proposition 2.4 [10] Let q be an almost periodic function. Then

σ(q(n)_{n∈ℤ}) ⊂ σ(q).

Theorem 2.4 [10] (Theorem 3.4 and Corollary 3.6 in [10]). Assume that equation (5) has a bounded solution and the following condition holds

σ_{Γ}(P)\σ(y) is closed,

where σ_{Γ}(P) = σ(P) ∩ Γ. Then there exists a bounded solution x of equation (5) such that σ(x) = σ(y). Moreover if σ(y) is countable and E doesn’t contain a copy of c_{0}, then there is an almost periodic solution of equation (5).

3 Main results

3.1 Periodic solutions

Theorem 3.1 Assume that (H_{1}) and (H_{2}) hold. If equation (1) has a bounded integral solution on the whole line and

σ_{Γ}(U_{B}(p, 0))\ {1} is closed,  (10)

where σ_{Γ}(U_{B}(p, 0)) = σ(U_{B}(p, 0)) ∩ Γ, then equation (1) has a p−periodic integral solution.

Proof. Let u be a bounded integral solution of equation (1) on the whole line. Then

u(t) = U_{B}(t, t − p)u(t − p) + lim_{λ→∞} ∫_{t−p}^{t} U_{B}(t, τ)B_{λ}f(τ)dτ, t ∈ ℝ,

which implies that

u(t) = U_{B}(t, t − p)u(t − p) + g(t), t ∈ ℝ,

where g is defined by

g(t) = lim_{λ→∞} ∫_{t−p}^{t} U_{B}(t, τ)B_{λ}f(τ)dτ, t ∈ ℝ.

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.

Then periodic integral solutions of equation (1) correspond to constant solutions of the following discrete equation

x_{n+1} = U_{B}(p, 0)x_{n} + g_{n+1}, n ∈ ℤ.  (11)

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.

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)).

Theorem 3.2 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.

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).

3.2 Almost periodic solutions

Let g be defined by

g(t) = lim_{λ→∞} ∫_{t−1}^{t} U_{B}(t, τ)B_{λ}f(τ)dτ, t ∈ ℝ.

Theorem 3.3 Assume that (H_{1}) and (H_{2}) hold. Furthermore we assume that B is 1−periodic, f is almost periodic and

σ_{Γ}(U_{B}(1, 0))\σ(g(n)_{n∈ℤ}) is closed.  (12)

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 of

w(t) = U_{B}(t, s)w(s) + lim_{λ→∞} ∫_{s}^{t} U_{B}(t, τ)B_{λ}θ(τ)dτ, t ≥ s,  (13)

where θ is an almost periodic function with values in E. Then w is almost periodic if and only if the sequence (w(n))_{n∈ℤ} is almost periodic.

Proof of Theorem 3.3. Theorem 2.4 implies that equation (11) has an almost periodic solution (v_{n})_{n∈ℤ} . Let v be defined by

v(t) = U_{B}(t, n)v_{n} + lim_{λ→∞} ∫_{n}^{t} U_{B}(t, τ)B_{λ}f(τ)dτ, t ∈ [n, n + 1) .

Then v is well defined, continuous and by Lemma 3.1 v is an almost periodic integral solution of equation (1). In the sequel we give some sufficient conditions for which condition (12) is satisfied.

Corollary 3.1 Assume that (H_{1}) and (H_{2}) hold. Furthermore we assume that B is 1−periodic and f is almost periodic. If T_{0}(t) is compact for t > 0, σ(f) 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.

Proof. We claim that U_{B}(1, 0) is compact if T_{0}(t) is compact for t > 0. In fact, let D be a bounded set in \overline{D(A)}, by formula (7), we have for any x_{0} ∈ D,

U_{B}(1, 0)x_{0} = T_{0}(1)x_{0} + lim_{λ→∞} ∫_{0}^{1} T_{0}(1 − τ)B_{λ} (B(τ)U_{B}(τ, 0)x_{0}) dτ.

Let ε > 0 such that 1 − ε > 0. Then

lim_{λ→∞} ∫_{0}^{1} T_{0}(1 − τ)B_{λ} (B(τ)U_{B}(τ, 0)x_{0}) dτ =

T_{0}(ε) lim_{λ→∞} ∫_{0}^{1−ε} T_{0}(1 − ε − τ)B_{λ} (B(τ)U_{B}(τ, 0)x_{0}) dτ + lim_{λ→∞} ∫_{1−ε}^{1} T_{0}(1 − τ)B_{λ} (B(τ)U_{B}(τ, 0)x_{0}) dτ.

Let χ denote the measure of noncompactness of sets which is defined for any bounded set B in E by χ(B) = inf {d > 0 : B has a finite cover of diameter < d}. Then it’s well known that χ(B) = 0 if and only if B is relatively compact, χ(B_{1} + B_{2}) ≤ χ(B_{1}) + χ(B_{2}), for any bounded sets B_{1} and B_{2}. Using the above properties, we can that

χ {U_{B}(1, 0)x_{0} : x_{0} ∈ D} ≤ aε, for some a > 0.

Letting ε → 0, we get χ {U_{B}(1, 0)x_{0} : x_{0} ∈ D} = 0 and U_{B}(1, 0) is compact. Consequently condition (12) is satisfied. To end the proof, we will show that σ(g(n)_{n∈ℤ}) is countable.

Lemma 3.2 [8] Let ξ : ℝ → E be uniformly continuous such that its range is relatively compact in E. Then for any s > 0, the limit

lim_{λ→∞} ∫_{t−s}^{t} U_{B}(t, τ)B_{λ}ξ(τ)dτ

exists uniformly for t in ℝ.

Let g_{λ} be defined by

g_{λ}(t) = ∫_{t−1}^{t} U_{B}(t, τ)B_{λ}f(τ)dτ, t ∈ ℝ.

Then g_{λ} is almost periodic and by Lemma 3.2 g_{λ} converges to g uniformly in t ∈ ℝ as λ → ∞, consequently g is almost periodic. Since Lemma 4.6 in [8] implies that σ(g_{λ}) ⊂ σ(f) and σ(g) ⊂ σ(f). Moreover Lemma 2.4 gives that σ(g(n)_{n∈ℤ}) ⊂ σ(g) ⊂ σ(f) and σ(g(n)_{n∈ℤ}) is also countable, by Theorem 3.3 we deduce that equation (1) has an almost periodic integral solution.

4 Partial functional differential equation with finite delay

In this section, we apply the previous results for equation (3).

Definition 4.1 A continuous function u : [s − r, ∞) → E is said to be an integral solution of equation (3) if and only if

i) ∫_{s}^{t} u(τ)dτ ∈ D(A), for t ≥ s, ii) u(t) = u(s) + A ∫_{s}^{t} u(τ)dτ + ∫_{s}^{t} (K(τ)u_{τ} + h(τ)) dτ, for t ≥ s, iii) u_{s} = φ.

Proposition 4.1 [2] For t ≥ s and φ ∈ C such that φ(0) ∈ \overline{D(A)}, equation (3) has a unique integral solution which is defined for t ≥ s.

By [2], the integral solution of equation (3) is given by

u(t) = T_{0}(t − s)φ(0) + lim_{λ→∞} ∫_{s}^{t} T_{0}(t − τ)B_{λ} (K(τ)u_{τ} + h(τ)) dτ, for t ≥ s.

Note that the phase space C_{0} of equation (3) is given by

C_{0} = \{φ ∈ C : φ(0) ∈ \overline{D(A)}\} .

Theorem 4.1 [1] Let U(t) be defined for every t ≥ 0, on C_{0} by

(U(t)φ) (θ) = \{ \begin{matrix}T_{0}(t + θ)φ(0) if t + θ ≥ 0 \\ φ(t + θ) if t + θ ≤ 0. \end{matrix}

Then (U(t) _{t≥0} is a strongly continuous semigroup on C_{0}, its generator is given by

\{ \begin{matrix}D(A_{U}) = \{ \begin{matrix}φ ∈ C^{1} ([−r, 0] ; E) : φ(0) ∈ D(A), φ′(0) ∈ \overline{D(A)} and \\ φ′(0) = Aφ(0) \end{matrix} \} \\ A_{U}φ = φ′. \end{matrix}

Define the space ⟨X_{o}⟩ by

⟨X_{o}⟩ = {X_{o}c : c ∈ E} ,

where the function X_{o}c is defined by

(X_{o}c) (θ) = \{ \begin{matrix}0 if θ ∈ [−r, 0[ \\ c if θ = 0. \end{matrix}

The space C ⊕ ⟨X_{o}⟩ is provided with the following norm

|φ + X_{0}c| = |φ| + |c| .

Theorem 4.2 [1] The continuous extension Ã_{U} of the operator A_{U} defined on C ⊕ ⟨X_{o}⟩ by

\{ \begin{matrix}D(Ã_{U}) = \{φ ∈ C^{1} ([−r, 0] , E) : φ(0) ∈ D(A) and φ′(0) ∈ \overline{D(A)}\} \\ Ã_{U}φ = φ′ + X_{0}(Aφ(0) − φ′(0)), \end{matrix}

is a Hille-Yosida operator. If u is an integral solution of equation (3), then x(t) = u_{t} is an integral solution of

\{ \begin{matrix}\frac{d}{dt}x(t) = (Ã_{U} + B̃(t)) x(t) + h̃(t), for t ≥ s \\ x(s) = φ \end{matrix}  (14)

where B̃(t)(φ + X_{0}c) = X_{0}K(t)φ and h̃(t) = X_{0}h(t), φ ∈ C, c ∈ E and t ∈ ℝ. Conversely if x is an integral solution of equation (14), then

u(t) = \{ \begin{matrix}x(t)(0) if t ≥ s \\ φ(t) if s − r ≤ t ≤ s \end{matrix}

is an integral solution of equation (3).

Let (V (t, s))_{t≥s} be the evolution family defined on C_{0} by

V (t, s)φ = x_{t}(., s, φ), for t ≥ s,

where x(., s, φ) is the integral solution of equation (3) with h = 0. Then by Theorem 3.1, we obtain

Proposition 4.2 Suppose that equation (3) has a bounded integral solution on the whole line. If h is p−periodic and 1 is an isolated point in σ_{Γ}(V (p, 0)), then equation (3) has a p−periodic integral solution. Moreover if h is anti p−periodic and −1 is an isolated point in σ_{Γ}(V (p, 0)), then equation (3) has an anti p−periodic integral solution.

Corollary 4.1 Assume that T_{0}(t) is compact whenever t > 0 and f is pperiodic. Then the existence of a bounded integral solution of (3) on the whole line implies the existence of a p-periodic integral solution of (3).

Proof. Recall that if T_{0}(t) is compact whenever t > 0 , then the monodromy operator is compact for p > r, for more details we refer to [1] . Consequently condition (10) is satisfied. If p ≤ r, then equation has a mp-periodic integral solution, for some m such that mp > r. Let z be the mp-periodic integral solution and K = \overline{co} \{z_{np} : n ∈ ℕ\} , then K is a convex compact set. Define the Poincaré map P_{0} on K by P_{0}ϕ = x_{p}(., 0, ϕ, h), where x_{.}(., 0, ϕ, h) is the integral solution of equation (3). Then P_{0}K ⊂ K and by Schauder’s fixed point theorem, we get that P_{0} has a fixed point which gives that equation (3) has a p-periodic integral solution.

We remark that Corollary 4.1 is extendable for a partial functional differential equation with infinite delay in which linear part is nondensely defined and the phase space is a uniform fading memory space. For more details we refer to [5, Theorem 6]. Acknowledgments

The first author would like to thank the Center for Advanced Mathematical Sciences of the American University of Beirut for hospitality where part of this work has been done.

The authors would like to thank the anonymous referees for their remarks about the evaluation of the previous version of the manuscript. Their valuable suggestions helped in improving the original version.

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