Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 3, 1-12; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE OF PSEUDO ALMOST PERIODIC SOLUTIONS

TO SOME CLASSES OF PARTIAL HYPERBOLIC EVOLUTION EQUATIONS

TOKA DIAGANA

Abstract. The paper examines the existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Namely, some sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those classes of hyperbolic evolution equations are given. As an application, we consider the existence of pseudo almost periodic solutions to the heat equations with delay.

1. Introduction

Let (𝕏, ∥ · ∥) be a Banach space and let A : D(A) ⊂ 𝕏 ↦ 𝕏 be a sectorial linear operator (see Definition 2.1). For α ∈ (0, 1), the space 𝕏_{α} denotes an abstract intermediate Banach space between D(A) and 𝕏. Examples of those 𝕏_{α} include, among others, the fractional spaces D((−A)^{α}) for α ∈ (0, 1), the real interpolation spaces D_{A}(α, ∞) due to J. L. Lions and J. Peetre, and the Hölder spaces D_{A}(α), which coincide with the continuous interpolation spaces that both G. Da Prato and P. Grisvard introduced in the literature.

In [7, 11, 12, 22], some sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to the abstract (semilinear) differential equations,

u′(t) + Au(t) = f(t, u(t)), t ∈ ℝ, and (1.1)

u′(t) + Au(t) = f(t, Bu(t)), t ∈ ℝ, (1.2)

where −A is a Hille-Yosida linear operator (respectively, the infinitesimal generator of an analytic semigroup, and the infinitesimal generator of a C_{0}-semigroup), B is a densely defined closed linear operator on 𝕏, and f : ℝ×𝕏 ↦ 𝕏 is a jointly continuous function, were given. Similarly, in [13], some reasonable sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to the class of partial evolution equations

\frac{d}{dt} [u(t) + f(t, Bu(t))] = Au(t) + g(t, Cu(t)), t ∈ ℝ (1.3)

where A is the infinitesimal generator of an exponentially stable semigroup acting on 𝕏, B, C are arbitrary densely defined closed linear operators on 𝕏, and f, g are some jointly continuous functions, were given.

The assumptions made in [13] require much more regularity for the operator A, that is, being the infinitesimal generator of an analytic semigroup. In this paper

EJQTDE, 2007 No. 3, p. 1 we address such an issue by studying pseudo almost periodic solutions to (1.3) in the case when A is a sectorial operator whose corresponding analytic semigroup (T (t))_{t≥0} is hyperbolic, equivalently,

2000 Mathematics Subject Classification. 44A35; 42A85; 42A75.

Key words and phrases. Sectorial operator; analytic semigroup; hyperbolic semigroup; almost periodic; pseudo almost periodic; hyperbolic evolution equation; heat equation with delay.

σ(A) ∩ iℝ = ∅,

where σ(A) denotes the spectrum of A.

Note that (1.3) in the case when A is sectorial corresponds to several interesting situations encountered in the literature. Applications include, among others, the existence and uniqueness of pseudo almost periodic solutions to the hyperbolic heat equation with delay.

As in [5, 14] in this paper we consider a general intermediate space 𝕏_{α} between D(A) and 𝕏. In contrast with the fractional power spaces considered in some recent papers of the author et al. [11, 12], the interpolation and Hölder spaces, for instance, depend only on D(A) and 𝕏 and can be explicitly expressed in many concrete cases. The literature related to those intermediate spaces is very extensive, in particular, we refer the reader to the excellent book by A. Lunardi [23], which contains a comprehensive presentation on this topic and related issues.

The concept of pseudo almost periodicity, which is the central question in this paper was introduced in the literature in the early nineties by C. Zhang [29, 30, 31] as a natural generalization of the well-known Bohr almost periodicity. Thus this new concept is welcome to implement another existing generalization of almost periodicity, that is, the concept of asymptotically almost periodicity due to Fréchet [6, 16].

The existence of almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions is one of the most attractive topics in qualitative theory of differential equations due to their significance and applications in physics, mathematical biology, control theory, physics and others.

Some contributions on almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions to abstract differential and partial differential equations have recently been made in [1, 2, 3, 7, 9, 11, 12, 13, 22]. However, the existence and uniqueness of pseudo almost periodic solutions to (1.3) in the case when A is sectorial is an important topic with some interesting applications, which is still an untreated question, is the main motivation of the present paper. Among other things, we will make extensive use of the method of analytic semigroups associated with sectorial operators and the Banach’s fixed-point principle to derive sufficient conditions for the existence and uniqueness of a pseudo almost periodic (mild) solution to (1.3).

2. Preliminaries

This section is devoted to some preliminary facts needed in the sequel. Throughout the rest of this paper, (𝕏, ∥·∥) stands for a Banach space, A is a sectorial linear operator (see Definition 2.1), which is not necessarily densely defined, and B, C are (possibly unbounded) linear operators such that A + B + C is not trivial, as each solution to (1.3) belongs to D(A + B + C) = D(A) ∩ D(B) ∩ D(C). Now if A is a linear operator on 𝕏, then ρ(A), σ(A), D(A), N(A), R(A) stand for the resolvent, spectrum, domain, kernel, and range of A. The space B(𝕏, 𝕐) denotes the Banach

EJQTDE, 2007 No. 3, p. 2 space of all bounded linear operators from 𝕏 into 𝕐 equipped with its natural norm with B(𝕏, 𝕏) = B(𝕏).

2.1. Sectorial Linear Operators and their Associated Semigroups.

Definition 2.1. A linear operator A : D(A) ⊂ 𝕏 ↦ 𝕏 (not necessarily densely defined) is said to be sectorial if the following hold: there exist constants ω ∈ ℝ, θ ∈ (\frac{π}{2} , π), and M > 0 such that

ρ(A) ⊃ S_{θ,ω} := {λ ∈ ℂ : λ ≠ ω, | arg(λ − ω)| < θ}, and (2.1)

∥R(λ, A)∥ ≤ \frac{M}{|λ − ω|}, λ ∈ S_{θ,ω}. (2.2)

The class of sectorial operators is very rich and contains most of classical operators encountered in the literature. Two examples of sectorial operators are given as follows:

Example 2.2. Let p ≥ 1 and let 𝕏 = L^{p}(ℝ) be the Lebesgue space equipped with its norm ∥ · ∥_{p} defined by

∥φ∥_{p} = (∫_{ℝ} |φ(x)|^{p}dx)^{1/p} .

Define the linear operator A on L^{p}(ℝ) by

D(A) = W^{2,p}(ℝ), A(φ) = φ′′, ∀φ ∈ D(A).

It can be checked that the operator A is sectorial on L^{p}(ℝ).

Example 2.3. Let p ≥ 1 and let Ω ⊂ ℝ^{d} be open bounded subset with C^{2} boundary ∂Ω. Let 𝕏 := L^{p}(Ω) be the Lebesgue space equipped with the norm, ∥ · ∥_{p} defined by,

∥φ∥_{p} = (∫_{Ω} |φ(x)|^{p}dx)^{1/p} .

Define the operator A as follows:

D(A) = W^{2,p}(Ω) ∩ W_{0}^{1,p}(Ω), A(φ) = Δφ, ∀φ ∈ D(A),

where Δ = ∑_{k=1}^{d} \frac{∂^{2}}{∂x_{k}^{2}} is the Laplace operator.

It can be checked that the operator A is sectorial on L^{p}(Ω).

It is well-known that [23] if A is sectorial, then it generates an analytic semigroup (T (t))_{t≥0}, which maps (0, ∞) into B(𝕏) and such that there exist M_{0}, M_{1} > 0 with

∥T (t)∥ ≤ M_{0}e^{ωt}, t > 0, (2.3)

∥t(A − ω)T (t)∥ ≤ M_{1}e^{ωt}, t > 0. (2.4)

Throughout the rest of the paper, we suppose that the semigroup (T (t))_{t≥0} is hyperbolic, that is, there exist a projection P and constants M, δ > 0 such that T (t)

EJQTDE, 2007 No. 3, p. 3 commutes with P, N(P) is invariant with respect to T (t), T (t) : R(Q) ↦ R(Q) is invertible, and the following hold

(2.5)  ∥T (t)P x∥ ≤ Me^{−δt}∥x∥  for t ≥ 0,
(2.6)  ∥T (t)Qx∥ ≤ Me^{δt}∥x∥  for t ≤ 0,

where Q := I − P and, for t ≤ 0, T (t) := (T (−t))^{−1}.

Recall that the analytic semigroup (T (t))_{t≥0} associated with A is hyperbolic if and only if

σ(A) ∩ iℝ = ∅,

see, e.g., [15, Prop. 1.15, pp.305].

Definition 2.4. Let α ∈ (0, 1). A Banach space (𝕏_{α}, ∥ · ∥_{α}) is said to be an intermediate space between D(A) and 𝕏, or a space of class J_{α}, if D(A) ⊂ 𝕏_{α} ⊂ 𝕏 and there is a constant c > 0 such that

(2.7)  ∥x∥_{α} ≤ c∥x∥^{1−α}∥x∥_{A}^{α},  x ∈ D(A),

where ∥ · ∥_{A} is the graph norm of A.

Concrete examples of 𝕏_{α} include D((−A^{α})) for α ∈ (0, 1), the domains of the fractional powers of A, the real interpolation spaces D_{A}(α, ∞), α ∈ (0, 1), defined as follows

\{ \begin{matrix}D_{A}(α, ∞) := \{x ∈ 𝕏 : [x]_{α} = sup_{0<t≤1} ∥t^{1−α}AT (t)x∥ < ∞\} \\ ∥x∥_{α} = ∥x∥ + [x]_{α}, \end{matrix}

the abstract Hölder spaces D_{A}(α) := \overline{D(A)}^{∥.∥_{α}} as well as the complex interpolation spaces [𝕏, D(A)]_{α}, see A. Lunardi [23] for details.

For a hyperbolic analytic semigroup (T (t))_{t≥0}, one can easily check that similar estimations as both (2.5) and (2.6) still hold with norms ∥ · ∥_{α}. In fact, as the part of A in R(Q) is bounded, it follows from (2.6) that

∥AT (t)Qx∥ ≤ C′e^{δt}∥x∥ for t ≤ 0.

Hence, from (2.7) there exists a constant c(α) > 0 such that

(2.8)  ∥T (t)Qx∥_{α} ≤ c(α)e^{δt}∥x∥  for t ≤ 0.

In addition to the above, the following holds

∥T (t)P x∥_{α} ≤ ∥T (1)∥_{B(𝕏,𝕏_{α})}∥T (t − 1)P x∥ for t ≥ 1,

and hence from (2.5), one obtains

∥T (t)P x∥_{α} ≤ M′e^{−δt}∥x∥, t ≥ 1,

where M′ depends on α. For t ∈ (0, 1], by (2.4) and (2.7)

∥T (t)P x∥_{α} ≤ M′′t^{−α}∥x∥.

Hence, there exist constants M(α) > 0 and γ > 0 such that

(2.9)  ∥T (t)P x∥_{α} ≤ M(α)t^{−α}e^{−γt}∥x∥  for t > 0.

EJQTDE, 2007 No. 3, p. 4 2.2. Pseudo Almost Periodic Functions. Let (𝕐, ∥ · ∥_{𝕐}) be another Banach space. Let BC(ℝ, 𝕏) (respectively, BC(ℝ × 𝕐, 𝕏)) denote the collection of all 𝕏valued bounded continuous functions (respectively, the class of jointly bounded continuous functions F : ℝ × 𝕐 ↦ 𝕏). The space BC(ℝ, 𝕏) equipped with its natural norm, that is, the sup norm defined by

∥u∥_{∞} = sup_{t∈ℝ} ∥u(t)∥

is a Banach space. Furthermore, C(ℝ, 𝕐) (respectively, C(ℝ × 𝕐, 𝕏)) denotes the class of continuous functions from ℝ into 𝕐 (respectively, the class of jointly continuous functions F : ℝ × 𝕐 ↦ 𝕏).

Definition 2.5. A function f ∈ C(ℝ, 𝕏) is called (Bohr) almost periodic if for each ε > 0 there exists l(ε) > 0 such that every interval of length l(ε) contains a number τ with the property that

∥f(t + τ) − f(t)∥ < ε for each t ∈ ℝ.

The number τ above is called an ε-translation number of f, and the collection of all such functions will be denoted AP (𝕏).

Definition 2.6. A function F ∈ C(ℝ × 𝕐, 𝕏) is called (Bohr) almost periodic in t ∈ ℝ uniformly in y ∈ 𝕐 if for each ε > 0 and any compact K ⊂ 𝕐 there exists l(ε) such that every interval of length l(ε) contains a number τ with the property that

∥F (t + τ, y) − F (t, y)∥ < ε for each t ∈ ℝ, y ∈ K.

The collection of those functions is denoted by AP (ℝ × 𝕐).

Set

AP_{0}(𝕏) := \{f ∈ BC(ℝ, 𝕏) : lim_{r→∞} \frac{1}{2r} ∫_{−r}^{r} ∥f(s)∥ds = 0\},

and define AP_{0}(ℝ × 𝕏) as the collection of functions F ∈ BC(ℝ × 𝕐, 𝕏) such that

lim_{r→∞} \frac{1}{2r} ∫_{−r}^{r} ∥F (t, u)∥dt = 0

uniformly in u ∈ 𝕐.

Definition 2.7. A function f ∈ BC(ℝ, 𝕏) is called pseudo almost periodic if it can be expressed as f = g + ϕ, where g ∈ AP (𝕏) and ϕ ∈ AP_{0}(𝕏). The collection of such functions will be denoted by P AP (𝕏).

Remark 2.8. The functions g and ϕ in Definition 2.7 are respectively called the almost periodic and the ergodic perturbation components of f. Moreover, the decomposition given in Definition 2.7 is unique.

Similarly,

Definition 2.9. A function F ∈ C(ℝ × 𝕐, 𝕏) is said to pseudo almost periodic in t ∈ ℝ uniformly in y ∈ 𝕐 if it can be expressed as F = G + Φ, where G ∈ AP (ℝ × 𝕐) and ϕ ∈ AP_{0}(ℝ × 𝕐). The collection of such functions will be denoted by P AP (ℝ × 𝕐).

EJQTDE, 2007 No. 3, p. 5

3. Main results

To study the existence and uniqueness of pseudo almost periodic solutions to (1.3) we need to introduce the notion of mild solution to it.

Definition 3.1. Let α ∈ (0, 1). A bounded continuous function u : ℝ ↦ 𝕏_{α} is said to be a mild solution to (1.3) provided that the function s → AT (t−s)P f(s, Bu(s)) is integrable on (−∞, t), s → AT (t−s)Qf(s, Bu(s)) is integrable on (t, ∞) for each t ∈ ℝ, and

u(t) = −f(t, Bu(t)) − ∫_{−∞}^{t} AT (t − s)P f(s, Bu(s))ds
+ ∫_{t}^{∞} AT (t − s)Qf(s, Bu(s))ds + ∫_{−∞}^{t} T (t − s)P g(s, Cu(s))ds
− ∫_{t}^{∞} T (t − s)Qg(s, Cu(s))ds

for each ∀t ∈ ℝ.

Throughout the rest of the paper we denote by Γ_{1}, Γ_{2}, Γ_{3}, and Γ_{4}, the nonlinear integral operators defined by

(Γ_{1}u)(t) := ∫_{−∞}^{t} AT (t−s)P f(s, Bu(s))ds, (Γ_{2}u)(t) := ∫_{t}^{∞} AT (t−s)Qf(s, Bu(s))ds,

(Γ_{3}u)(t) := ∫_{−∞}^{t} T (t − s)P g(s, Cu(s))ds, and

(Γ_{4}u)(t) := ∫_{t}^{∞} T (t − s)Qg(s, Cu(s))ds.

To study (1.3) we require the following assumptions: (H1) The operator A is sectorial and generates a hyperbolic (analytic) semigroup

(T (t))_{t≥0}. (H2) Let 0 < α < 1. Then 𝕏_{α} = D((−A^{α})), or 𝕏_{α} = D_{A}(α, p), 1 ≤ p ≤ +∞, or

𝕏_{α} = D_{A}(α), or 𝕏_{α} = [𝕏, D(A)]_{α}. We also assume that B, C : 𝕏_{α} ⟶ 𝕏 are bounded linear operators.

(H3) Let 0 < α < β < 1, and f : ℝ × 𝕏 ⟶ 𝕏_{β} be a pseudo almost periodic function in t ∈ ℝ uniformly in u ∈ 𝕏, g : ℝ × 𝕏 ↦ 𝕏 be pseudo almost periodic in t ∈ ℝ uniformly in u ∈ 𝕏.

(H4) The functions f, g are uniformly Lipschitz with respect to the second argument in the following sense: there exists K > 0 such that

∥f(t, u) − f(t, v)∥_{β} ≤ K∥u − v∥,

and

∥g(t, u) − g(t, v)∥ ≤ K∥u − v∥

for all u, v ∈ 𝕏 and t ∈ ℝ. In order to show that Γ_{1} and Γ_{2} are well defined, we need the following estimates.

EJQTDE, 2007 No. 3, p. 6 Lemma 3.2. Let 0 < α, β < 1. Then

(3.1)  ∥AT (t)Qx∥_{α} ≤ ce^{δt}∥x∥_{β}  for t ≤ 0,
(3.2)  ∥AT (t)P x∥_{α} ≤ ct^{β−α−1}e^{−γt}∥x∥_{β},  for t > 0.

Proof. As for (2.8), the fact that the part of A in R(Q) is bounded yields

∥AT (t)Qx∥ ≤ ce^{δt}∥x∥_{β}, ∥A^{2}T (t)Qx∥ ≤ ce^{δt}∥x∥_{β}, for t ≤ 0,

since X_{β} ↪ X. Hence, from (2.7) there is a constant c(α) > 0 such that

∥AT (t)Qx∥_{α} ≤ c(α)e^{δt}∥x∥_{β} for t ≤ 0.

Furthermore,

∥AT (t)P x∥_{α} ≤ ∥AT (1)∥_{B(𝕏,𝕏_{α})}∥T (t − 1)P x∥
≤ ce^{−δt}∥x∥_{β}, for t ≥ 1.

Now for t ∈ (0, 1], by (2.4) and (2.7), one has

∥AT (t)P x∥_{α} ≤ ct^{−α−1}∥x∥,

and

∥AT (t)P x∥_{α} ≤ ct^{−α}∥Ax∥,

for each x ∈ D(A). Thus, by reiteration Theorem (see [23]), it follows that

∥AT (t)P x∥_{α} ≤ ct^{β−α−1}∥x∥_{β}

for every x ∈ 𝕏_{β} and 0 < β < 1, and hence, there exist constants M(α) > 0 and γ > 0 such that

∥T (t)P x∥_{α} ≤ M(α)t^{β−α−1}e^{−γt}∥x∥_{β} for t > 0.
□

Lemma 3.3. Under assumptions (H1)-(H2)-(H3)-(H4), the integral operators Γ_{3} and Γ_{4} defined above map P AP (𝕏_{α}) into itself.

Proof. Let u ∈ P AP (𝕏_{α}). Since C ∈ B(𝕏_{α}, 𝕏) it follows that Cu ∈ P AP (𝕏). Setting h(t) = g(t, Cu(t)) and using the theorem of composition of pseudo almost periodic functions [3, Theorem 5] it follows that h ∈ P AP (𝕏). Now, write h = ϕ+ζ where ϕ ∈ AP (𝕏) and ζ ∈ AP_{0}(𝕏). Thus Γ_{3}u can be rewritten as

(Γ_{3}u)(t) = ∫_{−∞}^{t} T (t − s)P ϕ(s)ds + ∫_{−∞}^{t} T (t − s)P ζ(s)ds.

Set

Φ(t) = ∫_{−∞}^{t} T (t − s)P ϕ(s)ds,

and

Ψ(t) = ∫_{−∞}^{t} T (t − s)P ζ(s)ds

for each t ∈ ℝ. The next step consists of showing that Φ ∈ AP (𝕏_{α}) and Ψ ∈ AP_{0}(𝕏_{α}).

EJQTDE, 2007 No. 3, p. 7

Clearly, Φ ∈ AP (𝕏_{α}). Indeed, since ϕ ∈ AP (𝕏), for every ε > 0 there exists l(ε) > 0 such that for all ξ there is τ ∈ [ξ, ξ + l(ε)] with

∥Φ(t + τ) − Φ(t)∥ < μ . ε for each t ∈ ℝ,

where μ = \frac{γ^{1−α}}{M(α)Γ(1 − α)} with Γ being the classical gamma function.

Now using the expression

Φ(t + τ) − Φ(t) = ∫_{−∞}^{t} T (t − s)P (ϕ(s + τ) − ϕ(s)) ds

and (2.9) it easily follows that

∥Φ(t + τ) − Φ(t)∥_{α} < ε for each t ∈ ℝ,

and hence, Φ ∈ AP (𝕏_{α}). To complete the proof for Γ_{3}, we have to show that t ↦ Ψ(t) is in AP_{0}(𝕏_{α}). First, note that s ↦ Ψ(s) is a bounded continuous function. It remains to show that

lim_{r→∞} \frac{1}{2r} ∫_{−r}^{r} ∥Ψ(t)∥_{α} dt = 0.

Again using (2.9) one obtains that

lim_{r→∞} \frac{1}{2r} ∫_{−r}^{r} ∥Ψ(t)∥_{α} dt ≤ lim_{r→∞} \frac{M(α)}{2r} ∫_{−r}^{r} ∫_{0}^{+∞} s^{−α}e^{−γs} ∥ζ(t − s)∥ds dt

≤ lim_{r→∞} M(α) ∫_{0}^{+∞} s^{−α}e^{−γs} \frac{1}{2r} ∫_{−r}^{r} ∥ζ(t − s)∥dt ds = 0,

by using Lebesgue dominated Convergence theorem, and the fact that AP_{0}(𝕏) is invariant under translations. Thus Ψ belongs to AP_{0}(𝕏_{α}).

The proof for Γ_{4}u(·) is similar to that of Γ_{3}u(·). However one makes use of (2.8) rather than (2.9). □

Lemma 3.4. Under assumptions (H1)-(H2)-(H3)-(H4), the integral operators Γ_{1} and Γ_{2} defined above map P AP (𝕏_{α}) into itself.

Proof. Let u ∈ P AP (𝕏_{α}). Since B ∈ B(𝕏_{α}, 𝕏) it follows that the function t ↦ Bu(t) belongs to P AP (𝕏). Again, using the composition theorem of pseudo almost periodic functions [3, Theorem 5] it follows that ψ(·) = f(·, Bu(·)) is in P AP (𝕏_{β}) whenever u ∈ P AP (𝕏_{α}). In particular, ∥ψ∥_{∞,β} = sup_{t∈ℝ} ∥f(t, Bu(t))∥_{β} < ∞.

Now write ψ = w + z, where w ∈ AP (𝕏_{β}) and z ∈ AP_{0}(𝕏_{β}), that is, Γ_{1}ϕ = Ξ(w) + Ξ(z) where

Ξw(t) := ∫_{−∞}^{t} AT (t − s)P w(s)ds, and Ξz(t) := ∫_{−∞}^{t} AT (t − s)P z(s)ds.

Clearly, Ξ(w) ∈ AP (𝕏_{α}). Indeed, since w ∈ AP (𝕏_{β}), for every ε > 0 there exists l(ε) > 0 such that for all ξ there is τ ∈ [ξ, ξ + l(ε)] with the property:

∥w(t + τ) − w(t)∥_{β} < νε for each t ∈ ℝ,

EJQTDE, 2007 No. 3, p. 8 where ν = \frac{1}{cγ^{β−α}Γ(β − α)} and c being the constant appearing in Lemma 3.2.

Now, the estimate (3.2) yields

∥Ξ(w)(t + τ) − Ξ(w)(t)∥_{α} = ∥∫_{0}^{+∞} AT (s)P (w(t − s + τ) − w(t − s)) ds∥_{α}

≤ ∫_{0}^{+∞} s^{β−α−1}e^{−γs}∥w(t − s + τ) − w(t − s)∥_{β}ds

≤ ε

for each t ∈ ℝ, and hence Ξ(w) ∈ AP (𝕏_{α}).

Now, let r > 0. Then, by (3.2), we have

\frac{1}{2r} ∫_{−r}^{r} ∥(Ξz)(t)∥_{𝕏_{α}} dt ≤ \frac{1}{2r} ∫_{−r}^{r} ∫_{0}^{+∞} ∥AT (s)P z(t − s)∥_{α} ds dt

≤ \frac{1}{2r} ∫_{−r}^{r} ∫_{0}^{+∞} s^{β−α−1}e^{−γs}∥z(t − s)∥_{β} ds dt
≤ ∫_{0}^{+∞} s^{β−α−1}e^{−γs} \frac{1}{2r} ∫_{−r}^{r} ∥z(t − s)∥_{β} dt ds.

Obviously, lim_{r→∞} \frac{1}{2r} ∫_{−r}^{r} ∥(Ξz)(t)∥_{α}dt = 0, since t ↦ z(t − s) ∈ AP_{0}(𝕏_{β}) for every

s ∈ ℝ. Thus Ξz ∈ AP_{0}(𝕏_{α}).

The proof for Γ_{2}u(·) is similar to that of Γ_{1}u(·). However, one uses (3.1) instead of (3.2). □

Throughout the rest of the paper, the constant k(α) denotes the bound of the embedding 𝕏_{β} ↪ 𝕏_{α}, that is,

∥u∥_{α} ≤ k(α)∥u∥_{β} for each u ∈ 𝕏_{β}.

Theorem 3.5. Under the assumptions (H1)-(H2)-(H3)-(H4), the evolution equation (1.3) has a unique pseudo almost periodic mild solution whenever Θ < 1, where

Θ = Kϖ [k(α) + \frac{c}{δ} + c\frac{Γ(β − α)}{γ^{β−α}} + \frac{M(α)Γ(1 − α)}{γ^{1−α}} + \frac{c(α)}{δ}] ,

and ϖ = max(∥B∥_{B(𝕏_{α},𝕏)}, ∥C∥_{B(𝕏_{α},𝕏)}).

Proof. Consider the nonlinear operator 𝕄 on P AP (𝕏_{α}) given by

𝕄u(t) = −f(t, Bu(t)) − ∫_{−∞}^{t} AT (t − s)P f(s, Bu(s))ds
+ ∫_{t}^{∞} AT (t − s)Qf(s, Bu(s))ds + ∫_{−∞}^{t} T (t − s)P g(s, Cu(s))ds
− ∫_{t}^{∞} T (t − s)Qg(s, Cu(s))ds

for each t ∈ ℝ.

EJQTDE, 2007 No. 3, p. 9

As we have previously seen, for every u ∈ P AP (𝕏_{α}), f(·, Bu(·)) ∈ P AP (𝕏_{β}) ⊂ P AP (𝕏_{α}). In view of Lemma 3.3 and Lemma 3.4, it follows that 𝕄 maps P AP (𝕏_{α}) into itself. To complete the proof one has to show that 𝕄 has a unique fixed-point.

Let v, w ∈ P AP (𝕏_{α})

∥Γ_{1}(v)(t) − Γ_{1}(w)(t)∥_{α} ≤ ∫_{−∞}^{t} ∥AT (t − s)P [f(s, Bv(s)) − f(s, Bw(s))]∥_{α} ds
≤ cK∥B∥_{B(𝕏_{α},𝕏)}∥v − w∥_{∞,α} ∫_{−∞}^{t}(t − s)^{β−α−1}e^{−γ(t−s)}ds
= c\frac{Γ(β − α)}{γ^{β−α}}K∥B∥_{B(𝕏_{α},𝕏)}∥v − w∥_{∞,α}.

Similarly,

∥Γ_{2}(v)(t) − Γ_{2}(w)(t)∥_{α} ≤ ∫_{t}^{∞} ∥AT (t − s)Q [f(s, Bv(s)) − f(s, Bw(s))] ∥_{α}ds
≤ cK∥B∥_{B(𝕏_{α},𝕏)}∥v − w∥_{∞,α} ∫_{t}^{+∞} e^{δ(t−s)}ds
= \frac{cK∥B∥_{B(𝕏_{α},𝕏)}}{δ} ∥v − w∥_{∞,α}.

Now for Γ_{3} and Γ_{4}, we have the following approximations

∥Γ_{3}(v)(t) − Γ_{3}(w)(t)∥_{α} ≤ ∫_{−∞}^{t} ∥T (t − s)P [g(s, Cv(s)) − g(s, Cw(s))] ∥_{α}ds
≤ \frac{K∥C∥_{B(𝕏_{α},𝕏)}M(α)Γ(1 − α)}{γ^{1−α}}∥v − w∥_{∞,α},

and

∥Γ_{4}(v)(t) − Γ_{4}(w)(t)∥_{α} ≤ ∫_{t}^{+∞} ∥T (t − s)Q [g(s, Cv(s)) − g(s, Cw(s))] ∥_{α}ds
≤ Kc(α)∥C∥_{B(𝕏_{α},𝕏)}∥v − w∥_{∞,α} ∫_{t}^{+∞} e^{δ(t−s)}ds
= \frac{K∥C∥_{B(𝕏_{α},𝕏)}c(α)}{δ}∥v − w∥_{∞,α}.

Consequently,

∥𝕄v − 𝕄w∥_{∞,α} ≤ Θ . ∥v − w∥_{∞,α}.

Clearly, if Θ < 1, then (1.3) has a unique fixed-point by the Banach fixed point theorem, which obviously is the only pseudo almost periodic solution to (1.3). □

Example 3.6. For σ ∈ ℝ, consider the (semilinear) heat equation with delay endowed with Dirichlet conditions:

EJQTDE, 2007 No. 3, p. 10

(3.3) \frac{∂}{∂t}[φ(t, x) + f(t, φ(t − p, x))] = \frac{∂^{2}}{∂x^{2}} φ(t, x) + σφ(t, x) + g(t, φ(t − p, x))
(3.4)  φ(t, 0) = φ(t, 1) = 0

for t ∈ ℝ and x ∈ [0, 1], where p > 0, and f, g : ℝ × C[0, 1] ↦ C[0, 1] are some jointly continuous functions.

Take 𝕏 := C[0, 1], equipped with the sup norm. Define the operator A by

A(φ) := φ′′ + σφ, ∀φ ∈ D(A),

where D(A) := {φ ∈ C^{2}[0, 1], φ(0) = φ(1) = 0} ⊂ C[0, 1].

Clearly A is sectorial, and hence is the generator of an analytic semigroup. In addition to the above, the resolvent and spectrum of A are respectively given by

ρ(A) = ℂ − {−n^{2}π^{2} + σ : n ∈ ℕ} and σ(A) = {−n^{2}π^{2} + σ : n ∈ ℕ}

so that σ(A) ∩ iℝ = {∅} whenever σ ≠ n^{2}π^{2}.

Theorem 3.7. Under assumptions (H3)-(H4), if σ ≠ n^{2}π^{2} for each n ∈ ℕ, then the heat equation with delay (3.3)-(3.4) has a unique 𝕏_{α}-valued pseudo almost periodic mild solution whenever K is small enough.

Acknowledgement. The author wants to express many thanks to Professor Maniar for useful comments and suggestions on the manuscript.

References

1. E. Ait Dads, K. Ezzinbi, and O. Arino, Pseudo Almost Periodic Solutions for Some Differential Equations in a Banach Space, Nonlinear Anal 28 (1997), no. 7, pp. 1141–1155.

2. E. Ait Dads and O. Arino, Exponential Dichotomy and Existence of Pseudo Almost Periodic Solutions of Some Differential Equations, Nonlinear Anal 27 (1996), no. 4, pp. 369–386.

3. B. Amir and L. Maniar, Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain. Ann. Math. Blaise Pascal 6 (1999), no. 1, pp. 1–11.

4. M. Bahaj and O. Sidki, Almost Periodic Solutions of Semilinear Equations with Analytic Semigroups in Banach Spaces, Electron. J. Differential Equations 2002 (2002), no. 98, pp. 1–11.

5. S. Boulite, L. Maniar, and G. M. N’Guérekata, Almost Periodc Solutions for Hyperbolic Semilniear Evolution Equations, Semigroup Forum. Vol. 71 (2005), 231-240.

6. C. Corduneanu, Almost Periodic Functions, 2nd Edition, Chelsea-New York, 1989.

7. C. Cuevas and M. Pinto, Existence and Uniqueness of Pseudo Almost Periodic Solutions of Semilinear Cauchy Problems with Non-dense Domain, Nonlinear Anal 45(2001), no. 1, Ser. A: Theory Methods, pp. 73–83.

8. W. Desch, R. Grimmer, Ronald, and W. Schappacher, Well-Posedness and Wave Propagation for a Class of Integro-differential Equations in Banach Space. J. Differential Equations 74 (1988), no. 2, pp. 391–411.

9. T. Diagana, Pseudo Almost Periodic Solutions to Some Differential Equations, Nonlinear Anal 60 (2005), no. 7, pp. 1277–1286.

10. T. Diagana and E. Hernàndez M., Existence and Uniqueness of Pseudo Almost Periodic Solutions to Some Abstract Partial Neutral Functional-Differential Equations and Applications. To Appear in J. Math. Anal Appl. 327(2007), no. 2, 776-791.

11. T. Diagana, C. M. Mahop, and G. M. N’Guérékata, Pseudo Almost Periodic Solution to Some

Semilinear Differential Equations, Math. Comp. Modelling 43 (2006), no. 1-2, pp. 89-96.

EJQTDE, 2007 No. 3, p. 11

12. T. Diagana, C. M. Mahop, G. M. N’Guérékata, and B. Toni, Existence and Uniqueness of Pseudo Almost Periodic Solutions to Some Classes of Semilinear Differential Equations and Applications. Nonlinear Anal. 64 (2006), no. 11, pp. 2442-2453.

13. T. Diagana, Existence and Uniqueness of Pseudo Almost Periodic Solutions to Some Classes of Partial Evolution Equations. Nonlinear Anal. 66 (2007), no. 2, 384-395.

14. T. Diagana and G. M. N’Guérékata, Pseudo Almost Periodic Mild Solutions To Hyperbolic Evolution Equationa in Abstract Intermediate Banach Spaces. Appl Anal. 85(2006), Nos. 6-7, pp. 769-780.

15. K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate texts in Mathematics, Springer Verlag 1999.

16. A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics 377, Springer-Verlag, New York-Berlin, 1974.

17. J. A. Goldstein, Convexity, boundedness, and almost periodicity for differential equations in Hilbert space. Internat. J. Math. Math. Sci 2 (1979), no. 1, pp. 1–13.

18. E. Hernández and H. R. Henríquez, Existence of Periodic Solutions of Partial neutral Functional Differential Equations with Unbounded Delay. J. Math. Anal. Appl 221 (1998), no. 2, pp. 499–522.

19. E. Hernández, Existence Results for Partial Neutral Integro-differential Equations with Unbounded Delay. J. Math. Anal. Appl 292 (2004), no. 1, pp. 194–210.

20. E. Hernandez M., M. L. Pelicer, and J. P. C. dos Santos , Asymptotically almost periodic and almost periodic solutions for a class of evolution equations, Electron. J. Diff. Eqns 2004(2004), no. 61, pp. 1–15.

21. Y. Hino, T. Naito, N. V. Minh, and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces. Stability and Control: Theory, Methods and Applications, 15. Taylor and Francis, London, 2002.

22. H. X. Li, F. L. Huang, and J. Y. Li, Composition of pseudo almost-periodic functions and semilinear differential equations. J. Math. Anal. Appl 255 (2001), no. 2, pp. 436–446.

23. A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, PNLDE Vol. 16, Birkhäauser Verlag, Basel, 1995.

24. G. M. N’Guérékata, Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic / Plenum Publishers, New York-London-Moscow, 2001.

25. G. M. N’Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.

26. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.

27. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Texts in Appl. Math. 13, Springer-Verlag, New York-Berlin- Heidelberg-London-Paris, 1992.

28. S. Zaidman, Topics in Abstract Differential Equations, Pitman Research Notes in Mathematics Ser. II John Wiley and Sons, New York, 1994-1995.

29. C. Y. Zhang, Pseudo Almost Periodic Solutions of Some Differential Equations. J. Math. Anal. Appl 181 (1994), no. 1, pp. 62–76.

30. C. Y. Zhang, Pseudo Almost Periodic Solutions of Some Differential Equations. II. J. Math. Anal. Appl 192 (1995), no. 2, pp. 543–561.

31. C. Y. Zhang, Integration of Vector-Valued Pseudo Almost Periodic Functions, Proc. Amer. Math. Soc 121 (1994), no. 1, pp. 167–174.

(Received August 8, 2006)

Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, DC 20059, USA

E-mail address: tdiagana@howard.edu

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