ALMOST PERIODIC MILD SOLUTIONS OF A CLASS OF PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form (∗) x′ = f(t, x, xt) . To this end, we first associate with every almost periodic semilinear equation (∗∗) x′ = F (t, x) a nonlinear semigroup in the space of almost periodic functions. We then give sufficient conditions (in terms of the accretiveness of the generator of this semigroup) for the existence of almost periodic mild solutions of (∗∗) as fixed points of the semigroup. Those results are then carried over to equation (∗). The main results are stated under accretiveness conditions of the function f in terms of x and Lipschitz conditions with respect to xt.


Introduction
In this paper we are mainly concerned with the existence of almost periodic semilinear evolution equations of the form where A is the infinitesimal generator of a C 0 -semigroup T (t), t ≥ 0 and f is an everywhere defined continuous operator from R × X × C to X.
Recently, increasing interest in semilinear evolution equations can be observed (see e.g.[12], [18][19], [25][26][27], [32] and the references therein).This interest arises from a need to extend well-known results on ordinary differential equations to a class of partial differential equations.In this context we consider the existence of almost periodic mild solutions of equation (1).At this point we want to emphasize that our approach to the problem is somewhat different from that used in the papers [13][14][15][16][17], [28], [31].In fact, we first associate with an equation without delay (equation ( 2) below) a semigroup of nonlinear operators which then plays a role similar to that of monodromy operator for equations with periodic coefficients.Then we use the results thus obtained in order to study equation (1).Our method, in the case A = 0, requires a condition on f which is somewhat stronger than that used in the previous papers in order to guarantee the existence of the associated semigroups.However, our method in turn allows to impose accretiveness conditions on the function f in a more general context which seems to be more suitable for equations whose right-hand side f depends explicitly on t.

Almost Periodic Solutions of Differential Equations without Delay
In this section we deal with almost periodic mild solutions to the semilinear evolution equation where X is a Banach space, A is the infinitesimal generator of a C 0 -semigroup S(t), t ≥ 0 of linear operators of type ω, i.e.

S(t)x − S(t)y
and B is an everywhere defined continuous operator from R × X to X. Hereafter, by a mild solution x(t), t ∈ [s, τ ] of equation ( 2) we mean a continuous solution of the integral equation A strong solution (for the definition see [34], e.g.) of equation ( 2) is necessarily a mild solution of this equation.The inverse assertion, however, is not true for a general Banach space X (see [34] for a counter-example).On the other hand, under additional assumptions on X the mild solutions may be strong solutions.For more information on semilinear equations of the form (2) we refer to [12], [19], [24][25], [34].Before proceeding we recall some notions and results which will be frequently used later on.We define the bracket [•, •] in a Banach space Y as follows (see [5], [19] In particular, if γ = 0 , then F is said to be accretive.

Remark.
From this definition we may conclude that (F + γI) is accretive if and only if for all x, y ∈ D(F ), λ > 0, 1 ≥ λγ .

Definition 2. (Condition H1
). Equation ( 2) is said to satisfy condition H1 if i) A is the infinitesimal generator of a linear semigroup S(t), t ≥ 0 of type ω in X, ii) B is an everywhere defined continuous operator from R × X to X, iii) For every fixed t ∈ R, the operator (−B(t, •) + γI) is accretive in X.
The following well-known results are quoted for the reader's convenience (see [12], [19], [34]).2) satisfy condition H1.Then for every fixed s ∈ R and x ∈ X there exists a unique mild solution x(•) of equation ( 2) defined on [s, +∞).Moreover, if B is independent of t, then the mild solutions of equation ( 2) give rise to a semigroup of nonlinear operators T (t), t ≥ 0 having the following properties:

Theorem 1. Let equation (
Throughout this paper we shall denote by AP (X) the space of almost periodic X-valued functions in Bohr's sense with supremum norm, i.e. the space of continuous functions f from R to X such that the set {f (• + s) : R → X | s ∈ R} is precompact in the space of functions {f : R → X | f bounded and continuous} equipped with the supremum norm.
The following condition will be used frequently: The main point of our study is to associate with equation ( 2) an evolution semigroup which plays a role similar to that of the monodromy operator for equations with periodic cofficients.Hereafter we will denote by U (t, s), t ≥ s, the evolution operator corresponding to equation ( 2) which satisfies the assumptions of Theorem 1, i.e.U (t, s)x is the unique solution of equation (3).Proposition 1.Let the conditions H1 and H2 be satisfied.Then with equation ( 2) one can associate an evolution semigroup T h , h ≥ 0 acting on AP (X), defined as

Moreover, this semigroup has the following properties:
i) T h , h ≥ 0 is strongly continuous, and where Proof.We first look at the solutions to the equation It may be noted that S h , h ≥ 0 is a strongly continuous semigroup of linear oparators in AP (X) of type ω.Furthermore, for λ > 0, λγ < 1 and u, v ∈ AP (X), from the accretiveness of the operators −B(t, •) + γI we get This shows that (−B * + γI) is accretive .In virtue of Theorem 1 there exists a semigroup T h , h ≥ 0 such that If we denote [T t−s u](t) by x(t), we get where z = u(s).Consequently, from the uniqueness of mild solutions of equation ( 2) we get This completes the proof of the proposition.
The main idea underlying our approach is the following assertion.
Corollary 1.Let all assumptions of Proposition 1 be satisfied.Then a mild solution x(t) of equation ( 1), defined on the whole real line R, is almost periodic if and only if it is a common fixed point of the evolution semigroup T h , h ≥ 0 defined in Proposition 1 above.
Proof.Suppose that x(t), defined on the real line R, is an almost periodic mild solution of equation (2).Then from the uniqueness of mild solutions we get This shows that x is a fixed point of T h for every h > 0. Conversely, suppose that y(•) is any common fixed point of T h , h ≥ 0. Then This shows that y(•) is a mild solution of equation (2).
We now apply Corollary 1 in order to get sufficient conditions for the existence of almost periodic mild solutions of equation ( 2).
Corollary 2. Let all conditions of Proposition 1 be satisfied.Furthermore, let ω + µ be negative and −B * − µI be accretive.Then there exists a unique almost periodic mild solution of equation (2).
Proof.It is obvious that there exists a unique common fixed point of the semigroup T h , h ≥ 0. The assertion now follows from Corollary 1.

Remark.
A particular case in which we can check the accretiveness of −B * − µI is ω + γ < 0. In fact, this follows easily from the above estimates for u − v (see the estimate (8)).
Another "hyperbolic" case in which there exists a unique common fixed point for the semigroup T h , h ≥ 0 can be described as follows.Definition 4. A semigroup S(t), t ≥ 0 of linear operators in a Banach space X is said to be hyperbolic if there exist positive constants K, α and a bounded projection P of X with the following properties: i) P S(t) = S(t)P , ∀t ≥ 0, ii) (I − P )S(t)(I − P ) is a homeomorphism of KerP , ∀t ≥ 0, iii) [(I − P )S(t)(I − P )] −1 ≤ Ke −αt and P S(t)P ≤ Ke αt , ∀t ≥ 0.

Corollary 3. Let all conditions of Proposition 1 be satisfied. Moreover, let
A be the infinitesimal generator of a hyperbolic semigroup and let B(t, x) satisfy the estimate Then, for δ sufficiently small, equation ( 2) has a unique almost periodic mild solution.
Proof.It may be seen that Using Gronwall's inequality we get Consequently, Applying this in order to estimate the Lipschitz constant of T 1 − S 1 we get Since S 1 is hyperbolic, 1 does not belong to the spectrum σ(S 1 ).Consequently, applying the Inverse Function Theorem for Lipschitz continuous mappings (see e.g.[19,Proposition 2.3,p.67]) we observe that if δe ω < (S 1 − I) −1 , the maping T 1 − I is invertible and then it has a unique fixed point.Finally, this is a unique common fixed point for the semigroup T h , h ≥ 0. Now the assertion of the corollary follows from Corollary 1.

Almost Periodic Solutions of Differential Equations with Delays
In this section we apply the reults of the previous section in order to study the existence of almost periodic mild solutions of the equation ( 1) where A is defined as in Section 2, and f is an everywhere defined continuous mapping from R × X × C to X. Hereafter we call a continuous function x(t) defined on the real line R a mild solution of equation ( 1) if We refer the reader to [32] for more information on the existence and uniqueness of mild solutions of equations of the form (1). We should emphasize that our study is concerned only with the existence of almost periodic mild solutions of equation ( 1), and not with mild solutions in general.Consequently, the conditions guaranteeing the existence and uniqueness of mild solutions of equation ( 1) as general as in [32] are not supposed to be a priori conditions.

Definition 5. (Condition H3
). Equation ( 1) is said to satisfy Condition H3 if the following is true: i) For every g ∈ AP (X) the mapping F (t, x) = f (t, x, g t ) satisfies conditions H1 and H2 with the same constant γ. ii) There exists a constant µ with ω − µ < 0 such that −(µI Theorem 2. Let condition H3 hold.Then for δ sufficiently small (see the estimate (13) below), equation ( 1) has an almost periodic mild solution.
Proof.First we fix a function g ∈ AP (X).In view of Proposition 1 we observe that the equation has a unique almost periodic mild solution, where F (t, x) = f (t, x, g t ).We denote this solution by T g.Thus, we have defined an operator T acting on AP (X).We now prove that T is a strict contraction mapping.In fact, let us denote by U (t, s) and V (t, s) the Cauchy operators for given g, h ∈ AP (X), x ∈ X, t ≥ s.
Putting u(t) = U (t, s)x, v(t) = V (t, s)x for given s, x, from the assumptions we have where m(t, u(t) − v(t) ) = γ u(t) − v(t) + δ h − g .Using this we get Now let us fix arbitrary real numbers a ≤ b.Since the functions S(t − ξ)f (ξ, u(ξ), h ξ ) and S(t − ξ)f (ξ, v(ξ), g ξ ) are uniformly continuous on the set a ≤ ξ ≤ t ≤ b, for every > 0 there exists an η 0 = η 0 ( ) such that Applying this estimate repeatedly, we get where t 0 = s < t 1 < t 2 < ... < t n = t and |t i − t i−1 | = ∆ i .Thus, since is arbitrary, and since the function m is continuous, we get Applying Gronwall's inequality we get Now let us denote by T t h , T t g , t ≥ 0 the respective evolution semigroups corresponding to equations (10) and (11).Since T h and T g are defined as the unique fixed points u 0 , v 0 of T 1 h , T 1 g , respectively, we have where N = (e γ+ω − e γ )/ω.Finally, we have holds true, then T is a strict contraction mapping in AP (X).By virtue of the Contraction Mapping Principle T has a unique fixed point.It is easy to see that this fixed point is an almost periodic mild solution of equation ( 1).This completes the proof of the theorem.
Remarks. 1) In case ω = 0, γ = −µ we get the estimate which guarantees the existence of the fixed point of T .
2) If ω + γ < 0 , then we can choose µ = −γ, and therefore we get the accretiveness condition on −(F * + µI).However, in general, the condition ω + γ < 0 is a very strong restriction on the coefficients of equation ( 1), if f depends explicitly on t.

Examples
In applications one frequently encounters functions f from R×X×C → X of the form f (t, x, g t ) = F (t, x) + G(t, g t ), ∀t ∈ R, x ∈ X, g t ∈ C, where F satisfies condition ii) of Definition 5 and G(t, y) is Lipschitz continuous with respect to y ∈ C, i.e.
In order to describe a concrete example we consider a bounded domain Ω in R n with smooth boundary ∂Ω and suppose that A(x, D)u = |α|≤2ma α (x)D α u is a strongly elliptic differential operator in Ω.Then, defining the operatorAu = A(x, D)u, ∀u ∈ D(A) = W 2m,2 (Ω) ∩ W m,2 0 (Ω)we know from[27, Theorem 3.6] that the operator −A is the infinitesimal generator of an analytic semigroup of contractions on L 2 (Ω).Now let f, g : R×Ω×R → R be Lipschiz continuous and define the operators F (t, w)(x) = f (t, x, w(x)) and G(t, w)(x) = g(t, x, w(x)) where t ∈ R, x ∈ Ω and w ∈ L 2 (Ω).Then, for any positive constant r, the boundary value problem∂u(t, x) ∂t = A(x, D)u(t, x) + f (t, x, u(t, x)) + g(t, x, u(t − r, x)) in Ω , u(t, x) = 0 on ∂Ωfits into the abstract setting of equation (1).