BOUNDED AND ALMOST PERIODIC SOLUTIONS AND EVOLUTION SEMIGROUPS ASSOCIATED WITH NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS

We study evolution semigroups associated with nonautonomous functional differential equations. In fact, we convert a given functional differential equation to an abstract autonomous evolution equation and then derive a representation theorem for the solutions of the underlying functional differential equation. The representation theorem is then used to study the boundedness and almost periodicity of solutions of a class of nonautonomous functional differential equations.


Introduction
This paper is concerned with evolution semigroups associated with nonautonomous functional differential equations and their applications to study the asymptotic behavior of solutions to the underlying equations.Recently, this method has been extensively employed to study the asymptotic behavior of evolutionary processes determined by differential equations.Among the references listed in this paper we refer the reader to [1,13,15,17,19] and the references therein for applications of this method to study the wellposedness, stability, and exponential dichotomy of evolutionary processes determined by evolution equations without delay.In [18], we have demonstrated another useful application of evolution semigroup method to find spectral criteria for almost periodicity of solutions of linear periodic evolution equations.
In contrast to the above-mentioned works which are mainly concerned with linear or semilinear evolution equations without delay, in this paper, we deal with nonlinear nonautonomous functional differential equations.We describe thoroughly the evolution semigroups associated with underlying equations and prove a representation theorem for solutions of equations under consideration.We then demonstrate an application of the obtained results to study the boundedness and almost periodicity of solutions to functional differential equations.As a result we get a sufficient condition for the existence of bounded and almost periodic solutions which is an extension to fully nonlinear functional differential equations of previous results by other authors (see [10,12,16,20,21] for related results and methods, [6,25] for more information on ordinary differential equations with almost periodic coefficients, and [8] for various conditions for almost periodicity of solutions of equations with infinite delay).In this paper, we put an emphasis on an application of evolution semigroup method to the study of boundedness and almost periodicity of solutions to nonlinear nonautonomous functional differential equations via nonlinear semigroup techniques.For other various applications of nonlinear semigroup theory to nonlinear functional differential equations, we refer the reader to [5,8,9,22,24] and the references therein for some fundamental notions and results which may have direct relations with the present paper.We now give a brief outline of our constructions.Under the assumptions of the global existence and uniqueness theorem, the Functional Differential Equations (FDE) under consideration generates an evolutionary process {U(t, s), t ≥ s}.To this process we associate the so-called evolution semigroup of (possibly nonlinear) operators {T h , h ≥ 0} defined by the formula where v belongs to a suitable space of functions.The key point of our study is the analysis of the semigroup {T h , h ≥ 0} and its generator in the framework of the Crandall-Liggett theorem [3, Theorem 1] and the Brezis-Pazy theorem [2,Corollary 4.3].It turns out that the infinitesimal generator of this semigroup can be computed explicitly and that it generates the semigroup in the sense of Crandall and Liggett [3].
As an application of this, we then consider the asymptotic behavior of solutions of the FDE under consideration.We exploit the resemblance of the operator T h with the monodromy operator of a differential equation with periodic coefficients and derive a sufficient condition for the existence of exponentially stable bounded and almost periodic solutions.Our condition is given in terms of the accretiveness of the operator A defined below.In the case of equations without delay the operator A turns out to be the differential operator −d/dt + f (t,•).Thus, in this case, the accretiveness of A follows from that of f (t,•) for all t.In light of this, our result seems to be new for nonautonomous functional differential equations (see [10,12,16,21,20] for closely related results and methods).Moreover, our condition is given in other form than that of the above-mentioned papers, which has a direct relation to those of Bohl-Perron type for exponential dichotomy of linear equations (see [4,Chapters III and IV] for more information).

Preliminaries
In this section, we collect some well-known notions as well as some results which will be used throughout this paper.
Definition 2.1.Let Y be an arbitrary Banach space and A a (single-valued) operator acting in Y .Then A is said to be accretive if the estimate is true for all x, y ∈ D(A) and λ > 0.

B. Aulbach and N. Van Minh 247
The following result about accretive operators will be useful later on.Its proof can be found in [3].Lemma 2.2.Let A be an operator acting in a Banach space Y .Then the operator (ωI − A) is accretive if and only if for all x, y ∈ D(A) and λ > 0. The results being derived in this paper rely upon the following general theorems which are due to M. Crandall and T. Liggett [3] and H. Brezis and A. Pazy [2], respectively.For the reader's convenience we state them in full details.
Theorem 2.4 (M.Crandall and T. Liggett [3,Theorem 1]).Let ω be a real number and let B be a densely defined operator acting in Y such that ωI + B is accretive.In addition, suppose that there exists a positive number λ 0 such that R(I + λB) = Y for all 0 < λ < λ 0 .Then −B generates a strongly continuous semigroup {S(t), t ≥ 0} on Y which for all x ∈ Y , n ∈ N and t ≥ 0 satisfies the two conditions: Then for each x ∈ Y , the limit exists uniformly in every bounded interval of (0, ∞).
Under suitable conditions on the accretiveness and the range, one can prove the closedness of the operator B in Theorems 2.4 and 2.5.
Remark 2.6.The closedness of the operator B in Theorem 2.4 can be shown as follows.Suppose that {x n } ⊂ Y , x n → y, Bx n → z as n → ∞.Then we have to show that y ∈ D(B) and By = z.In fact, since the inverse (I +λB) −1 exists on Y for sufficiently small positive λ and is Lipschitz continuous, we have This shows that y is an element of D(B).Furthermore, from this we get (2.7) Since λ > 0 we have By = z and the assertion follows.
Now we consider functional differential equations of the form where t ∈ R, x ∈ X, and X is a Banach space, (2.9) Here and in what follows, we use the notation from [14].Along with (2.8) we also deal with the Cauchy problem is, under some regularity conditions on f (cf.[14]), equivalent to the integral equation Definition 2.7.A mapping f : R × C → X is said to be admissible if the following conditions are satisfied: ) is Lipschitz continuous with respect to y uniformly for t ∈ R, that is, there exists a positive constant L such that B. Aulbach and N. Van Minh 249 (iii) there exists a positive constant N such that Theorem 2.8.Let f be admissible.Then the Cauchy problem (2.10) is equivalent to the integral equation (2.11) and it has a unique solution x t (φ), t ≥ s which satisfies the following estimates: Proof.The equivalence of (2.10) and (2.11) as well as the unique existence of solutions is assured by standard arguments (cf.[7,22]).In order to derive the estimates (2.14) and (2.15), we adapt the corresponding proof from [22].To this end we first define and continue for any n ∈ N inductively by setting (2.17) Then the limit exists uniformly on every compact interval of the form [s, t 0 ] and u(t) is the unique solution of the Cauchy problem (2.10) (see [22]).From the assumptions we obtain the estimate and by induction we continue to get for each n ∈ N, where M = N(1 + φ C ).Consequently, we arrive at the estimate which is true for any n ∈ N. Thus, finally we get For the proof of (2.15) we refer to [22].

Evolution semigroups associated with nonautonomous functional differential equations
In this section, we prove a representation theorem for solutions of (2.8), that is the main results of this paper.Throughout we assume that the function f in (2.8) is admissible.
Our study is mainly concerned with the so-called evolution semigroup of operators {T h , h ≥ 0} associated with (2.8) defined by the formulas where x t (φ) is determined by (2.10) and consists of all bounded and uniformly continuous mappings v : R × [−r, 0] → X with supremum norm.Later on, for every v ∈ we will use the abbreviated notation v(t) = v(t, •).Using the existence and the uniqueness of the solutions of (2.10) it is easy to check that {T h , h ≥ 0} is indeed a semigroup.
Proposition 3.1.Let f be admissible.Then the family of operators {T h , h ≥ 0} defined by (3.2) is a strongly continuous semigroup of operators of type L on , where L is the Lipschitz constant stemming from the admissibility of f .Proof.First of all, we show that for every h ≥ 0 the operator T h acts on .By definition we have (3.3) Thus, from (2.14) we get the estimate Now we are going to show that (T h v)(t, θ ) is uniformly continuous with respect to (t, θ).In fact, we have sup B. Aulbach and N. Van Minh 251 where E 1 , E 2 , and E 3 are defined by ) Now we let (t, θ ) − (t θ ) tend to 0. In view of the uniform continuity of v ∈ the expressions (3.6) and (3.8) tend to 0. In order to prove that also (3.7) vanishes, it is sufficient to prove that the limit exists.To this end we suppose, without loss of generality, that t < t.Using (2.15) and the admissibility of f , we get (3.10) Now we state and prove a claim which will be needed in the present proof as well as later on.
Claim 3.2.Under the assumptions and notations of Proposition 3.1, the following holds: (3.12) The right-hand side of (3.12) equals max sup Due to the uniform continuity of v, in order to prove (3.11), it is sufficient to show that lim This relation, however, is clear, because in view of (2.14) we have lim ) and then (3.9).Thus, the function T h v(t, θ) is uniformly continuous with respect to (t, θ) and this means that the operator T h acts on .Next we show that the semigroup {T h , h ≥ 0} is strongly continuous.In fact, by definition we have to show that lim From the definition and the uniform continuity of v, it suffices to show that Proof.We have ) is uniformly continuous with respect to t, we get lim On the other hand, we have Applying (3.11) and taking into account the uniform continuity of v we get (3.18).This completes the proof of the lemma.
Remark 3.6.In Definition 3.5 it follows from the properties (i) and (ii) that the function v is differentiable on the line R × {0} and that then (iii) holds.In order to prove the differentiability of v, we put

.23)
A simple computation provides the identity g(0) = 0 as well as Hence, we obtain the estimate Similarly, we obtain Finally, we get the estimate In view of this relation and the uniform continuity of Dv(t, θ) with respect to (t, θ), we observe that v is differentiable on R × {0}.Consequently, we get From the definition of T h , it is clear that for every θ < 0 we get Bv(t, θ) = Dv(t, θ).
Since Bv is an element of it is not difficult to show that Dv(t, θ) exists for all (t, θ) ∈ R × [−r, 0] using the following elementary claim following from the mean value theorem.Continuing the proof of Proposition 3.7 we put θ = 0. Then by definition, we get By virtue of Lemma 3.4, the derivative (∂v/∂t)(t, 0) exists and is uniformly continuous.Furthermore, we get Using the uniform continuity of Dv(t, θ), it is not difficult to see that Consequently, since f satisfies condition H, the function (∂v/∂t)(t, 0) is bounded and uniformly continuous on R × {0}.Thus, it is easy to show that lim (3.38) According to Lemma 3.4, we then get From the uniform continuity of the functions Dv(t, θ) and (∂v/∂t)(t, 0) and in view of the relations (3.18) and (3.28), we have Thus, we get (3.42) On the other hand, in view of (3.40) and Lemma 3.4 we have lim B. Aulbach and N. Van Minh 257 Now combining (3.37), (3.42), and (3.43) and using the uniform continuity of Dv(t, θ), we observe that This finally means that v ∈ D(B) and Bv = Av.Therefore, the proof of the proposition is complete.
Now we continue to study the operator A by proving the following assertion.
Proof.Assuming that φ ∈ , we are going to show that there exists a unique ψ ∈ D(A) such that By definition, it may be seen that ψ belongs to D(A) and that ψ solves (3.45) if and only if We first solve (3.46) by the method of characteristics and then show that ψ belongs to D(A).Setting we see that w(θ) = ψ(t, θ) and w(0) = ψ(t + θ, 0) as well as Thus from the equation we get Now solving (3.50), we obtain On the other hand, since ψ(t, 0) is the bounded solution of the equation Finally, for ψ we get the integral equation (3.54) Now, we solve (3.54) by considering an operator K acting on where Kψ(t, θ) is defined as the right-hand side of (3.54).It is clear that Kψ ∈ .Furthermore, we have (3.55) Hence, K is a strict contraction which therefore has a unique fixed point.For simplicity we denote it by ψ as well.Now, it is not difficult to show that ψ as a unique solution of (3.54) is also a unique solution of (3.46).This shows that for 0 < λ < L we get R(I −λA) = and that (I −λA) −1 exists (as a single-valued operator on ).Now we are going to show that (I − λA) −1 is Lipschitz continuous with constant (1 − λL) −1 .To this end let ψ 1 , ψ 2 be the solutions of (3.54) corresponding to φ 1 , φ 2 , respectively.Therefore, we have (3.56) Thus, we get Thus it is sufficient to show that lim where ψ λ denotes the solution of the integral equation (3.54).Since ψ λ is the unique fixed point of the operator K, we have (3.61) Thus, On the other hand, we have This shows that (3.58) holds.The proof of the proposition is therefore complete.Now we are in a position to apply Theorems 2.4 and 2.5 in order to get the main result of this section.

Theorem 3.11 (representation theorem). If f satisfies condition H, then the operator
A generates the semigroup {T h , h ≥ 0} defined by (3.2) in the Crandall-Liggett sense, that is,

.64)
Furthermore, T h represents the solution x t (φ) of (2.10) in the sense that Proof.The theorem is a direct consequence of Propositions 3.1, 3.7, and 3.9 and Theorems 2.4 and 2.5.

Bounded and almost periodic solutions and evolution semigroups
In this section, we apply Theorem 3.
Remark 4.2.As noted in the introduction, the condition of Theorem 4.1 on the accretiveness of the operator ωI − A in the case of equations without delay turns out to be that of the differential operator ωI − {−d/dt + f (t,•)} in the corresponding function space.A sufficient condition for this is the accretiveness of the operator {ωI − f (t,•)} in the phase space X.In fact, this can be easily proved by using a fundamental result on continuous perturbation of linear accretive operator (see [23]).Here, it may be noted that the operator −d/dt in the function space C u (R, X) of uniformly continuous and bounded X-valued functions is accretive.Hence, Theorem 4.1 is a FDE-analog of Medvedev's result (see [10,12,16,20,21]).
A consequence of the representation theorem and Theorem 4.1 is the following which concerns the almost periodicity.To this end we recall the following notion.Now we are in a position to apply Theorem 4.1 to see that the unique fixed point of the semigroup {T h , h ≥ 0} should be in ap .This completes the proof.
In conclusion we remark that Theorem 4.1 and Corollary 4.4 are closely related to recent results by S. Kato et al., see for example, [11,12], and also Seifert [20,21], Kartsatos [10] in which a result by Medvedev [16], similar to Theorem 4.1 but for equations without delay, plays the key role to prove the existence and uniqueness of the bounded solution to the underlying equation.Our constructions can be easily extended to the case in which the semigroup {T h , h ≥ 0} leaves invariant some subspace of ap .This subspace can be determined as in the proof of Corollary 4.4.A simple model of this is the subspace of periodic functions.

Claim 3 . 8 .
Let f : [a, b] → X be a continuous function which is continuously differentiable on [a, b) such that lim x→b − f (x) = c.(3.31)Then f is differentiable from the left at b with derivative f (b) = c.
.33) Thus v is an element of D(A) and Bv = Av.Conversely, supposing v ∈ D(A) we now show that v ∈ D(B) and Av = Bv.In fact, by definition we have to show that sup t,θ

BCorollary 4 . 4 . 6 )
. Aulbach and N.Van Minh 261    Definition 4.3.A function u : R → Y is said to be almost periodic if for every given sequence {τ n } n∈N , there exists a subsequence {τ n k } k∈N such that the sequence of functions {u(• + τ n k )} k∈N is uniformly convergent on R as k → ∞.Let all assumptions of Theorem 4.1 be satisfied.Furthermore, for every fixed y, let f (t,y) be almost periodic with respect to t.Then (2.8) has a unique exponentially stable solution which is almost periodic.Proof.We define the following closed subspace:ap = {v ∈ : v(•) is almost periodic}.(4.3)We will use the representation theorem to show that {T h , h ≥ 0} leaves ap invariant, that is, for every φ ∈ ap we have T h φ ∈ ap .In fact, we first show that(I − λA) −1 φ ∈ ap for 0 < λ < 1 L .(4.4)We return to the integral equation (3.54) which determines ψ as the solution of the integral equation (3.54) with ψ = (I −λA) −1 φ.Now we show that ψ is almost periodic.To this end, it suffices to prove that the integral operator defined by the right-hand side of (3.54) (that is, the operator K) leaves ap invariant.But this can be easily seen by considering the inhomogeneous equation Note that g(τ ) is almost periodic with respect to τ .In fact, this is equivalent to the almost periodicity of f (τ, u(τ, •)) for every fixed u ∈ ap .In turn, this can be proved in the same way as in[4, Chapter VII, Lemma 4.1].According to the representation theorem, .11) .17)This, however, follows from (2.14) and the admissibility of f .Thus the proposition is proved.An admissible function f is said to satisfy condition H if for every v ∈ the function taking t into f (t, v(t, •)) is uniformly continuous.Throughout this paper, we use condition H on f to get the following assertion.
11 to study the asymptotic behavior of solutions of the functional differential equation (2.8).Theorem 4.1.Let f satisfy condition H and let ω be a negative number such that ωI − A is accretive.Then (2.8) has a unique solution which is defined on R and bounded as well as exponentially stable.