A Massera Type Criterion for Almost Automorphy of Nonautonomous Boundary Differential Equations ∗

For abstract linear nonautonomous boundary differential equations with an almost automorphic forcing term, a Massera type criterion is established for the existence of an almost automorphic solution with the help of the spectrum of monodromy operator, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.


Introduction
A classical result of Massera in his landmark paper [1] says that a necessary and sufficient condition for an ω-periodic linear scalar ordinary differential equation to have an ω-periodic solution is that it has a bounded solution on the positive half line.Since then, there has been an increasing interest in extending this classical result to various classes of functions (such as anti-periodic functions [2], quasi-periodic functions [3], almost periodic functions [4,5,6], almost automorphic functions [7,8]) and also to various classes of dynamical systems (such as ordinary differential equations [1], functional differential equations [9,10], quasi-linear partial differential equations [11], dynamic equations on time scale [12]).
Recently, there has been an increasing interest in the almost automorphy of dynamical systems, which is first introduced by Bochner [13] and is more general than the almost periodicity and attracts more and more attention.One can see [14,15] for a complete background on almost automorphic functions and see the important Memoirs [16] for almost automorphic dynamics.Many different kinds of criteria are established for the existence of almost automorphic solutions of various kinds of dynamical systems [14,16,17,18,19,20,21,22,23].
Particularly, some Massera type criteria are derived for some neutral functional differential equations [7], evolution equations [8], and differential equation with piecewise constant arguments [24].In this note, we will make an attempt to give an extension of the classical result of Massera to almost automorphic solutions of nonautonomous boundary differential equations (or sometimes, nonautonomous boundary Cauchy problems), which are an abstract formulation of partial differential equations with boundary conditions modeling natural phenomena such as retarded differential (difference) equations, dynamic population equations, and boundary control problems, and has been widely studied (see [25] and references cited therein).
The paper is organized as follows.Section 2 introduces some notations, assumptions and preliminary results on almost automorphic functions and nonautonomous boundary Cauchy problems.Section 3 investigates the almost automorphy of bounded solutions of a nonautonomous boundary differential equations in a Banach spaces and establishes a necessary and sufficient criterion of Massera type in term of spectral countability condition and boundedness of solutions for the existence of almost automorphic solution.

Preliminaries
We begin in this section by fixing some notations, assumptions and recalling a few basic results on almost automorphic functions and nonautonomous inhomogeneous boundary Cauchy problems.

Notations
Let N, Z, R, and C stand for the set of natural numbers, integers, real numbers, and complex numbers, respectively.Let X, Y be two Banach spaces and L(X, Y ) denote the space of all bounded linear operators from X to Y .C(R, X) stands for the set of continuous functions from R to X with the supreme norm.l ∞ (Z, X) denotes the space of all bounded (twosided) sequences in a Banach space X with supreme norm, i.e., u := sup n∈Z u(n) for u = {u(n)} n∈Z ∈ l ∞ (Z, X). c 0 denotes the Banach space of all numerical sequence x = {x n } ∞ n=1 satisfying lim n→∞ x n = 0, endowed with the supreme norm.For A being a linear operator on X, D(A), σ(A) and ρ(A) stand for the domain, the spectrum and the resolvent set of A, respectively.

Almost automorphic functions and sequences
We recall the definition of almost automorphic functions and some of their properties.Definition 2.1.(Bochner [13]) A function f ∈ C(R, X) is said to be almost automorphic in Bochner's sense if for every sequence of real numbers (σ n ) n∈N , there exists a subsequence The collection AA(R, X) of all almost automorphic X-valued functions is a Banach space under the supreme norm.In addition, if the convergence in the definition above is uniform EJQTDE, 2011 No. 73, p. 2 in t ∈ R, then f ∈ AP (R, X), the space of all almost periodic functions with values in X.A typical example [14] of almost automorphic function but not almost periodic reads Therefore, AP (R, X) ⊂ AA(R, X) with strict inclusion.
• the range ℜ f := {f (t) : t ∈ R} is relatively compact in X, thus f is bounded in norm.
Similarly as for functions, we define below the almost automorphy of sequences.
The set aa(Z, X) of all almost automorphic sequences in X forms a closed subspace of l ∞ (Z, X).It is well known that the range of an almost automorphic sequence is precompact.Also, if the convergence in Definition 2.2 is uniform in p ∈ Z, then the almost automorphic sequence is almost periodic.Moreover, if u is an almost automorphic function defined on R, then u| Z is an almost automorphic sequence.
Consider the linear difference equation where B is a bounded linear operator.
Lemma 2.2.[23] Let X be a Banach space and not contain any subspace being isomorphic Definition 2.4.[25,27] A family of linear bounded operators U := {U(t, s) : t ≥ s, t, s ∈ R} on a Banach space X is called a (strong continuous) evolution family if (i) U(t, s) = U(t, r)U(r, s) and U(s, s) = I (I is the identity on X) for t ≥ r ≥ s and t, r, s ∈ R; (ii) the mapping ) is strongly continuous.Moreover, the evolution family is said to be q-periodic if there exists a positive constant q > 0 such that U(t + q, s + q) = U(t, s) for all t ≥ s.

Consider the linear inhomogeneous nonautonomous boundary Cauchy problem
where the first equation is defined in a Banach space X called state space and the second equation is in a "boundary space" ∂X.We now introduce the setting of our abstract boundary Cauchy problems.Let X, D, ∂X be Banach spaces such that D is dense and continuously embedded in X.Consider the operators A m (t) ∈ L(D, X), L(t) ∈ L(D, ∂X) for t ∈ R, subject to the following hypotheses: is stable with stability constants (M, ω 0 ).
(H 3 ) the operator L(t) : D → ∂X is surjective for t ∈ R and t → L(t)x is 1-periodic continuous differentiable for all x ∈ D.
(H 4 ) there exist constants γ > 0 and ω ∈ R such that (H 5 ) there are positive constants c 1 and c 2 such that (H 6 ) f : R → X and g : R → ∂X are continuous.Under the above assumptions, it follows that there exists a 1-periodic evolution family U := {U(t, s) : t ≥ s, t, s ∈ R} generated by (A(t), D(A(t))) t∈R having exponential growth, that is, U(t, s) ≤ Me ω 0 (t−s) , t ≥ s.
We emphasize that in this paper, for the sake of simplicity of the notations we assume the 1-periodicity, and this does not mean any restriction on the period of the operators or evolution family.For the 1-periodic evolution family (U(t, s)) t≥s , we have a family of 1periodic monodromy operators P (t) := U(t + 1, t), t ∈ R, i.e., P (t + 1) = P (t), t ∈ R. Denote P := P (0) = U(1, 0), then σ(P (t))\{0} = σ(P )\{0}, i.e., the characteristic multipliers of the monodromy operators are independent of t, and the resolvent R(λ, P (t)) is strongly continuous for λ ∈ ρ(P ).
Other consequences of our assumptions will be needed in the sequel.For the proof, see [25] and the references cited therein.
Lemma 2.3.[25] Under the above assumptions, we have the following properties for t ∈ R and λ, µ ∈ ρ(A(t)): Next, we introduce the following definition of a mild solution to the inhomogeneous boundary Cauchy problem (2.2) given by the variation of constants formula.Definition 2.5.A continuous function u : R → X is called a mild solution of (2.2) if it satisfies the following equation In [25], the wellposedness of nonautonomous boudary Cauchy problems (2.2) is well studied and there is a unique mild solution of (2.2) under the above assumptions.

Massera type criteria for almost automorphy of (2.2)
In this section, we explore the almost automorphy of bounded solutions of (2.2) and establishes a necessary and sufficient criterion of Massera type for the existence of almost automorphic solution for (2.2).
EJQTDE, 2011 No. 73, p. 5 Proof.Since f (t) ∈ AA(R, X), g(t) ∈ AA(R, ∂X), then for any sequence {s k } with s k ∈ Z, there exists a subsequence {n k } ⊂ {s k } and functions f 1 , g 1 such that Using the fact that L λ,t and (U(t, s)) t≥s are 1-periodic, by Lemma 2.3, we have then by Lebesgue's Dominated Convergence Theorem, one has Similarly, we have lim where [•] is the integer part function.Then in this way, the function v(t) is well-defined on R. We claim that lim In fact, u(t where U(t, τ )λL λ,τ g 1 (τ )dτ.
Then by (3.2), one has and Using the fact that L λ,t and (U(t, s)) t≥s are 1-periodic, we have By Lebesgue's dominated convergence theorem, lim The proof of this claim is divided into two cases.Case 1.
where I 2 (k), J 2 (k), F 2 (k) are defined below.By the 1-periodicity of (U(t, s)) t≥s and the fact that [t + t k ] = [t + t 0 ], we have The strong continuity of (U(t, s)) t≥s implies that lim From the the strong continuity of (U(t, s)) t≥s and the precompactness of the range of f , it follows that lim k→∞ J 2 (k) = 0.
For J 3 (k) and F 3 (k), using the fact that L λ,t and (U(t, s)) t≥s are 1-periodic, one has From the strong continuity of (U(t, s)) t≥s , it follows that lim k→∞ EJQTDE, 2011 No. 73, p. 9 Similarly, we have The proof is complete.Theorem 3.1.Assume that f (t) ∈ AA(R, X), g(t) ∈ AA(R, ∂X), and X dose not contain any subspace isomorphic to c 0 and σ Γ (P ) is countable.Then (2.2) has an almost automorphic mild solution on R if and only if it admits a bounded mild solution on R. By Lemma 3.1, h(n) ∈ aa(Z, X).Since {u(n)} n∈Z is a bounded solution of (3.5), X dose not contain any subspace isomorphic to c 0 , and σ Γ (P ) is countable, by Lemma 2.2, u(n) ∈ aa(Z, X).Then, by Lemma 3.2, u(t) ∈ AA(R, X).