EXISTENCE OF ALMOST AUTOMORPHIC SOLUTIONS TO SOME CLASSES OF NONAUTONOMOUS HIGHER-ORDER DIFFERENTIAL EQUATIONS

In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential equations with Stepanov almost automorphic forcing terms. A few illustrative examples are discussed at the very end of the paper.


Introduction
The main motivation of this paper comes from the work of Andres, Bersani, and Radová [8], in which the existence (and uniqueness) of almost periodic solutions was established for the class of n-order autonomous differential equations where f, p : R → R are (Stepanov) almost periodic, f is Lipschitz, and a k ∈ R for k = 1, ..., n are given real constants such that the real part of each root of the characteristic polynomial associated with the (linear) differential operator on the left-hand side of Eq. (1.1), that is, The method utilized in [8] makes extensive use of a very complicated representation formula for solutions to Eq. (1.1).For details on that representation formula, we refer the reader to [9] and [10] and the references therein.
Let H be a Hilbert space.In this paper, we study a more general equation than Eq.(1.1).Namely, using similar techniques as in [14,27], we study and obtain some reasonable sufficient conditions, which do guarantee the existence of almost automorphic solutions to the class of nonautonomous n-order differential equations where A : D(A) ⊂ H → H is a (possibly unbounded) self-adjoint linear operator on H whose spectrum consists of isolated eigenvalues 0 < λ 1 < λ 2 < ... < λ l → ∞ as l → ∞ 1991 Mathematics Subject Classification.43A60; 34B05; 34C27; 42A75; 47D06; 35L90.Key words and phrases.exponential dichotomy; Acquistapace and Terreni conditions; evolution families; almost automorphic; Stepanov almost automorphic, nonautonomous higher-order differential equation.EJQTDE, 2010 No. 22, p. 1 with each eigenvalue having a finite multiplicity γ j equals to the multiplicity of the corresponding eigenspace, the functions a k : R → R (k = 0, 1, ..., n − 1) are almost automorphic with inf t∈R a 0 (t) = γ 0 > 0, and the function f : R×H → H is Stepanov almost automorphic in the first variable uniformly in the second variable.
Indeed, assuming that u is differentiable n times and setting .
, then Eq. (1.2) can be rewritten in the Hilbert space X n in the following form (1.4) z ′ (t) = A(t)z(t) + F (t, z(t)), t ∈ R, where A(t) is the family of n × n-operator matrices defined by (1.5) whose domains D(A(t)) are constant in t ∈ R and are precisely given by Moreover, the semilinear term F appearing in Eq. (1.4) is defined on R × X n α for some α ∈ (0, 1) by , where X n α is the real interpolation space between X n and D(A(t)) given by Under some reasonable assumptions, it will be shown that the linear operator matrices A(t) satisfy the well-known Acquistapace-Terreni conditions [3], which do guarantee the existence of an evolution family U (t, s) associated with it.Moreover, it will be shown that U (t, s) is exponentially stable under those assumptions.
The existence of almost automorphic solutions to higher-order differential equations is important due to their (possible) applications.For instance when n = 2, we have thermoelastic plate equations [14,27] or telegraph equation [31] or Sine-Gordon equations [26].Let us also mention that when n = 2, some contributions on the maximal regularity, bounded, almost periodic, asymptotically almost periodic solutions to abstract second-order differential and partial differential equations have recently been made, among them are [11], [12], [44], [45], [46], and [47].In [8], the existence of almost periodic solutions to higher-order differential equations with constant coefficients in the form Eq. (1.1) was obtained in particular in the case when the forcing term is almost periodic.However, to the best of our knowledge, the existence of almost automorphic solutions to higher-order nonautonomous equations in the form Eq. (1.2) in the case when the forcing term is Stepanov almost automorphic is an untreated original question, which in fact constitutes the main motivation of the present paper.
The paper is organized as follows: Section 2 is devoted to preliminaries facts needed in the sequel.In particular, facts related to the existence of evolution families as well as preliminary results on intermediate spaces will be stated there.In addition, basic definitions and classical results on (Stepanov) almost automorphic functions are also given.In Sections 3 and 4, we prove the main result.In Section 5, we provide the reader with an example to illustrate our main result.

Preliminaries
Let H be a Hilbert space equipped with the norm • and the inner product •, • .In this paper, A : D(A) ⊂ H → H stands for a self-adjoint (possibly unbounded) linear operator on H whose spectrum consists of isolated eigenvalues with each eigenvalue having a finite multiplicity γ j equals to the multiplicity of the corresponding eigenspace.Let {e k j } be a (complete) orthonormal sequence of eigenvectors associated with the eigenvalues {λ j } j≥1 .
Clearly, for each u ∈ D(A) where u, e k j e k j .
Note that {E j } j≥1 is a sequence of orthogonal projections on H.Moreover, each u ∈ H can written as follows: It should also be mentioned that the operator −A is the infinitesimal generator of an analytic semigroup {T (t)} t≥0 , which is explicitly expressed in terms of those orthogonal projections E j by, for all u ∈ H, In addition, the fractional powers A r (r ≥ 0) of A exist and are given by Let (X, • ) be a Banach space.If L is a linear operator on the Banach space X, then: • D(L) stands for its domain; • ρ(L) stands for its resolvent; • σ(L) stands for its spectrum; • N (L) stands for its null-space or kernel; and • R(L) stands for its range.We set Q = I − P for a projection P .If Y, Z are Banach spaces, then the space B(Y, Z) denotes the collection of all bounded linear operators from Y into Z equipped with its natural topology.This is simply denoted by B(Y) when Y = Z.
2.1.Evolution Families.Hypothesis (H.1).The family of closed linear operators A(t) for t ∈ R on X with domain D(A(t)) (possibly not densely defined) satisfy the so-called Acquistapace-Terreni conditions, that is, there exist constants ω ∈ R, θ ∈ π 2 , π , K, L ≥ 0 and µ, ν ∈ (0, 1] with µ + ν > 1 such that (2.1) Note that in the particular case when A(t) has a constant domain D = D(A(t)), it is well-known [6,38] that Eq. (2.2) can be replaced with the following: There exist constants L and 0 < µ ≤ 1 such that It should mentioned that (H.1) was introduced in the literature by Acquistapace and Terreni in [2,3] for ω = 0.Among other things, it ensures that there exists a unique evolution family U = U (t, s) on X associated with A(t) satisfying (a) U (t, s)U (s, r) = U (t, r); ), and a constant C depending only on the constants appearing in (H.1); and (e) ∂ + s U (t, s)x = −U (t, s)A(s)x for t > s and x ∈ D(A(s)) with A(s)x ∈ D(A(s)).It should also be mentioned that the above-mentioned proprieties were mainly established in [1,Theorem 2.3] and [49, Theorem 2.1], see also [3,48].In that case we say that A(•) generates the evolution family U (•, •).
One says that an evolution family U has an exponential dichotomy (or is hyperbolic) if there are projections P (t) (t ∈ R) that are uniformly bounded and strongly continuous in t and constants δ > 0 and N ≥ 1 such that (f) U (t, s)P (s) = P (t)U (t, s); ≤ N e −δ(t−s) for t ≥ s and t, s ∈ R. According to [40], the following sufficient conditions are required for A(t) to have exponential dichotomy.
(i) Let (A(t), D(t)) t∈R be generators of analytic semigroups on X of the same type.Suppose that is finite, and The semigroups (e τ A(t) ) τ ≥0 , t ∈ R, are hyperbolic with projection P t and constants N, δ > 0.Moreover, let This setting requires some estimates related to U (t, s).For that, we introduce the interpolation spaces for A(t).We refer the reader to the following excellent books [6], [23], and [29] for proofs and further information on theses interpolation spaces.
Let A be a sectorial operator on X (for that, in assumption (H.1), replace A(t) with A) and let α ∈ (0, 1).Define the real interpolation space which, by the way, is a Banach space when endowed with the norm . For convenience we further write Moreover, let XA := D(A) of X.In particular, we have the following continuous embedding for all 0 < α < β < 1, where the fractional powers are defined in the usual way.
EJQTDE, 2010 No. 22, p. 6 In general, D(A) is not dense in the spaces X A α and X.However, we have the following continuous injection (2.5) Given the family of linear operators A(t) for t ∈ R, satisfying (H.1), we set for 0 ≤ α ≤ 1 and t ∈ R, with the corresponding norms.Then the embedding in Eq. (2.4) holds with constants independent of t ∈ R.These interpolation spaces are of class J α ([29, Definition 1.1.1]) and hence there is a constant c(α) such that We have the following fundamental estimates for the evolution family U.
[14] For x ∈ X, 0 ≤ α ≤ 1 and t > s, the following hold: (ii) There is a constant m(α), such that In addition to above, we also need the following assumptions: Hypothesis (H.2).The evolution family U generated by A(•) has an exponential dichotomy with constants N, δ > 0 and dichotomy projections P (t) for t ∈ R.

2.2.
Stepanov Almost Automorphic Functions.Let (X, • ), (Y, • Y ) be two Banach spaces.Let BC(R, X) (respectively, BC(R × Y, X)) denote the collection of all X-valued bounded continuous functions (respectively, the class of jointly bounded continuous functions F : R × Y → X).The space BC(R, X) equipped with the sup norm for each u ∈ X.
Definition 2.5.Let p ∈ [1, ∞).The space BS p (X) of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f : R → X such that f b belongs to L ∞ R; L p ((0, 1), X) .This is a Banach space with the norm The collection of all almost automorphic functions from R to X will be denoted AA(X).

Similarly
The collection of all almost automorphic functions from R × Y to X will be denoted AA(R × Y).
We have the following composition result: Then, then the function defined by EJQTDE, 2010 No. 22, p. 8 We also have the following composition result, which is a straightforward consequence of the composition of pseudo almost automorphic functions obtained in [43].
Theorem 2.9.[43] We will denote by AA u (X) the closed subspace of all functions f ∈ AA(X) with g ∈ C(R, X).Equivalently, f ∈ AA u (X) if and only if f is almost automorphic and the convergence in Definition 2.7 are uniform on compact intervals, i.e. in the Fréchet space C(R, X).Indeed, if f is almost automorphic, then, its range is relatively compact.Obviously, the following inclusions hold: where AP (X) is the Banach space of almost periodic functions from R to X. Definition 2.10.[36] The space AS p (X) of Stepanov almost automorphic functions (or S p -almost automorphic) consists of all f ∈ BS p (X) such that f b ∈ AA L p (0, 1; X) .That is, a function f ∈ L p loc (R; X) is said to be S p -almost automorphic if its Bochner transform f b : R → L p (0, 1; X) is almost automorphic in the sense that for every sequence of real numbers (s ′ n ) n∈N , there exists a subsequence (s n ) n∈N and a function g ∈ L p loc (R; X) such that t+1 t f (s n + s) − g(s) p ds 1/p → 0, and The collection of those S p -almost automorphic functions F : R × Y → X will be denoted by AS p (R × Y).
We have the following straightforward composition theorems, which generalize Theorem 2.8 and Theorem 2.9, respectively: Theorem 2.13.Let F : R× Y → X be a S p -almost automorphic function.Suppose that u → F (t, u) is Lipschz in the sense that there exists L ≥ 0 such Theorem 2.14.Let F : R × Y → X be a S p -almost automorphic function, where.Suppose that F (t, u) is uniformly continuous in every bounded subset K ⊂ X uniformly for t ∈ R. If g ∈ AS p (Y), then Γ : R → X defined by Γ(•) := F (•, g(•)) belongs to AS p (X).

Main results
Consider the nonautonomous differential equation where F : R × X α → X is S p -almost automorphic.Definition 3.1.A function u : R → X α is said to be a bounded solution to Eq. (3.1) provided that Throughout the rest of the paper, we set S 1 u(t) := S 11 u(t) − S 12 u(t), where for all t ∈ R.
To study Eq.(3.1), in addition to the previous assumptions, we require that p > 1, 1 p + 1 q = 1, and that the following assumptions hold: Moreover, F is Lipschitz in the following sense: there exists L > 0 for which then the integral operator S 1 defined above maps AA(X α ) into itself.
Define for all n = 1, 2, ..., the sequence of integral operators and hence from the Hölder's inequality and the estimate Eq. (2.7) it follows that Using Eq. (3.3), we then deduce from Weirstrass Theorem that the series defined by is uniformly convergent on R.Moreover, D ∈ C(R, X α ) and for all t ∈ R.
Let us show that Φ n ∈ AA(X α ) for each n = 1, 2, 3, ... Indeed, since ϕ ∈ AS p (X β ) ⊂ AS p (X α ), for every sequence of real numbers (τ ′ n ) n∈N there exist a subsequence (τ n k ) k∈N and a function ϕ such that Define for all n = 1, 2, 3, ..., the sequence of integral operators Using Lebesgue Dominated Convergence Theorem, one can easily see that Similarly, using [15] it follows that Similarly In view of the above, it follows that S 1 ∈ AA(X α ).
Lemma 3.3.The integral operator S 1 defined above is a contraction whenever L is small enough.
Proof.Let v, w ∈ AA(X α ).Now, Similarly, Consequently, and hence S 1 is a contraction whenever L is small enough.Proof.The proof makes use of Lemma 3.2, Lemma 3.3, and the Bananch fixed-point principle.

Almost Automorphic Solutions to Some Higher-Order Differential Equations
We have previously seen that each u ∈ H can be written in terms of the sequence of orthogonal projections E n as follows: u = A l (t)P l z, where From Eq. (1.3) it easily follows that there exists ω ∈ π 2 , π such that if we define On the other hand, one can show without difficulty that A l (t) = K −1 l (t)J l (t)K l (t), where J l (t), K l (t) are respectively given by and EJQTDE, 2010 No. 22, p. 16 For λ ∈ S ω and z ∈ X, one has Hence, EJQTDE, 2010 No. 22, p. 17 Moreover, for z := > 0. Thus, there exists C 1 > 0 such that K l (t)P l z ≤ C 1 d l n (t) z for all l ≥ 1 and t ∈ R.
Using induction, one can compute K −1 l (t) and show that for z := z for all l ≥ 1 and t ∈ R.
EJQTDE, 2010 No. 22, p. 18 Now, for z ∈ X, we have Let λ 0 > 0. Define the function It is clear that η is continuous and bounded on the closed set On the other hand, it is clear that η is bounded for λ > λ 0 .Thus η is bounded on S ω .If we take Therefore, Consequently, for all t ∈ R. Hence, for t, s, r ∈ R, computing A(t) − A(s) A(r) −1 and assuming that there exist L k ≥ 0 (k = 0, 1, 2, ..., n − 1) and µ ∈ (0, 1] such that it easily follows that there exists C > 0 such that (A(t) − A(s))A(r) −1 z ≤ C t − s µ z .
In summary, the family of operators A(t) t∈R satisfy Acquistpace-Terreni conditions.Consequently, there exists an evolution family U (t, s) associated with it.Let us now check that U (t, s) has exponential dichotomy.For that, we will have to check that (i)-(j) hold.First of all note that For every t ∈ R, the family of linear operators A(t) generate an analytic semigroup (e τ A(t) ) τ ≥0 on X given by On the other hand, we have Using the continuity of a k (k = 0, ..., n − 1) and the equality it follows that the mapping J ∋ t → R(λ, A(t)) is strongly continuous for λ ∈ S ω where J ⊂ R is an arbitrary compact interval.
for each t ∈ R, one can easily see that, for the topology of B(X), the following hold for each t ∈ R, and hence t → A −1 (t) is almost automorphic with respect to operator-topology.
It is now clear that if f satisfies (H.5) and if L is small enough, then the higherorder differential equation Eq. (1.4) has an almost automorphic solution Therefore, If f = f 1 + f 2 satisfies (H.5) and if the Lipschitz constant of f 1 is small enough, then Eq. (1.2) has at least one almost automorphic solution u ∈ H α .

Examples of Second-Order Boundary Value Problems
In this section, we provide with a few illustrative examples.Precisely, we study the existence of almost automorphic solutions to modified versions of the so-called (nonautonomous) Sine-Gordon equations (see [26]).
In this section, we take n = 2 and suppose a 0 and a 1 , in addition of being almost automorphic, satisfy the other previous assumptions.Moreover, we let α = Precisely, we are interested in the following system of second-order partial differential equations where a 1 , a 0 : R × J → R are almost automorphic positive functions and Q : R × J × L 2 (J) → L 2 (J) is S p -almost automorphic for p > 1.
Let us take Av = −v ′′ for all v ∈ D(A) = H 1 0 (J) ∩ H 2 (J) and suppose that Q : R × J × L 2 (J) → H β 0 (J) is S p -almost automorphic in t ∈ R uniformly in x ∈ J and u ∈ L 2 (J) Moreover, Q is Lipschitz in the following sense: there exists L ′′ > 0 for which Q(t, x, u) − Q(t, x, v) for all u, v ∈ L 2 (J), x ∈ J and t ∈ R. Consequently, the system Eq.(5.3) -Eq.(5.4) has unique solution u ∈ AA(H 1 0 (J)) when K ′′ is small enough.

A Slightly Modified Version of the Nonautonomous Sine-Gordon
Equations.Let Ω ⊂ R N (N ≥ 1) be a open bounded subset with C 2 boundary Γ = ∂Ω and let H = L 2 (Ω) equipped with its natural topology • L 2 (Ω) .Here, we are interested in a slightly modified version of the nonautonomous Sine-Gordon studied in the previous example, that is, the system of second-order partial differential equations given by ∂ 2 u ∂t Therefore, the system Eq.(5.5) -Eq.(5.6) has a unique solution u ∈ AA(H 1 0 (Ω)) when L ′′′ is small enough.

22 5. 1 .2 u ∂t 2 + c ∂u ∂t − d ∂ 2 u ∂x 2 +
No. 22, p. Nonautonomous Sine-Gordon Equations.Let L > 0 and and let J = (0, L).Let H = L 2 (J) be equipped with its natural topology.Our main objective here is to study the existence of almost automorphic solutions to a slightly modified version of the so-called Sine-Gordon equation with Dirichlet boundary conditions, which had been studied in the literature especially by Leiva[26] in the following form∂ k sin u = p(t, x), t ∈ R, x ∈ J (5.1) u(t, 0) = u(t, L) = 0, t ∈ R (5.2)where c, d, k are positive constants, p : R × J → R is continuous and bounded.
Definition 2.7.(Bochner) A function F ∈ C(R × Y, X) is said to be almost automorphic if for every sequence of real numbers (s ′ n ) n∈N , there exists a subsequence (s n ) n∈N such that G(t, u) := lim n→∞ F (t + s n , u) is well defined for each t ∈ R, and lim n→∞ Therefore the sequence Φ n ∈ AA(X α ) for each n = 1, 2, ... and hence D ∈ AA(X α ).Consequently t → S 11 (t) belong to AA(X α ).The proof for t → S 12 (t) is similar to that of t → S 11 (t) and hence omitted.