Almost automorphic solutions to second-order semilinear evolution equations

Nonlinear Analysis 71 (2009) e432–e435

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Almost automorphic solutions to second-order semilinear evolution equations Gaston M. N’Guérékata ∗ Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA

article

info

MSC: 44A35 42A85 42A75 Keywords: Almost automorphic function Uniform spectrum Semilinear evolution equations

abstract In this paper we give some sufficient conditions for ensuring the existence and uniqueness of a mild almost automorphic solution to a second-order semilinear evolution equation in a Banach space. We also present some properties of the (new) notion of a uniform spectrum of bounded functions. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction In this paper we are concerned with the existence and uniqueness of an almost automorphic solution to the semilinear equation d2

x(t ) = Au(t ) + f (t , x(t )), t ∈ R, (1) dt where A : D(A) ⊂ X 7→ X is a densely defined and closed linear operator, which is also the infinitesimal generator of a holomorphic semigroup, and f : R × X 7→ X is almost automorphic and jointly continuous. The existence of almost automorphic, almost periodic, asymptotically almost periodic, and pseudo-almost periodic solutions is one of most attractive topics in the qualitative theory of differential equations due to their significance and applications in physical sciences. The concept of almost automorphy (a.a. for short), which is the central issue in this paper, was first introduced in the literature by Bochner in the earlier sixties; it is a natural generalization of the notion of almost periodicity (see [7,8]). In the last decade, several authors including Nguyen van Minh, J. Liang, Ti-Jun Xiao, T. Diagana, D. Bugajewski, K. Ezzinbi, L. Maniar, B. Basit, J. A. Goldstein, A. Pankov, J. Liu, H. S. Ding, G. M. N’Guérékata and others, have produced extensive literature on the theory of almost automorphy and its applications to differential equations. We would like to mention particularly the papers [2–6,1], related to the present work. The main result in this paper is a generalization of Theorem 3.3 in [5] to the case where f is not necessarily lipschitzian. We start the paper with a presentation of the notion of a uniform spectrum of bounded functions introduced recently in [2] and used as a tool alternative to the Carleman spectrum to prove the existence of almost automorphic mild solutions to linear evolution equations with almost automorphic forcing term. We point out the fact that although the concept of a uniform spectrum of bounded functions was introduced in the framework of almost automorphy, it has been used in [6] to prove the existence and uniqueness of bounded continuous (not necessarily almost automorphic) mild solutions to the first-order equation x˙ (t ) = Ax(t ) + f (t )

∗

Tel.: +1 443 885 3965; fax: +1 443 885 8216. E-mail address: gaston.n’[email protected].

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.11.004

G.M. N’Guérékata / Nonlinear Analysis 71 (2009) e432–e435

e433

and complete second-order evolution equation x¨ (t ) = Bx˙ (t ) + Ax(t ) + f (t ) where A and B are sectorial operators and f is bounded and continuous. 2. Uniform spectrum of functions in BC (R, X) Let us consider the following simple ordinary differential equation in a complex Banach space X: x0 (t ) − λx = f (t ),

(2)

where f ∈ BC (R, X), the space of all bounded and continuous R → X functions. If Re λ 6= 0, the homogeneous equation associated with this has an exponential dichotomy; so, (2) has a unique bounded solution which we denote by xf ,λ (·). Moreover, from the theory of ordinary differential equations, it follows that for every fixed ξ ∈ R,

xf ,λ (ξ ) :=

Z   

ξ

  −

=

Z   

eλ(ξ −t ) f (t )dt

−∞ Z

+∞

ξ

eλ(ξ −t ) f (t )dt

(if Re λ < 0) (3)

(if Re λ > 0).

0

e−λη f (ξ + η)dη −∞ Z

  −

(if Re λ < 0) (4)

+∞

e−λη f (ξ + η)dη

(if Re λ > 0).

0

It is well known that the differentiation operator D is a closed operator on BC (R, X). Moreover ρ(D ) ⊃ C \ iR and xf ,λ = (D − λ)−1 f for every λ ∈ C \ iR and f ∈ BC (R, X). Hence, for every λ ∈ C with Re λ 6= 0 and f ∈ BC (R, X) the function [(λ − D )−1 f ](t ) = S[ (t )f (λ) ∈ BC (R, X). Moreover,

(λ − D )−1 f is analytic on C \ iR.

Definition 2.1 ([2]). Let f ∈ BC (R, X). Then: (i) α ∈ R is said to be uniformly regular with respect to f if there exists a neighborhood U of iα in C such that the function (λ − D )−1 f , as a complex function of λ with Re λ 6= 0, has an analytic continuation into U. (ii) The set of ξ ∈ R such that ξ is not uniformly regular with respect to f ∈ BC (R, X) is called the uniform spectrum of f and is denoted by spu (f ). Observe that, if f ∈ BUC (R, X), then α ∈ R is uniformly regular if and only if it is regular with respect to f (cf. [5]). We now list some properties of uniform spectra of functions in BC (R, X). Proposition 2.2 ([2]). Let us have g , f , fn ∈ BC (R, X) such that fn → f as n → +∞ and let Λ ⊂ R be a closed subset satisfying spu (fn ) ⊂ Λ for all n ∈ N. Then the following assertions hold: (i) (ii) (iii) (iv) (v) (vi)

spu (f ) = spu (f (h + ·)); spu (α f (·)) ⊂ spu (f ), α ∈ C; sp(f ) ⊂ spu (f ); spu (Bf (·)) ⊂ spu (f ), B ∈ L(X); spu (f + g ) ⊂ spu (f ) ∪ spu (g ); spu (f ) ⊂ Λ.

Note that by the following important result (see [6] for the proof), the concept of a uniform spectrum coincides with the concept of a Carleman spectrum for functions in BC (R, X). Proposition 2.3. Let f ∈ BC (R, X). Then spu (f ) = spc (f ), where spc (f ) denotes the Carleman spectrum of f . We also have the following: (n)

Proposition 2.4. Let f ∈ Cb (X). Then spu (f (i) ) ⊂ spu (f (i−1) ), Proof. See [1].



for every i = 1, 2, . . . , n.

e434

G.M. N’Guérékata / Nonlinear Analysis 71 (2009) e432–e435

3. Almost automorphic functions 3.1. Almost automorphic functions Definition 3.1 (Bochner). A function f ∈ C (R, X) is said to be almost automorphic if for any sequence of real numbers (s0n ), there exists a subsequence (sn ) such that lim lim f (t + sn − sm ) = f (t )

(5)

m→∞ n→∞

for any t ∈ R. Theorem 3.2. Assume that f , f1 , and f2 are almost automorphic and λ is any scalar; then the following hold true. (i) (ii) (iii) (iv)

λf and f1 + f2 are almost automorphic, fτ (t ) := f (t + τ ), t ∈ R, is almost automorphic, f¯ (t ) := f (−t ), t ∈ R, is almost automorphic, the range Rf of f is precompact, so f is bounded.

Proof. See [7, Theorems 2.1.3 and 2.1.4] for proofs.



Theorem 3.3. If {fn } is a sequence of almost automorphic X-valued functions such that fn 7→ f uniformly on R, then f is almost automorphic. Proof. See [7, Theorem 2.1.10] for the proof.



If we equip AA(X), the space of almost automorphic functions, with the sup norm

kf k∞ = sup kf (t )k t ∈R

then it turns out to be a Banach space. Recall that AP (X) and BC (R, X) denote the spaces of all X-valued almost periodic functions and all X-valued bounded and continuous functions, respectively. The following inclusions are obvious: AP (X) ⊂ AA(X) ⊂ BC (R, X). We refer the reader to [7,8] for information on the properties of the almost automorphic functions. In particular, we will use the following: Theorem 3.4. Let us define F : R 7→ X by F (t ) = precompact.

Rt 0

f (s)ds where f ∈ AA(X). Then F ∈ AA(X) iff RF = {F (t )| t ∈ R} is

In the case where X is a uniformly convex Banach space, this result holds if RF is bounded [7]. It is important to note that unlike almost periodic functions, almost automorphic functions may not be uniformly continuous, as shown in the following example due to B. M. Levitan: Example 3.5. The following function: f (t ) := sin

1

√

2 + cos t + cos

2t

is almost automorphic, but not uniformly continuous. Therefore, it is not almost periodic. 4. The result We consider the second-order semilinear evolution equation (1) d2 dt

x(t ) = Ax(t ) + f (t , x(t )),

t ∈R

(6)

such that: H1. A is an (unbounded) linear operator which generates a holomorphic semigroup of linear operators on the Banach space X , and f is a function R × X → X satisfying H2. f is almost automorphic with respect to the first variable, uniformly with respect to the second variable, H3. f (t , .) is uniformly continuous on each bounded subset K ⊂ X , uniformly for t ∈ R.

G.M. N’Guérékata / Nonlinear Analysis 71 (2009) e432–e435

e435

Definition 4.1. A continuous function x : R → X is said to be a mild solution to Eq. (6) if t

Z

(t − ξ )x(ξ )dξ ∈ D(A),

∀t ∈ R,

0

and x(t ) = u + t v + A

t

Z

(t − ξ )x(ξ )dξ + 0

t

Z

(t − ξ )f (ξ , x(ξ ))dξ ,

t ∈ R,

0

for some fixed u, v ∈ X . Note that a classical solution to Eq. (6) is a function x ∈ C 2 (R, X ) such that x(t ) ∈ D(A), ∀t ∈ R, and Eq. (6) is satisfied. Clearly a classical solution is a mild solution for Eq. (6). Now we state and prove: Theorem 4.2. Under assumptions H1–H3, if we assume that σ (A) ∩ −R2 = ∅, where σ (A) denotes the spectrum of the operator A, then Eq. (6) has a unique almost automorphic mild solution. Proof. let Γ (·) := f (·, φ(·)), where φ ∈ AA(X ). Then in view of H2 and H3 and Lemma 2.2 [3], Γ ∈ AA(X ). Now take L to be an operator defined as in Definition 3.3 [5]. It is invertible on AA(X ). As in the proof of Theorem 3.20 [5], denote the Nemytskii operator by F , such that F h(t ) = f (t , h(t )) with h ∈ AA(X ). If we let D(L) ∈ AA(X ) be equipped with the graph norm, then the operator L − F : D(L) → AA(X ) is invertible; therefore Eq. (6) has a unique mild solution x ∈ AA(X ).  References [1] J-B. Baillon, J. Blot, G.M. N’Guérékata, D. Pennequin, On C (n) -almost periodic solutions to some nonautonomous differential equations in Banach spaces, Comment. Math. Prace Mat. 46 (2) (2006) 263–273. [2] T. Diagana, G.M. N’Guérékata, Nguyen Van Min, Almost automorphic solutions of evolution equations, Proc. Amer. Math. Soc. 132 (2004) 3289–3298. [3] J. Liang, J. Zhang, T-J. Xiao, Composition of pseudo almost automorphic functions, preprint. [4] J. Liu, G.M. N’Guérékata, Nguyen Van Min, A Massera type theorem for almost automorphic solutions of evolution equations, J. Math. Anal. Appl. 299 (2004). [5] J. Liu, G.M. N’Guérékata, Nguyen Van Min, Almost automorphic solutions of second order evolution equations, Appl. Anal. 84 (11) (2005) 1173–1184. [6] J. Liu, G.M. N’Guérékata, Nguyen Van Min, Vu Quoc Phong, Bounded solutions of parabolic equations in continuous function spaces, Funkcialaj Ekvacioj 49 (2006) 337–355. [7] G.M. N’Guérékata, Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, London, Moscow, 2001. [8] G.M. N’Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.