DSpace at VNU: C-semigroups and almost periodic solutions of evolution equations

journal of differential equations 152, 358 376 (1999) Article ID jdeq.1998.3531, available online at http:ÂÂwww.idealibrary.com on Evolution Semigroups and Spectral Criteria for Almost Periodic Solutions of Periodic Evolution Equations Toshiki Naito and Nguyen Van Minh Department of Mathematics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan E-mail: naitoÄe-one.uec.ac.jp; minhÄim.uec.ac.jp Received November 17, 1997; revised June 30, 1998 We investigate spectral criteria for the existence of (almost) periodic solutions to linear 1-periodic evolution equations of the form dxÂdt=A(t) x+f (t) with (in general, unbounded) A(t) and (almost) periodic f Using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set e isp( f ), then the above equation has an almost periodic (mild) solution x f which is unique if one requires sp(x f )/[*+2?k, k # Z, * # sp( f )] We emphasize that our method allows us to treat the equations without assumption on the existence of Floquet representation This improves recent results on the subject In addition we discuss some particular cases, in which the spectrum of monodromy operator does not intersect the unit circle, and apply the obtained results to study the asymptotic behavior of solutions Finally, an application to parabolic equations is 1999 Academic Press considered Key Words: periodic equation; monodromy operator; spectrum of bounded function; (almost) periodic solution; exponential dichotomy INTRODUCTION AND PRELIMINARIES Let us consider the following linear evolution equations dx =A(t) x, dt (1) dx =A(t) x+f (t), dt (2) and where x # X, X is a complex Banach space, A(t) is a (unbounded) linear operator acting in X for every fixed t # R such that A(t)=A(t+1) for all t # R, f : R Ä X is an almost periodic function Under suitable conditions Eq (1) is well posed (see, e.g., [P]), i.e., one can associate with Eq (1) 358 0022-0396Â99 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved SEMIGROUPS AND ALMOST PERIODICITY 359 an evolutionary process (U(t, s)) t s which satisfies the conditions in Definition Once the evolution equations (1) and (2) are well posed, the asymptotic behavior of solutions at infinity is of particular interest, which has been a central topic discussed for the past two decades We refer the reader to the books [Hal1, He, LZ, Na, Ne1], and the surveys [Bat, V4], and the references therein for more complete information on the subject One of the interesting topics in the study of the asymptotic behavior of solutions to well-posed evolution equations is to find (spectral) criteria for periodicity and almost periodicity of solutions to the well posed Eqs (1) and (2) We refer the reader to the books [LZ, HMN, Pr2], and recent papers [AB1, Bas, RV, V1, V2, V4, VS] for more information on this subject In this direction, if A(t)=A \t # R, i.e., A(t) does not depend on the time t, it is of particular interest to find conditions on the spectra of A and f so that Eq (2) has an almost periodic solution which is unique in some subspace of AP(X) (the space of all X-valued almost periodic continuous functions in Bohr's sense) To this end, let A generate a strongly continuous semigroup (T(t)) t Then if _(T(1)) & [e i*, * # sp( f )]=0 _f # L 1(R), suppFf/(!&=, !+=), f V u{0], (5) SEMIGROUPS AND ALMOST PERIODICITY 361 where f V u(s) := | + f (s&t) u(t) dt & It coincides with the set _(u) consisting of ! # R such that the Fourier Carleman transform of u u^(*)= { | e &*tu(*) dt, (Re*>0), & (6) | (Re*0, U( } , }&h) f ( } ) # M (iv) If C(t)is a norm-continuous 1-periodic operator valued function and f # M, then C( } ) f ( } ) # M (v) by (3) M is invariant under the group of translations (S(h)) h # R defined In what follows, technically, we need the following condition on the evolutionary process under consideration: SEMIGROUPS AND ALMOST PERIODICITY 365 Definition The 1-periodic strongly continuous evolutionary process (U(t, s)) t s is said to satisfy condition C if the map taking t into U(t, t&h) is continuous in operator topology for every fixed positive h In the sequel we will be concerned mainly with the following concrete examples of subspaces of AP(X) which satisfy condition H: Example Let us denote by P(1) the subspace of AP(X) consisting of all 1-periodic functions It is clear that P(1) satisfies condition H Example Let (U(t, s)) t s satisfy condition C Hereafter, for every given f # AP(X), we shall denote by M( f ) the subspace of AP(X) consisting of all almost periodic functions u such that sp(u)/ [*+2?n, n # Z, * # sp( f )] Then M( f ) satisfies condition H In fact, by Theorem 1, it is a closed subspace of AP(X), and moreover it satisfies conditions (ii), (v) of the definition We now check that conditions (iii) and (iv) are also satisfied This can be done in the same way as in [V1, Lemma 4.3] The following corollary will be the key tool to study the unique solvability of the inhomogeneous equation (9) in various subspaces M of AP(X) satisfying condition H Corollary Let M satisfy condition H Then, if # \(T | M ), the inhomogeneous equation (9) has a unique solution in M for every f # M Proof Under the assumption, the evolutionary semigroup (T h ) h leaves M invariant The generator A of (T h | M ) h can be defined as the part of L in M Thus, the corollary is an immediate consequence of Lemma and the spectral inclusion e _(A) /_(T | M ) Let M be a subspace of AP(X) invariant under the evolution semigroup (T h ) h associated with the given 1-periodic evolutionary process (U(t, s)) t s in AP(X) Below we will use the following notation: P M v(t) :=P(t) v(t), \t # R, v # M If M=AP(X) we will denote P M =P In the sequel we need the following Lemma Let (U(t, s)) t s be a 1-periodic strongly continuous evolutionary process and M be an invariant subspace of the evolution semigroup (T h ) h associated with it in AP(X) Then the following assertions hold 366 NAITO AND MINH (i) If M :=P(1), then _(P M )"[0]/_(P)"[0] (ii) If the 1-periodic strongly (U(t, s)) t s satisfies condition C, then continuous evolutionary process _(P M )"[0]=_(P)"[0] for all invariant subspaces M satisfying condition H Proof (i) Let * # \(P)"[0] Then we show that * # \(P M )"[0] In fact, by the strong continuity of the underlying process and Lemma it may be noted that the map taking t into R(*, P(t)) :=(*&P(t)) &1 is strongly continuous (see, e.g., [Ra, Proposition 12]) Hence suppose that v # P(1) We have to solve the equation (*&P(t)) x(t)=v(t) in P(1) Since (*&P(t)) &1 v(t) is continuous and 1-periodic, it is a solution to the above equation Moreover, it is unique Thus (i) is proved (ii) For u, v # M, consider the equation (*&P M ) u=v It is equivalent to the equation (*&P(t)) u(t)=v(t), t # R If * # \(P M )"[0], for every v the first equation has a unique solution u, and &u& &R(*, P M )& &v& Take a function v # M of the form v(t)= ye i+t, for some + # R; the existence of such a + is guaranteed by the axioms of condition H Then the solution u satisfies &u& &R(*, P M )& &y& Hence, for every y # X the solution of the equation (*&P(0)) u(0)= y has a unique solution u(0) such that &u(0)& sup &u(t)& t &R(*, P M )& sup &v(t)& &R(*, P M )& &y& t This implies that * # \(P)"[0] and &R(*, P(t))& &R(*, P M )& Conversely, suppose that * # \(P)"[0] By Lemma for every v the second equation has a unique solution u(t)=R(*, P(t)) v(t) On the other hand, it may be noted from the assumption that the map taking t into R(*, P(t)) is norm continuous By definition of condition H, the function taking t into (*&P(t)) &1 v(t) belongs to M Since R(*, P(t)) is a norm continuous, 1-periodic function, it holds that r :=sup[&R(*, P(t))&: t # R]< This means that &u(t)& r&v(t)& r sup t &v(t)&, or &u& r &v& Hence * # \(P M ), and &R(*, P M )& r K We now illustrate Corollary in some concrete situations 2.1 Unique Solvability of the Inhomogeneous Equations in P(1) In this subsection we will consider the unique solvability of Eq (9) in P(1) SEMIGROUPS AND ALMOST PERIODICITY 367 Proposition Let (U(t, s)) t s be 1-periodic strongly continuous Then the following assertions are equivalent : (i) # \(P), (ii) Eq (9) is uniquely solvable in P(1) for a given f # P(1) Proof Suppose that (i) holds true Then we show that (ii) holds by applying Corollary To this end, we show that _(T | P(1) )"[0]/ _(P)"[0] To see this, we note that T | P(1) =P P(1) In view of Lemma 3, # \(T | P(1) ) By Example and Corollary (ii) holds true also The proof of the converse conclusion is suggested by [Pr1, p 849] In fact, we suppose that Eq (9) is uniquely solvable in P(1) We now show that # \(P) For every x # X put f (t)=U(t, 0) g(t) x for t # [0, 1], where g(t) is any continuous function of t such that g(0)= g(1)=0, and | g(t) dt=1 Thus f (t) can be continued to a 1-periodic function on the real line which we denote also by f (t) for short Put Sx=[L &1(& f )](0) Obviously, S is a bounded operator We have [L &1(& f )](1)=U(1, 0)[L &1(&f )](0)+ | U(1, !) U(!, 0) g(!) x d! Sx=PSx+Px Thus (I&P)(Sx+x)=Px+x&Px=x So, I&P is surjective From the uniqueness of solvability of (9) we get easily the injectiveness of I&P In other words, # \(P) K Remarks Proposition in the autonomous case has been proved in [Pr2, Example 12.1, p 315], see also [Pr1, p 849] Another autonomous form of Proposition can also be found in [LM, Theorem 2.2, p 179] which implies Gearhart's spectral mapping theorem for C -semigroups in Hilbert spaces, and a spectral mapping theorem for C -semigroups in Banach spaces (see [LM]) See also [VS] for another proof of this autonomous result 368 NAITO AND MINH 2.2 Unique Solvability in AP(X ) and Exponential Dichotomy This subsection will be devoted to the unique solvability of Eq (9) in AP(X) and its applications to the study of exponential dichotomy Let us begin with the following lemma which is a consequence of Proposition Lemma Let (U(t, s)) t s be 1-periodic strongly continuous Then the following assertions are equivalent : (i) S & _(P)=