DSpace at VNU: On the Asymptotic Periodic Solutions of Abstract Functional Differential Equations

Funkcialaj Ekvacioj, 47 (2004) 307–327 On the Asymptotic Periodic Solutions of Abstract Functional DiÔerential Equations By Takeshi Nishikawa,1 Nguyen Van Minh2 and Toshiki Naito1 (1University of Electro-Communications, Japan and 2Hanoi University of Science, Vietnam) Abstract The paper is concerned with conditions for all mild solutions of abstract functional diÔerential equations with finite delay in a Banach space to be periodic and asymptotic periodic, where forcing term is a continuous 1-periodic function The obtained results extend various recent ones on the subject Key Words and Phrases Abstract functional diÔerential equation, Asymptotic periodic solutions 2000 Mathematics Subject Classification Numbers Primary 34K14, 34K30; Secondary 34G10, 34C27 Introduction This paper is concerned with equations of the form 1ị dutị ẳ Autị ỵ Fut þ f ðtÞ; dt t A R; where A is the generator of a strongly continuous semigroup of bounded linear operators ðTðtÞÞtb0 on a given Banach space X, F is a bounded linear operator from the phase space C :¼ Cẵr; 0; Xị to X, ut is an element of C which is dened as ut yị ẳ ut ỵ yÞ for Àr a y a 0, and f is an X-valued continuous 1periodic function with Fourier coe‰cients: f~k ¼ e2ikpt f tịdt; k ẳ 0; G1; G2; : The main problem which we consider in this paper is to find conditions for all solutions of Eq (1) to be periodic or asymptotic periodic This problem has a long history and has been considered in part by many authors, see e.g [4, 5, 11, 13, 16, 29, 30, 35, 39] and the references therein On the other hand, it arises naturally from recent studies on the existence of (almost) periodic solutions of evolution equations (see e.g [6, 7, 10, 17, 20, 22, 24, 25, 26, 27, 31, 32, 33, 38, 39, 40]) By the superposition principle, it is closely related to the 308 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito conditions for the inhomogeneous equations to have at least one periodic solution, and for all solutions of the corresponding homogeneous equations to be (asymptotic) periodic solutions Our plan of the paper is first to prove a new criterion for Eq (1) to have a periodic mild solution Next, using this result we can apply known results on (almost) periodic C0 -semigroups to the homogeneous equations By the superposition principle, the combination of these two steps allows us to study the inhomogeneous equation (1) The obtained results Theorems 3.4, 3.7, 3.15, 3.16, 3.18 extend the known ones in [15, 17, 22, 29, 30] and complement the ones in [4, 7, 10, 21, 27, 28, 31, 40, 41] To prove the main results in this paper we will make use of the harmonic analysis of bounded functions (see [1, 3, 23, 38] and the references therein for more details) The applications of the method of sums of commuting operators into the study of almost periodic solutions of functional diÔerential equations can be consulted in [27] This method is based on a result by Arendt, Rabiger, Sourour [2] a summary of which is given in the next section To study the homogeneous equations we will need the splitting theorem of Glicksberg and DeLeeuw For the reader’s convenience we summarize some notions and results in the Apprendix Preliminaries 2.1 Notation and Definitions In this paper we use the following notations: N; Z; R; C stand for the set of natural, integer, real, complex numbers, respectively; X will denote a given complex Banach space If T is a linear operator on X, then DðTÞ stands for its domain Given two Banach spaces Y; Z by LðY; ZÞ we will denote the space of all bounded linear operators from Y to Z and LX; Xị :ẳ LXị As usual, sðTÞ; rðTÞ; Rðl; TÞ are the notations of the spectrum, resolvent set and resolvent of the operator T The notations BUCðR; XÞ; APðXÞ will stand for the space of all X-valued bounded uniformly continuous functions on R and its subspace of almost periodic functions in Bohr’s sense (see, [23]); APðXÞ is a Banach space with supremum norm We will denote by B the operater acting on BUCðR; XÞ defined by the formula ẵButị :ẳ Fut , Eu A BUCR; Xị We will denote by SðtÞ the translation group on BUCðR; Xị, i.e., Stịvsị :ẳ vt ỵ sị, Et; s A R, v A BUCR; Xị with innitesimal generator D :ẳ d=dt which is dened on DDị :ẳ BUC R; XÞ Let M be a subspace of BUCðR; XÞ, A be a linear operator on X We shall denote by AM the operator M C f 7! Af ðÁÞ with DAM ị ẳ f f A M j Et A R; f ðtÞ A DðAÞ; Af ðÁÞ A Mg When M ẳ BUCR; Xị we shall use the notation A :¼ AM For translation invariant subspaces M H BUCðR; XÞ we will denote by DM the infinitesimal generator of the translation group ðSðtÞjM Þt A R in M 309 Asymptotic Periodic Solutions Definition 2.1 A function f : R ! X is said to be asymptotic periodic if there exists a periodic function f : R ! X such that lim t!y ð f ðtÞ À f tịị ẳ A C0 -semigroup Ttịịtb0 is called compact for t > t0 if TðtÞ is a compact operator for every t > t0 ðTðtÞÞtb0 is called compact if TðtÞ is compact for each t > 2.2 Commuting Operators In this subsection we recall the notion of two commuting operators and some related results on the spectral properties of their sum Definition 2.2 Let A and B be operators on a Banach space G with nonempty resolvent set We say that A and B commute if one of the following equivalent conditions hold: i) Rðl; AÞRðm; BÞ ¼ Rðm; BÞRðl; AÞ for some (all) l A rðAÞ, m A rðBÞ, ii) If x A DðAÞ, Rðm; BÞx A DAị and ARm; Bịx ẳ Rm; BịAx for some (all) m A rðBÞ For y A ð0; pÞ, R > we denote Sy; Rị ẳ fz A C : jzj b R; jargzj a yg i) Definition 2.3 Let A and B be commuting operators Then A is said to be of class Sp=2 ỵ y; Rị if there are positive constants y; R such that < y < p=2, and 2ị Sp=2 ỵ y; Rị H rAị and sup klRl; Aịk < y; l A Sp=2ỵy; RÞ ii) A and B are said to satisfy condition P if there are positive constants y; y ; R; y < y < p=2, such that A and B are of class Sp=2 ỵ y; Rị; Sp=2 À y ; RÞ, respectively If in addition, an operator A satisfying (i) in the above definition has dense domain, it generates an analytic (strongly continuous) semigroup In this case A is said to be sectorial As usual, A ỵ B is dened by A ỵ Bịx ẳ Ax ỵ Bx with domain DA ỵ Bị ẳ DAị V DðBÞ In this paper we will use the following norm, defined by A on the space X, kxkTA :¼ kRðl; Aịxk, where l A rAị It is seen that diÔerent l A rðAÞ yields equivalent norms We say that an operator C on X is A-closed if its graph is closed with respect to the topology induced by TA on the product X  X In this case, A-closure of C is denoted by C A Theorem 2.4 Assume that A and B commute assertions hold: i) If one of the operators is bounded, then 3ị sA ỵ Bị H sAị ỵ sBị: Then the following 310 ii) Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito If A and B satisfy condition P, then A ỵ B is A-closable, and sA ỵ Bị A ị H sAị ỵ sBị: ð4Þ In particular, if DðAÞ is dense in X, then A ỵ Bị A ẳ A ỵ B, where A þ B denotes the usual closure of A þ B Proof 2.3 For the proof we refer the reader to [2, Theorems 7.2, 7.3] r Functional diÔerential equations Definition 2.5 Let A be a closed linear operator on X An X-valued continuous function u on R is said to be a mild solution of Eq (1) on R if for every s, t t utị ẳ usị ỵ A uxịdx ỵ ẵFux ỵ f xịdx; Et b s: s s If A is the generator of a C0 -semigroup, by [20, Lemma 2.11] this condition is equivalent to the condition that, for every s, t utị ẳ Tt sịusị ỵ Tt xịẵFux ỵ f xịdx Et b s: s Consider the homogeneous equation of Eq (1) dutị ẳ Autị ỵ Fut : dt 5ị One can dene the solution semigroup ðV ðtÞÞtb0 on C which is defined by V tịf :ẳ wt , f A C, where wị is the unique solution of the Cauchy problem ( Ðt wtị ẳ Tt sịf0ị ỵ Tt xịẵFwx dx; Et b 0; 6ị w0 ẳ f: Let G be the generator of ðV ðtÞÞtb0 The characteristic operator Dlị of Eq (5) is dened by Dlịx :ẳ lI À A À Fe lÁ Þx; ð7Þ x A DðAÞ: Moreover, we dene the sets rDị :ẳ fl A C : bD1 lị A LXịg; sDị :ẳ rDị c and si Dị :ẳ fx A R : ix A sDịg Lemma 2.6 si Dị H si A ỵ Bị :ẳ fx A R : ix A sA ỵ Bịg: 311 Asymptotic Periodic Solutions Proof We will follow the manner in the proof of [34, Proposition 3.6] Let il A rðA ỵ Bị Set G ẳ il A Bị1 For f A BUCR; Xị, we set uf ẳ Gf Then il A Bịuf ẳ f : Since for all x A R, ASxị ẳ SxịA and BSxị ẳ SxịB, we have il A BịSxịuf ẳ Sxịil A Bịuf ẳ Sxị f : Therefore SxịGf ẳ GSxị f for x A R, f A BUCðR; XÞ On the other hand, for fl :ẳ e il x, x A Xị, we have Sxị fl ¼ e ilx fl Thus, we have dGfl SðhÞGfl ðtÞ À Gfl ðtÞ SðhÞ fl ðtÞ À fl tị tị ẳ lim ẳ lim G h!0 h!0 h h dt ẳ G lim h!0 Shị fl tị fl tị e ilh fl tị fl tị ẳ G lim ẳ ilGfl tị; h!0 h h that is, Gfl tị ẳ e ilt y for some y A X Since Gfl A DAị, Gfl tị ẳ ye ilt A DðAÞ that is y A DðAÞ From the definition of G it follows that Gfl ¼ e ilÁ y satises ẵil A Bịe il ytị ẳ e ilt x, t A R Hence we have ily À Ay F e il yị ẳ x: Thus Dilị is surjective Let x ¼ Then by using the relation e ilt y ¼ Ge ilt 0, we get y ẳ Thus Dilị is injective Consequently there exists DðilÞÀ1 A LðXÞ, i.e., il A rðDÞ r Lemma 2.7 Let A be the generator of a compact semigroup isi Dị ẳ sGị V iR, which is a finite set Then, Proof For the proof we refer the reader to [42, Lemma 5.5] and [19, Proposition 3.2] r 2.4 We denote by F the Fourier transform, i.e ỵy Ff ÞðsÞ :¼ eÀist f ðtÞdt Spectrum of a function Ày ðs A R, f A L ðRÞÞ the following set The Beurling spectrum of u A BUCðR; XÞ is dened to be 8ị spuị :ẳ fx A R : Ee > b f A L ðRÞ; supp Ff H x e; x ỵ eị; f u 0g; where f usị :ẳ é ỵy Ày f ðs À tÞuðtÞdt 312 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito Example 2.8 If f ðtÞ is a 1-periodic function with the corresponding P Fourier series f @ k A Z f~k e 2ikpt , then spð f ị ẳ f2kp : f~k 0g Theorem 2.9 Under the notation as above, spðuÞ coincides with the set consisting of x A R such that the Fourier-Carleman transform of u (Ðy Àlt uðtÞdt; ðRe l > 0Þ e é y lt u^lị ẳ 9ị e uðÀtÞdt; ðRe l < 0Þ has no holomorphic extension to any neighborhood of ix Proof For the proof we refer the reader to [38, Proposition 0.5, p 22] r We list some properties of the spectrum of a function, which we will need in the sequel Theorem 2.10 Let f ; gn A BUCðR; XÞ, n A N such that gn ! f as n ! y Then i) spð f ị is closed, ii) sp f ỵ hịị ¼ spð f Þ, iii) If a A Cnf0g, spðaf Þ ¼ spð f Þ, iv) If spðgn Þ H L for all n A N, spð f Þ H L, v) If A is a closed linear operator, f ðtÞ A DðAÞ Et A R and Af ðÁÞ A BUCðR; XÞ, then spðAf Þ H spð f Þ, vi) spðc à f Þ H spð f Þ V supp Fc, Ec A L ðRÞ Proof For the proof we refer the reader to [38, p 20–21] r As an immediate consequence of the above theorem we have the following: Corollary 2.11 Let L be a closed subset of R Then the set LXị :ẳ fg A APXị : spðgÞ H Lg is a closed subspace of APðXÞ which is invariant under translations The following theorem is very important to derive main results in the next section Theorem 2.12 Proof A function f is 1-periodic if and only if spð f Þ H 2pZ For the proof we refer the reader to [3, Theorem 4.8.8] r Main results 3.1 Conditions for all solutions to be periodic We begin this subsection by proving a necessary and su‰cient condition for the existence of 1-periodic 313 Asymptotic Periodic Solutions solutions to the inhomogeneous equation (1) We will extend the following theorem for ordinary diÔerential equations, which is derived instantly by [17, Theorem 1.2] Proposition 3.1 Let L be a d  d matrix and f ðtÞ is a C d -valued, 1periodic continuous function Then the equation z_tị ẳ Lztị ỵ f ðtÞ has a 1periodic solution if and only if, for every k A Z, the equation ð2ikp À LÞx ¼ f~k has a solution x A C d To this purpose we will use the following two lemmas Let SðRÞ be the family of rapidly decreasing functions on R Lemma 3.2 Let A be a closed linear operator and f A SðRÞ If u is a bounded mild solution of Eq (1) on R, then f à u is a classical solution to Eq (1) with forcing term f à f Proof This lemma is proved in the similar manner in the proof of [20, Lemma 2.12] In fact, let us dene Utị :ẳ t usịds; Etị :ẳ t ẵFus ỵ f sịds; t A R: 0 Then, by denition, we have utị ẳ u0ị ỵ AUtị ỵ Etị; t A R: From the closedness of A, we have f uịtị ẳ y fxịdxu0ị ỵ Af Utịị ỵ f Eịtị; t A R: Ày Since f is a rapidly decreasing function, all convolutions above are innitely diÔerentiable From the closedness of A, we have that dðf à UÞ=dtðtÞ A DðAÞ, t A R,   d df Uị Af Uịtịị ẳ A tị dt dt ẳ Af uịtịị; and d f uịtị ẳ Af uịtịị ỵ f Buịịtị ỵ f f ịtị; dt t A R: 314 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito By denition of Bu, we have f Buịtị ẳ y fsịFuts ds y F fsịuts ịds y ẳ y ẳF y fsịuts ds: y Since uts yị ẳ ut s ỵ yị, y A ẵr; 0, by the denition of a Riemann integral, it follows that ð y Ày fsịuts  y fsịut s ỵ yịds ds yị ẳ y ẳ f uịt ỵ yị: Hence, f Buịịtị ẳ F f uịt ị, and d f uịtị ẳ Af uịtị ỵ F f uịt ị ỵ f f ịtị: dt Lemma 3.3 k A Z, r If Eq (1) has a 1-periodic mild solution u, then, for every D2ikpị~ uk ẳ f~k : Proof Let u be a 1-periodic mild solution to Eq (1) If u is a classical solution, it is easy to see that D2ikpị~ uk ẳ f~k If u is not a classical solution, we set w ¼ u à f, g ¼ f à f, where f is a rapidly decreasing smooth scalar function f such that the Fourier transform Ff has support concentrated on ð2kp e; 2kp ỵ eị and is equal to on a neighborhood of 2kp Then by Lemma 3.2, w is a classical solution to Eq (1) with f replaced by g; hence ~ k ¼ g~k Moreover we have D2ikpịw g~k ẳ Ff2kpị f~k ẳ f~k ; ~ k ẳ u~k similarly Consequently we have D2ikpị~ and w uk ¼ f~k r Theorem 3.4 Let A be the generator of an analytic semigroup Then, Eq (1) has a 1-periodic mild solution if and only if for every k A Z, the equation 10ị D2ikpịx ẳ f~k Asymptotic Periodic Solutions 315 has solutions x A X If xk is a solution of Eq (10) for k A Z, then Py 2ikpt is the Fourier series of a 1-periodic mild solution of Eq (1) k¼Ày xk e Proof It is su‰cient to prove the su‰ciency To this end, let us consider the operator D À A À B as a sum of commuting operators D and ÀA À B (see [34, Lemma 3.1]) By [20, Lemma 2.8], A is sectorial, and B is a bounded linear operator Hence by [36, Corollary 2.2], A ỵ B is sectorial, so si A ỵ Bị is a bounded subset of R Meanwhile, if L is a closed subset of the real line, then sDLXị ị ẳ iL by [20, Lemma 2.6] Moreover by [20, Theorem 2.8], it is seen that if sðDLðXÞ ị V sA ỵ Bị ẳ q, for every f A LðXÞ, Eq (1) has a unique solution u A LðXÞ Let N be a natural su‰ciently large number such that 11ị si A ỵ Bị H ẵN; N : Thus, if sp f ị H RnẵN; N , then Eq (1) has a unique solution u with spðuÞ H sp f ị Therefore, we decompose f ẳ f1 ỵ f2 as follows: f1 tị :ẳ N X f~k e 2ikpt ; kẳN f2 tị :ẳ f tị f1 ðtÞ: By the above remark, Eq (1) with f replaced by f2 has a unique 1-periodic mild solution u2 by Theorem 2.12 On the other hand, for every ÀN a k a N there exists an x~k such that Dð2ikpÞ~ xk ¼ f~k by the assumption of this theorem Thus Eq (1) with f replaced by f~k e 2ikpt has at least one 1-periodic solution x~k e 2ikpt Consequently, Eq (1) with f replaced by f1 has at least one PN 1-periodic solution u1 tị ẳ kẳN x~k e 2ikpt By the superposition principle, Eq (1) has at least one 1-periodic mild solution u ẳ u1 ỵ u2 Let xk be a solution of Eq (10), k A Z, and uðtÞ the 1-periodic mild solution for Eq (1) in the above Since the relation (11) holds, si Dị H ẵN; N by Lemma 2.6; hence, if jkj > N xk ẳ u~k Set vtị ẳ X ðxk À u~k Þe 2ikpt : jkjaN Since Dð2ikpÞðxk u~k ị ẳ 0, vtị is a solution of the homogeneous equation of Eq (1) Thus, wtị :ẳ utị þ vðtÞ is a 1-periodic mild solution of Eq (1) If ~ k ¼ u~k ¼ xk If jkj a N, we have that jkj > N, we have that w 316 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito ~ k ẳ u~k ỵ ~vk ẳ u~k ỵ xk À u~k ¼ xk : w ~ k ¼ xk for every k A Z Hence, w r Remark 3.5 Since si ðDÞ is bounded, Eq (10) should have solutions at most at finitely many k A Z, jkj a N, where N depends only of A; F Remark 3.6 By the same argument we can prove the above theorem for equations of more general form: ð12Þ x_ tị ẳ Axtị ỵ ỵy dBhịxt ỵ hị ỵ f ðtÞ; t A R; Ày where A is the generator of an analytic semigroup This result extends a main result of [17] and [22] to the infinite dimensional case The analyticity of the semigroup generated by A cannot be dropped due to the failure of the spectral mapping theorem for linear semigroups in the infinite dimensional case (see e.g [8], [36]) This theorem can be generalized to cover the general case of functional equations discussed in [27] For periodic functional equations with infinite delay we refer the reader to [40] for a general criterion for the existence of periodic solutions In the case that instead of an analytic semigroup the operator A generates a compact semigroup, all conclusions of Theorem 3.4 hold true from the decomposition of the variation of constants formula Theorem 3.7 Let A be the generator of a compact semigroup Then, Eq (1) has a 1-periodic mild solution if and only if for every k A Z, the equation (10) has solutions x A X If xk is a solution of Eq (10) for k A Z, then Py 2ipkt is the Fourier series of a 1-periodic mild solution of Eq (1) k¼Ày xk e Proof It su‰ces to prove the su‰ciency To this end, we use the results in the paper [19] The space C is decomposed as C ẳ S l U; V tịS H S; V ðtÞU H U; where S is a stable subspace for V ðtÞ and U is finite dimensional Let uðtÞ be a mild solution on R of Eq (1), and P S : C 7! S and P U : C 7! U be projections corresponding to the decomposition The solution uðtÞ is a 1-periodic solution if and only if P S ut and P U ut are 1-periodic Let yðtÞ be the S-valued function defined in [21, p 346] Then yðtÞ is 1-periodic, and P S ut is 1-periodic if and only if P S ut ¼ yðtÞ by [21, Propostition 4.1] Hence uðtÞ is a 1-periodic mild solution if and only if P U ut is 1-periodic Let dim U ¼ d, and F ¼ ðf1 ; ; fd Þ be a basis vector of U Then P U ut ẳ Fztị by a vector zðtÞ A C d Asymptotic Periodic Solutions 317 By [21, Proposition 4.2] there is a d-column vector x à ¼ colðx1à ; ; xdà Þ, xià A X à , i ¼ 1; ; d, such that zðtÞ is a solution of the ordinary diÔerential equation: z_tị ẳ Lztị ỵ hx à ; f ðtÞi; ð13Þ where L is a d  d matrix then 2ikp B sðLÞ Set f1 ðtÞ :¼ N X Let N be a positive integer such that, if jkj > N, f~k e 2ikpt ; f2 tị :ẳ f tị f1 tị: kẳN Consider Eq (13) with f tị replaced by f2 tị g~k ẳ  Set gtị ẳ hx ; f2 tịi Then 0; jkj a N; à ~ hx ; fk i; jkj > N: Hence for every k A Z, the equation 2ikp Lịx ẳ g~k has a solution x A C d By Proposition 3.1, Eq (13) with f ðtÞ replaced by f2 ðtÞ has a 1-periodic solution Thus, Eq (1) with forcing term f2 has a 1-periodic mild solution By repeating the argument of the proof of Theorem 3.4, Eq (1) has at least one 1-periodic mild solution Hence Eq (1) has at least one 1-periodic mild solution Since si ðDÞ is bounded by Lemma 2.7, the rest part is proved by repeating the argument in the proof of Theorem 3.4 r Remark 3.8 In the paper [21] the variation of constants formula is proved under the following condition: F is represented as 14ị Ff ẳ dBhịfhị; Ef A C; r where B : ẵr; ! LðXÞ is of bounded variation with a given positive real r > Note that this result has been improved by dropping this condition in the recent paper [28] Let us consider conditions for all mild solutions of Eq (1) to be 1periodic We first consider conditions for all mild solutions of the equations without delay to be 1-periodic For this purpose we shall prove the following lemma Lemma 3.9 Let A be the generator of a C0 -semigroup ðTðtÞÞtb0 , and f be 1-periodic If all mild solutions of 318 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito 15ị utị ẳ Ttịu0ị ỵ t Tt xị f ðxÞdx; Et b 0; are 1-periodic, all mild solutions of Eq (15) on ðÀy; yÞ are also 1-periodic Proof Let u be a mild solution of Eq (15) on ðÀy; yÞ Since u is a mild solution of Eq (15) on ẵ0; yị, utị ẳ ut ỵ 1ị for t b from the assumption of this lemma Take a t0 < arbitrary We can choose n0 A N such that t0 ỵ n0 b Dene a function vtị :ẳ ut n0 ị Since f tị is 1-periodic, the translated function vðtÞ is also a mild solution of Eq (15) on y; yị Hence, vt ỵ 1ị ẳ vtị for t b In particularly vt0 ỵ n0 ỵ 1ị ẳ vt0 ỵ n0 ị, which implies ut0 ỵ 1ị ẳ ut0 ị Therefore utị is also a 1-periodic for t < r Theorem 3.10 Let A be the generator of an analytic semigroup Then, it is necessary and su‰cient for all mild solutions of Eq (15) on ẵ0; yị to be 1periodic that the following conditions are satisfied: i) For every k A Z the equation 16ị ii) 2ikp Aịx ẳ f~k has a solution x A X , sAị ẳ sp Aị H 2ipZ, and the corresponding eigenvectors spans a dense subspace in X Proof Necessity: Since all mild solutions of Eq (15) on ẵ0; yị are 1periodic, by Lemma 3.9 all mild solutions of Eq (15) on ðÀy; yÞ are also 1-periodic By Lemma 3.3, for every k A Z, Eq (16) is solvable The superposition principle yields that all mild solutions of u_ tị ẳ Autị are 1periodic, i.e., Ttịịtb0 is 1-periodic Hence, by [8, Theorem 2.26] sAị ẳ sp ðAÞ H 2ipZ and the set of all eigenvectors of A span a dense subset in X Su‰ciency: Since for every k A Z Eq (16) is solvable, by Theorem 3.4, Eq (15) has at least one 1-periodic mild solution on the whole line On the other hand, since sðAÞ ¼ sp ðAÞ H 2ipZ and the set of all eigenvectors of A is a dense subset in X, by [8, Theorem 2.26], ðTðtÞÞtb0 is 1-periodic, i.e., all mild solutions of u_ tị ẳ Autị are 1-periodic Hence, the superposition principle yields that all mild solutions of Eq (15) on ½0; yÞ are 1-periodic r Remark 3.11 However, if < r, it is impossible that all mild solutions of Eq (1) are 1-periodic In fact, if for f A C, V tịf ẳ V t ỵ 1ịf, Et b 0, then we have f ẳ V 1ịf Then since < r, we have V 1ịfị1ị ẳ f1ị Since V 1ịf :ẳ w1 , where wị is the mild solution of Eq (6), we have V 1ịfị1ị ẳ w0ị ¼ fð0Þ Thus we have fð0Þ ¼ fðÀ1Þ In other words, if fð0Þ fðÀ1Þ, 319 Asymptotic Periodic Solutions then V ð1Þf f Hence ðV ðtÞÞtb0 is not 1-periodic, i.e., there should be some mild solutions which are not 1-periodic Remark 3.12 If A is the generator of a compact semigroup, then from the well known knowledge of abstract functional diÔerential equations it follows that the solution semigroup V ðtÞ is compact for t > r (see [42, Proposition 2.4]) Consequently, since dim C ¼ y, the identity V kịf ẳ f, Ef A C, for some k A N never holds, i.e., there should be some mild solutions which are not 1-periodic However, there may happen that all mild solutions of Eq (1) are asymptotic periodic as shown in the next subsection 3.2 Conditions for all solutions to be asymptotic periodic 3.2.1 Necessary conditions We have the following necessary conditions for all mild solutions to be asymptotic 1-periodic To this purpose we will use the following proposition Proposition 3.13 Let f be a 1-periodic function, and u an asymtotic 1periodic mild solution on ẵ0; yị of Eq (1) If utị is decomposed as utị ẳ u tị þ u ðtÞ for t b Àr such that lim t!y u tị ẳ and u ðtÞ is 1-periodic on ðÀy; yÞ, then u ðtÞ is a mild solution on ẵ0; yị of Eq (5) and u ðtÞ is a mild solution on ðÀy; yÞ of Eq (1) respectively Proof We will follow the manner in the proof of [1, Proposition 3.4] Ðt Since u is a mild solution on ẵ0; yị of Eq (1), uxịdx A DAị, Et A ẵ0; yị and t t utị ẳ u0ị ỵ A uxịdx ỵ ẵFux þ f ðxފdx: 0 Take an n A N such that n > r Then uxỵn A C for x > 0, and we have u ðt ỵ nị ỵ u tị ẳ u t ỵ nị ỵ u t ỵ nị ẳ ut þ nÞ ð tþn  ð tþn 1 ẳ u nị ỵ u nị ỵ A u xịdx ỵ u xịdx n ỵ tỵn n Fux0 dx ỵ tỵn n ẳ u nị ỵ u 0ị ỵ A n Fux1 dx ỵ  t ỵ 0 Fuxỵn dx ỵ t f xịdx n u x ỵ nịdx ỵ t tỵn Fux1 dx ỵ t t f ðxÞdx: u ðxÞdx  320 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito Since A is closed and F is bounded, by taking the limit as n ! y we have that Ðt u ðxÞdx A DðAÞ and 1 u tị ẳ u 0ị ỵ A t u xịdx ỵ t Fux1 dx þ ðt f ðxÞdx; t b 0: Therefore, u is a mild solution on ẵ0; yị of Eq (1) However, since u and f are 1-periodic functions, u is a mild solution of Eq (1) on y; yị By linearity, u tị ẳ utị u ðtÞ, t b Àr; is a mild solution on ẵ0; yị of Eq (5) r Using Proposition 3.13 and the superposition principle, we derive the following lemma Lemma 3.14 All mild solutions of Eq (1) are asymptotic 1-periodic if and only if the following conditions satisfied: i) Eq (1) has a 1-periodic mild solution, ii) All mild solutions of Eq (5) are asymptotic 1-periodic Proposition 3.15 Let A be the generator of a C0 -semigroup If all mild solutions of Eq (1) on ẵ0; ỵyị are asymptotic 1-periodic, the following conditions hold: i) For every k A Z, Eq (10) has a solution x A X, ii) The solution semigroup ðV ðtÞÞtb0 is uniformly bounded, iii) sp ðGÞ V iR H 2ipZ Proof By Proposition 3.13 there exists a 1-periodic mild solution u on R Hence by Lemma 3.3, Condition (i) is satisfied Moreover by Lemma 3.14, all mild solutions of Eq (5) are asymptotic 1periodic For a given f A C, we denote by wðt; fÞ the solution of Eq (5) such that w0 ¼ f Then wt; fị is decomposed uniquely as wt; fị ẳ w t; fị ỵ w t; fị, for t b Àr, such that lim t!y w ðt; fÞ ¼ and w ðt; fÞ is a 1-periodic function on R Then by Proposition 3.13, we have i i w t; fị ẳ w 0; fị ỵ A t i w x; fịdx ỵ t Fwxi fịdx; t b 0; i ẳ 0; 1: We set n o D0 ¼ f A C : lim V sịf ẳ ; s!y D1 ẳ ff A C : V 1ịf ẳ fg: Then D0 and D1 are subspaces of C and D0 V D1 ¼ f0g Moreover it is clear that V ðtÞD0 H D0 and V ðtÞD1 H D1 For any f A C, we set f ¼ w00 Asymptotic Periodic Solutions 321 ¼ and f ¼ w01 Then we have lim s!y V sịf ẳ and V tịf ẳ wt1 ẳ wtỵ1 1 V t ỵ 1ịf Hence, f ẳ f ỵ f , f A D0 , f A D1 If n A N, then lim V nịf ẳ lim V nịf ỵ lim V nịf ẳ f n!y n!y n!y for every f A C; hence M :ẳ supn A N kV nịk is bounded For t b 0, we have V tị ẳ V ẵtịV t ẵtị Since a t ẵt < 1, it follows that kV tịk a kV ẵtịk kV t ẵtịk a M sup 0as r Hence, sðGÞ V fl : Re l b 0g consists of finite number of normal eigenvalues, l1 ; l2 ; ; lp Moreover, C is decomposed as follows: C ¼ S l U; U ¼ U1 l U2 l Á Á Á l Up ; where Uj ẳ NG lj I ị mj ị, j ¼ 1; 2; ; p, and S is the stable subspace of 322 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito V ðtÞ Condition (ii) implies V Uj tị is uniformly bounded for j ẳ 1; 2; ; p; thus, Re lj ¼ Since, for f A Uj , m À1 j X k t kV Uj tịfk ẳ G lj ị k f ; kẳ0 k! kV Uj tịfk is bounded if and only if ðG À lj Þf ¼ 0; that is mj ¼ 1, and V Uj tịf ẳ e lj t f Since lj A 2ipZ by Condition (iii), ðV Uj ðtÞÞtb0 is 1-periodic Hence, V tịf ẳ V S tịP S f ỵ V U ðtÞP U f is asymptotic 1-periodic Therefore, by Lemma 3.14, this shows that all mild solutions of Eq (1) are asymptotic 1periodic r If ðTðtÞÞtb0 is an analytic semigroup, we need an additional condition this purpose, we now prove the following lemma To Lemma 3.17 Let z be a bounded mild solution of z_tị ẳ Gztị Then we have spzị H si Gị :ẳ fx A R : ix A sðGÞg Proof We will follow the manner in the proof of [20, Lemma 2.21] Since z is a mild solution of z_tị ẳ Gztị, the Fourier-Carleman transform of z satises lI Gị^ zlị ẳ z0ị; where Re l 0 Assume ix A rðGÞ Then ðlI À GÞÀ1 is holomorphic in a neighborhood of ix Hence z^ðlÞ has a holomorphic extension on a neighborhood of ix, i.e., x B spðzÞ r Theorem 3.18 Let A be the generator of an analytic semigroup Assume further that the following conditions are satisfied: i) For every k A Z, Eq (10) has a solution x A X, ii) The solution semigroup ðV ðtÞÞtb0 is uniformly bounded, iii) si ðGÞ H 2pZ, Ðt iv) For every o A si ðGÞ the limit lim t!y t À1 eÀios V ðsÞf ds exists for every f A C; or RG ioị ỵ NG À ioÞ is dense in C for all o A si ðGÞ Then, all mild solutions of Eq (1) on ½0; yÞ are asymptotic 1-periodic Proof First, the existence of a 1-periodic mild solution uðÁÞ to Eq (1) is guaranteed by Condition (i) On the other hand, Condition (ii), (iii) and (iv) imply that the semigroup ðV ðtÞÞtb0 is asymptotic almost periodic by Theorem 3.21 in Appendix By Theorem 3.22 in Appendix there exists a decomposition of the space C ¼ C0 l C1 such that for every f A C0 lim t!y V tịf ẳ and C1 is ðV ðtÞÞtb0 -invariant and ðV ðtÞjC1 Þtb0 can be extended to a bounded group For every f A C1 denote ztị :ẳ V tịjC1 f Then, z is bounded and 323 Asymptotic Periodic Solutions uniformly continuous on R By Lemma 3.17, we have spðzÞ H si ðGÞ Hence, by Condition (iii) it is 1-periodic Consequentry ðV ðtÞÞtb0 is asymptotic 1periodic By Lemma 3.14, this shows that all mild solutions of Eq (1) should be the sum of the 1-periodic mild solution uðÁÞ and asymptotic 1-periodic mild solution of Eq (5) r To illustrate the above abstract results we will give below an example in which the conditions (ii) and (iii) in Prroposition 3.16 can be verified Example 3.19 Consider the equation 17ị < wt x; tị ẳ wxx x; tị ỵ wx; tị p=2ịwx; t 1ị ỵ f ðx; tÞ; a Ex a p; Et b 0; : w0; tị ẳ wp; tị ẳ 0; Et > 0; where wðx; tÞ; f ðx; tÞ are scalar-valued functions L ½0; pŠ and AT : X ! X by the formula ð18Þ We define the space X :ẳ 00 < AT y ẳ y ỵ y; DAT ị ẳ fy A X : y; y are absolutely continuous; y 00 A X; : y0ị ẳ ypị ẳ 0g: We dene F : C ! X by the formula F jị ẳ p=2ịj1ị evolution equation we are concerned with is the following 19ị dutị ẳ AT utị ỵ Fut ỵ f tị; dt In this case the uðtÞ A X; where AT is the infinitesimal generator of a compact and analytic semigroup ðTðtÞÞtb0 in X (see [42, p 414]) Moreover, the eigenvalues of AT are À n , n ¼ 1; 2; ; and since the set si ðDÞ is determined from the set of imaginary solutions of the equation ð20Þ l ỵ p=2el ẳ n ; n ¼ 1; 2; ; a simple computation shows si Dị ẳ fp=2; p=2g ẳ: L As is shown in [42, Lemma 5.8], ðG À lI ÞÀ1 has simple poles at L, where G is the generator of the solution semigroup ðV ðtÞÞtb0 The space C is decomposed as C ẳ NG ip=2ị l NG ỵ ip=2ị l QL , where QL ẳ RG ip=2ị V RG ỵ ip=2ị There exsist positive K and o such that kV ðtÞfk a KeÀot kfk for f A QL ; V tịf ẳ eGipt=2 f for f A NðG À ðGip=2ÞÞ Hence, ðV ðtÞÞtb0 is asymptotic 4periodic Let f ðx; tÞ be 4-periodic Then spð f Þ H pZ=2 (here we consider f as the function t 7! f tị :ẳ f ; tị A X) By our theory, it is necessary and 324 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito su‰cient for all mild solutions of Eq (19) to be asymptotic 4-periodic that the following equations are solvable ( é4 Dip=2ịu ẳ 14 eipt=2 f tịdt; é4 Dip=2ịu ẳ 14 e ipt=2 f ðtÞdt: Moreover, if u1 ; uÀ1 are solutions of equations in the above, respectively, then u1 e ipt=2 ỵ u1 eipt=2 ỵ X ikpt=2 e D ikp=2ị eÀikpt=2 f ðtÞdt k0G1 is the Fourier series of a 4-periodic mild solution of Eq (19) Appendix In this Appendix for the reader’s convenience, we collect some known notions and results on asymptotic almost periodic semigroups and the splitting Theorem of Glicksberg and DeLeeuw which we have used above (more details can be found in [35, Chap 5, § 7]) Definition 3.20 A C0 -semigroup ðTðtÞÞtb0 on X is said to be asymptotic almost periodic if for each x A X the set fTtịx; t A ẵ0; ỵyịg is relatively compact in X (Originally, in [35, Chap 5] this notion is referred to as the notion of almost periodic semigroups To distiguish this notion from our mentioned one we refer to it as the notion of asymptotic almost periodic semigroups.) Theorem 3.21 ([35, Theorem 5.7.10]) Let ðTðtÞÞtb0 be a uniformly bounded C0 -semigroup on a Banach space X, with generator A, and assume that sðAÞ V iR is countable Then the following assertions are equivalent: i) ðTðtÞÞtb0 is asymptotic almost periodic, Ðt ii) For every io A sðAÞ V iR the limit lim t!y t À1 eÀios TðsÞx ds exists for every x A X, iii) For every io A sAị V iR, RA ioị ỵ NðA À ioÞ is dense in X The following is referred to as the splitting Theorem of Glicksberg and DeLeeuw Theorem 3.22 ([35], Theorem 5.7.7]) Let ðTðtÞÞtb0 be an asymptotic almost periodic C0 -semigroup on a Banach space X Then there exists a direct sum decomposition X ¼ X0 l X1 of Ttịịtb0 -invariant subspaces, where n o X0 ẳ x A X : lim kTtịxk ẳ t!y Asymptotic Periodic Solutions 325 and X1 is the closed linear span of all eigenvectors of the generator A with purely imaginary eigenvalues Moreover, the restriction of ðTðtÞÞtb0 to X1 extends to an almost periodic C0 -group on X1 If ðTðtÞÞtb0 is contractive, this group is isometric References [ ] Arendt, W., Batty, C J K., Asymptotically almost periodic solutions of inhomogeneous cauchy problem on the half-line, Bull London Math Soc., 31 (1999), 291304 [ ] Arendt, W., Raăbiger, F., Sourour, A., Spectral properties of the operators equations AX þ XB ¼ Y , Quart J Math Oxford (2), 45 (1994), 133–149 [ ] Arendt, W., Batty, C J K., Hieber, M., Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Problems, Birkhaăser, Basel-Boston-Berlin, 2001 [ ] Bart, H., Goldberg, S., Characterizations of almost periodic strongly continuous groups and semigroups, Math Ann., 236 (1978), no 2, 105–116 [ ] Cassell, J S., Zhan Yuan Hou, Asymptotically diagonal linear diÔerential equations with retardation, J London Math Soc., (2) 47 (1993), no 3, 473–483 [ ] Chow, S N., Hale, J K., Strongly limit-compact maps, Funkc Ekvac., 17 (1974), 31–38 [ ] Daners, D., Medina, P K., Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Math Ser volume 279, Longman New York, 1992 [ ] Engel, K J., Nagel, R., One-parameter Semigroups for linear Evolution Equations Springer, Berlin, 1999 [ ] Fink, A M., Almost Periodic DiÔerential Equations, Lecture Notes in Math., 377, Springer, Berlin-New York, 1974 [10] Furumochi, T., Naito, T., Nguyen Van Minh, Boundedness and almost periodicity of solutions of partial functional diÔerential equations, J DiÔerential Equations, 180 (2002), 125152 [11] Gonzalez, C., Jimenez-Melado, A., Asymptotic behavior of solutions of diÔerence equations in Banach spaces, Proc Amer Math Soc., 128 (2000), no 6, 1743–1749 [12] Gyori, I., Krisztin, T., Oscillation results for linear autonomous partial delay diÔerential equations, J Math Anal Appl., 174 (1993), no 1, 204–217 [13] Haddock, J R., Nkashama, M N., Wu, J., Asymptotic constancy for pseudomonotone dynamical systems on function spaces, (English English summary) J DiÔerential Equations, 100 (1992), no 2, 292–311 [14] Hale, J K., Verduyyn Lunel, S M., Introdcution to Functional DiÔerential Equations, Springer-Verlag, 1993 [15] Hara, T., Miyazaki, R., Morii, T., Asymptotic behavior solutions for diÔerential-diÔerence equations with N delays and Hopf bifurcation, Funkcialaj Ekvacioj, 43 (2000), 193–212 [16] Hatvani, L., Asymptotic constancy of the solutions of a nonlinear delay equation (English) Sivasundaram, S (ed.) et al., Advances in nonlinear dynamics Langhorne, PA: Gordon and Breach Stab Control Theory Methods Appl., (1997), 63–70 [17] Hatvani, L., Krisztin, T., On the existence of periodic solutions for linear inhomogeneous and quasilinear functional diÔerential equations, J DiÔerential Equations, 97 (1992), 115 [18] Hino, Y., Murakami, S., Periodic solutions of a linear volterra system DiÔerential equations (Xanthi, 1987), 319–326, Lecture Notes in Pure and Appl Math., 118, Dekker, New York, 1987 326 Takeshi Nishikawa, Nguyen Van Minh and Toshiki Naito [19] Hino, Y., Murakami, S., Naito, T., Functional DiÔerential Equations with Innite Delay, Lecture Notes in Math., 1473, Springer-Verlag, Berlin-New York, 1991 [20] Hino, Y., Naito, T., Nguyen Van Minh, Shin, J S., Almost Periodic Solutions of DiÔerential Equations in Banach Spaces, Gordon and Breach, Amsterdam-London, 2001 [21] Hino, Y., Murakami, S., Naito, T., Nguyen Van Minh, A variation of constants formula for functional diÔerential equations in the phase space, J DiÔerential Equations, 179 (2002), 336355 [22] Langenhop, C., Periodic and almost periodic solutions of volterra integral diÔerential equations with innite memory, J DiÔerential Equations, 58 (1985), 391–403 [23] Levitan, B M., Zhikov, V V., Almost Periodic Functions and DiÔerential Equations, Cambridge University Press 1982 [24] Li, Y., Lin, Z., Li, Z., A Massera type criterion for linear functional diÔerential equations with advanced and delay J Math Anal Appl., 200 (1996), 715–725 [25] Li, Y., Cong, F., Lin, Z., Liu, W., Periodic solutions for evolution equations, Nonlinear Anal., 36 (1999), 275–293 [26] Massera, J L., The existence of periodic solutions of systems of diÔerential equations, Duke Math J., 17 (1952), 457–475 [27] Murakami, S., Naito, T., Nguyen Van Minh, Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spaces, J DiÔerential Equations, 164 (2000), 240285 [28] Murakami, S., Naito, T., Nguyen Van Minh, Massera’s theorem for almost periodic solutions of functional diÔerential equations, To appear in J Math Soc Japan [29] Murakami, K., Asymptotic constancy and periodic solutions for linear autonomous delay diÔerential equations, Funkcial Ekvac., 39 (1996), no 3, 519–540 [30] Murakami, K., Asymptotic constancy for systems of delay diÔerential equations Proceedings of the Second World Congress of Nonlinear Analysts, Part (Athens, 1996) Nonlinear Anal., 30 (1997), no 7, 4595–4606 [31] Naito, T., Nguyen Van Minh, Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J DiÔerential Equations, 152 (1999), 358376 [32] Naito, T., Nguyen Van Minh, Miyazaki, R., Shin, J S., A decomposition theorem for bounded solutions and the existence of periodic solutions to periodic diÔerential equations, J DiÔerential Equations, 160 (2000), 263282 [33] Naito, T., Nguyen Van Minh, Shin, J S., New spectral criteria for almost periodic solutions of evolution equations, Studia Mathematica, 145 (2001), 97–111 [34] Naito, T., Nguyen Van Minh, Liu, J., On the bounded solutions of volterra equations, Applicable Analysis To appear [35] van Neerven, J., The asymptotic behaviour of semigroups of linear operators Operator theory: Advances and Applications, 88 Birkhauser Verlag, Basel, 1996 [36] Pazy, A., Semigroups of Linear Operators and Applications to Partial DiÔerential Equations, Applied Math Sci., 44, Spriger-Verlag, Berlin-New York, 1983 [37] Pruăss, J., Bounded solutions of Volterra equations, SIAM Math Anal., 19 (1987), 133–149 [38] Pruăss, J., Evolutionary Integral Equations and Applications, Birkhaăuser, Basel, 1993 [39] Vu, Q P., The operator equation AX À XB ¼ C with unbounded operators A and B and related abstract Cauchy problems, Math Z., 208 (1991), no 4, 567–588 [40] Shin, J S., Naito, T., Semi-fredholm operators and periodic solutions for linear functional diÔerential equations, J DiÔerential Equations, 153 (1999), 407–441 [41] Shin, J S., Naito, T., Nguyen Van Minh, Existence and uniqueness of periodic solutions to periodic linear functional diÔerential equations with nite delay, Funkcial Ekvac., 153 Asymptotic Periodic Solutions 327 (1999), 407–441 [42] Travis, C C., Webb, G F., Existence and stability for partial functional diÔerential equations, Trans Amer Math Soc., 200 (1974), 394–418 [43] Wu, J., Theory and Applications of Partial Functional DiÔerential Equations, Applied Math Sci., 119, Springer, Berlin-New York, 1996 [44] Yoshizawa, T., Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Math., Sciences 14, Springer-Verlag, New York, 1975 nuna adreso: Takeshi Nishikawa Department of Computer Science and Information Mathematics University of Electro-Communications Chofu, Tokyo 182-8585 Japan E-mail: takeshi0324@proof.ocn.ne.jp Nguyen Van Minh Department of Mathematics Hanoi University of Science Hanoi, Vietnam E-mail: nvminh@netnam.vn Toshiki Naito Department of Mathematics University of Electro-Communications Chofu, Tokyo 182-8585 Japan E-mail: naito@e-one.uec.ac.jp (Ricevita la 15-an de aprilo, 2003) (Reviziita la 18-an de septembro, 2003) ... Naito conditions for the inhomogeneous equations to have at least one periodic solution, and for all solutions of the corresponding homogeneous equations to be (asymptotic) periodic solutions Our... dropping this condition in the recent paper [28] Let us consider conditions for all mild solutions of Eq (1) to be 1periodic We first consider conditions for all mild solutions of the equations without... functions (see [1, 3, 23, 38] and the references therein for more details) The applications of the method of sums of commuting operators into the study of almost periodic solutions of functional