DSpace at VNU: Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows

This article was downloaded by: [Colorado College] On: 29 October 2014, At: 16:12 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Applicable Analysis: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gapa20 Discrete and continuous versions of Barbashin-type theorem of linear skewevolution semiflows Pham Viet Hai a a Department of Mathematics , College of Science, Vietnam National University , Hanoi, Vietnam Published online: 23 Feb 2011 To cite this article: Pham Viet Hai (2011) Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows, Applicable Analysis: An International Journal, 90:12, 1897-1907, DOI: 10.1080/00036811.2010.534728 To link to this article: http://dx.doi.org/10.1080/00036811.2010.534728 PLEASE SCROLL DOWN FOR 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systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Applicable Analysis Vol 90, No 12, December 2011, 1897–1907 Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows Pham Viet Hai* Department of Mathematics, College of Science, Vietnam National University, Hanoi, Vietnam Downloaded by [Colorado College] at 16:12 29 October 2014 Communicated by R.P Gilbert (Received 12 July 2010; final version received 19 October 2010) This article is concerned with a well-known theorem of Barbashin which states that an evolution family fU(t, s)}t!s!0, or simple U, is uniformly exponentially stable if and only if U satisfies the integral condition  R t supt!0 Uðt, Þd In fact, the author formulated the above result for non-autonomous differential equations in the frame work of finitedimensional spaces The aim of this article is to give discrete and continuous versions of Barbashin-type theorem for the case linear skewevolution semiflows Giving up disadvantages in Barbashin’s proof, we shall extend this problem, based on the recent methods Thus we obtain necessary and sufficient conditions for uniform exponential stability, generalizing a classical stability theorem due to Barbashin Keywords: exponential stability; Barbashin theorem; linear skew-evolution semiflows AMS Subject Classifications: 34D05; 34D20 Introduction The understanding of the asymptotic behaviour of evolution equations is one of the most important problems of modern mathematical analysis There are many ways to study the problem: input-output criterions, discrete-time methods, the Dato– Pazy–Rolewicz theorem, and so on The results of this area have become increasingly important We shall abbreviate research works, which appear in recent times For more details about these results, we can see references The earliest study on the input-output method or the so-called admissibility may be [1], which is concerned with the problem of conditional stability of a system x0 ¼ A(t)x and its connection with the existence of bounded solutions of the equation x0 ẳ A(t)x ỵ f(t) After the seminal researches of Perron, there have been a great number of works devoted to this problem, such as [1–9] For the case of discrete-time systems analogous results were first obtained in 1934 by Ta Li [5] A nice proof for *Email: phamviethai86@gmail.com ISSN 0003–6811 print/ISSN 1563–504X online ß 2011 Taylor & Francis http://dx.doi.org/10.1080/00036811.2010.534728 http://www.tandfonline.com Downloaded by [Colorado College] at 16:12 29 October 2014 1898 P.V Hai Ta Li’s result is presented in [6] Perron’s ideas have been successfully extended by Massera, Schaăffer and by Daleckii, Krein, respectively, in infinite-dimensional spaces, [7,8] Especially, Latushkin and Schnaubelt established the relation between the exponential dichotomy of a strongly continuous cocycle over a flow and the dichotomy of the associated discrete cocycle, employing an evolution semigroup technique, [4] The authors extended some important theorems in the field of evolution families, proving that the uniform exponential dichotomy of a linear skew-product semiflow is equivalent to the hyperbolicity of its evolution semigroup on C0(€, X) This result can be interpreted as a generalization of a dichotomy theorem due to Minh, Raăbiger and Schnaubelt, [2] A significant step has been made by Henry in [10] The author characterized the dichotomy of a sequence of bounded linear operators (Tn)n2Z in terms of the existence and uniqueness of bounded solutions for xnỵ1 ẳ Tnxn ỵ fn, for every bounded sequence ( fn)n2Z Moreover, the author showed the relation between the discrete dichotomy and the exponential dichotomy for an evolution family Another approach was given in [11] where, Novo and Obaya constructed continuous separation of state spaces on the compact positively invariant subset M under assumptions that skew-product semiflows are eventually strongly monotone, which has consanguineous relations with the exponential dichotomy of linear skew-product semiflows Recently, a great number of articles about the Dato–Pazy–Rolewicz theorem were published, [10,12–23] This theorem was the starting point for outstanding results concerning the exponential stability A new idea has been presented by Preda, Pogan and Preda, [12] The authors characterized the uniform exponential stability of evolution families in terms of the existence of some functionals on sequence R1 (function) In fact, these functionals are generalizations of the integration Pspaces or series j¼0 This interesting idea provides us a way to attack Barbashin’s theorem For more details about this result, we refer to [24] This article is orgainized as follows In Section 2, for the reader’s convenience, we recall some concepts and results on linear skew-evolution semiflows Section is devoted to the proof of main results First, the discrete version of Barbashin’s theorem is proved And then we prove Barbashin’s theorem by using the discrete version Thus, we obtain a connection between the discrete version and the continuous version Notations and preliminaries Let X be a Banach space, L(X) the Banach algebra of all bounded linear operators acting on X, (€, d ) a metric space The norm on X, L(X) will be denoted by kk and T :ẳ ft, sị R2ỵ , t ! s ! 0g Definition 2.1 The mapping  : T  € ! € is called an evolution semiflow on € if: (1) (t, t, ) ¼  for all , t (2) (t, s, (s, r, )) ¼  (t, r, ) for all t ! s ! r ! 0,  € (3)  is continuous Given an evolution semiflow, the linear skew-evolution semiflow can be defined as follows [18,19,25] 1899 Applicable Analysis Definition 2.2 A pair  ¼ (È, ) is called a linear skew-evolution semiflow on E :¼ X  € if  is an evolution semiflow on € and È : T  € ! L(X) has the following properties: (1) È(t, t, ) ¼ I, the identity operator on X for all (t, ) Rỵ (2) È(t, r, ) ¼ È(t, s, (s, r, ))È(s, r, ) for all t ! s ! r ! and  € (3) There are M, !40 such that kẩ(t ỵ s, s, )xk Me!t kxk t, s, , xị R2ỵ X for all Remark Downloaded by [Colorado College] at 16:12 29 October 2014 (1) The mapping È given Definition 2.2 is called the cocycle associated to the linear skew-evolution semiflow  (2) In what follows, we shall denote by M, ! the constants defined in Definition 2.2 Example 2.3 (1) One can easily check that C0-semigroups, evolution families and linear skewproduct semiflows are particular cases of linear skew-evolution semiflows (2) If  ¼ (È, ) is a linear skew-evolution semiflow on E then for every R the pair  ¼ (È, ), where È(t, s, ) ¼ eÀ(tÀs)È(t, s, ), is also a linear skewevolution semiflow on E (3) Let € be a compact metric space,  an evolution semiflow on € and A : € ! L(X) a continuous mapping If È(t, t0, )x is the solution of the Cauchy problem u0 tị ẳ At, t0 , ịịutị, t ! t0 , then the pair  ¼ (È, ) is a linear skew-evolution semiflow We denote by Cs(€, L(X)) the space of all strongly continuous bounded mapping H : € ! L(X), which is a Banach space with respect to the norm   kHk :ẳ sup Hị: 2 THEOREM 2.4 Let  ¼ (È, ) be a linear skew-evolution semiflow on E If P Cs(€, L(X)), there is an unique linear skew-evolution semiflow P ¼ (ÈP,  P) on E such that Z ẩP t, s, ịx ẳ ẩt, s, ịx ỵ t ẩt, , , s, ịịP, s, ÞÞÈP ð, s, Þx d ð1Þ s for t ! s ! and (, x) €  X Proof First, we shall show that for every  € and t ! s ! 0, the integral equation (1) has a solution which is a bounded linear operator on X Therefore, we define the sequence < ẩ0 t, s, ịx ẳ ẩt, s, ịx, Zt : ẩnỵ1 t, s, ịx ẳ ẩt, , , s, ịịP, s, ÞÞÈn ð, s, Þx d: s 1900 P.V Hai One can easily check that n   Èn ðt, s, Þ Me!ðtÀsÞ ðMkPkðt À sÞÞ : n! ð2Þ It makes sense to define ẩP t, s, ị :ẳ X ẩn t, s, ị, nẳ0 Downloaded by [Colorado College] at 16:12 29 October 2014 for every , t ! s So ÈP(t, t, ) ¼ I, for all , t Using (2), we have for every , t ! s, ÈP(t, s, ) L(X) and   ÈP t, s, ị Me!ỵMkPkịtsị : Moreover, ẩP t, s, ịx ẳ ẩt, s, ịx ỵ ẳ ẩt, s, ịx ỵ X ẩn t, s, ịx nẳ1 Z t X Èðt, , ð, s, ÞÞPðð, s, ÞÞÈnÀ1 ð, s, ịx d nẳ1 Zt s ẩt, , , s, ịịP, s, ịịẩP , s, ịx d: ẳ ẩt, s, ịx þ s It is easy to show that ÈP verifies the cocycle identity Finally, we prove the uniqueness Suppose that È0P is a cocycle which verifies the conditions of Theorem 2.4 Then we have Zt     ÈP ðt, s, Þx À È0 ðt, s, Þx MkPke!ðtÀÞ ÈP ð, s, Þx À È0P ð, s, Þxd: P s From Gronwall’s lemma, it follows that ÈP ¼ È0P g COROLLARY 2.5 (1) If P Cs (€, L(X)),  is an evolution semiflow and (T(t))t!0 is a C0-semigroup, there is a unique linear skew-evolution semiflow P ¼ (ÈP,  P) on E such that Zt ÈP ðt, s, ịx ẳ Tt sịx ỵ Tt ịP, s, ÞÞÈP ð, s, Þx d s for t ! s ! and (, x) €  X (2) If P Cs(€, L(X)),  is an evolution semiflow and (U(t, s))t!s!0 is an evolution family, there is a unique linear skew-evolution semiflow P ¼ (ÈP,  P) on E such that Zt Uðt, ÞPðð, s, ÞÞÈP ð, s, ịx d ẩP t, s, ịx ẳ Ut, sịx ỵ s for t ! s ! and (, x) €  X Applicable Analysis 1901 Definition 2.6 The linear skew-evolution semiflow  ¼ (È, ) is said to be uniformly exponentially stable if there are K40 and 40 such that   ẩt ỵ s, s, ịx Ket kxk, for all t, s, , xị R2ỵ €  X A condition for the uniform exponential stability of linear skew-evolution semiflows is given by the following lemma LEMMA 2.7 If there are two constants p and c (0, 1) such that   ẩ p ỵ m, m, Þ c, Downloaded by [Colorado College] at 16:12 29 October 2014 for all (, m) €  N, then the linear skew-evolution semiflow  ¼ (È, ) is uniformly exponentially stable Proof g See [18] Main results As said above, to investigate the problem of Barbashin, P we use the method introduced in [12,20] That is, we shall generalize the series j¼0 to obtain a class of functionals We next use the discrete-time method to find the discrete version of Barbashin’s theorem and then convert the result to the continuous version Throughout this section, we shall denote S ỵ (Mỵ) the set of all positive sequences (functions) and s1 s2 if s1(j) s2(j), for every j N or Rỵ N is the set of all nondecreasing functions b : Rỵ ! Rỵ with the property b(t)40, for all t40 We need the following notion Definition 3.1 (1) F(s1) (2) H(N) is the set of all functions F : S ỵ ! [0, 1] with the property F(s2) provided that s1 s À Á F X f0, ,ng ¼ 1: n!1   lim inf Remark (1) If F1, F2 H(N) then F1 ỵ F2 H(N) for every , 40 (2) If F1 F2, F1 H(N) and F2 satisfies the condition (1) of Definition 3.1 then F2 H(N) Example 3.2 Let F1, F2 : S ỵ ! [0, 1] be maps defined by F1 sị :ẳ X s j ị, jẳ0 F2 sị :ẳ Y ỵ s j ịị: jẳ0 1902 P.V Hai One can check that F1, F2 H(N) Definition 3.3 (1) G(s1) (2) H(Rỵ) is the set of all functionals G : Mỵ ! [0, 1] with the property G(s2) provided that s1 s2 G X ẵ0,n lim inf ẳ 1: n!1   Remark Downloaded by [Colorado College] at 16:12 29 October 2014 (1) If G1, G2 H(Rỵ), then G1 ỵ G2 H(Rỵ) for every , 40 (2) If G1 G2, G1 H(Rỵ) and G2 satisfies the condition (1) of Definition 3.3, then G2 H(Rỵ) Example 3.4 The mapping Z G f ị :ẳ f ị d belongs to H(Rỵ) We start with the discrete-time version of Barbashin’s theorem THEOREM 3.5 The linear skew-evolution semiflow  is uniformly exponentially stable if and only if there exist F H(N), K40, b N and a non-decreasing sequence (tn) & Rỵ such that sup Fð’b ð, m, n, :ÞÞ K, n, m, N,  € where Á & À b ẩm ỵ tn , m ỵ tj , m ỵ tj , m, ÞÞ , for j f0, , ng, 0, for j 2= f0, , ng: P Proof Necessity Let F H(N) be the functional defined by Fsị ẳ nẳ0 snị, tj ẳ j and b(t) ẳ t Definition 2.6 guarantees that there are K, 40 such that kÈ(t, s, )k KeÀ(tÀs) It follows that the uniform boundedness is well-defined from inequalities b , m, n, j ị ẳ Fb , m, n, :ịị ẳ n  X  ẩm þ n, m þ j, ðm þ j, m, ÞÞ jẳ0 n X jẳ0 Kenj ị ẳ n X jẳ0 KeÀj X KeÀj 1: j¼0 Sufficiency To prove the converse, we divide the proof into two steps Step Let us first prove the uniform boundedness of È(m þ n, m, ) This means, we need to show that there is L satisfying the inequality   sup ẩm ỵ tn , m, ị L 1: n, m,  1903 Applicable Analysis From the condition (2) of Definition 3.1, there is k N* satisfying the condition À Á K F X f0, , kg ! bð1Þ for every  Rỵ We consider two cases as follows If n   ẩm ỵ tn , m, Þ Me!tk : k then Downloaded by [Colorado College] at 16:12 29 October 2014 If n ! k then taking j {0, , k} randomly, one can easily check     ẩm ỵ tn , m, ị ẳ ẩm ỵ tn , m ỵ tj , m ỵ tj , m, ịịẩm ỵ tj , m, ị    ẩm ỵ tn , m ỵ tj , m ỵ tj , m, ịịẩm þ tj , m, Þ   Me!tk Èðm þ tn , m ỵ tj , m ỵ tj , m, ịị,   ẩm ỵ tn , m, ị ’b ð, m, n, :Þ ! b X f0, ,kg , Me!tk      ẩm ỵ tn , m, Þ X K!F b f0, ,kg Me!tk   ẩm ỵ tn , m, ị K , !b Me!tk b1ị which implies   ẩm ỵ tn , m, ị Me!tk : The uniform boundedness of ẩ(m ỵ tn, m, ) is proved since we only take L ! Me!tk Step We prove that the conditions of Lemma 2.7 works Using the condition (2) of Definition 3.1 again, we get the natural number r such that À Á K F X f0, , rg ! : bð2LÞ For m N and j {0, , r}, it is clear that     ẩm ỵ tr , m, ị ẳ ẩm ỵ tr , m ỵ tj , m ỵ tj , m, ịịẩm ỵ tj , m, ị    ẩm ỵ tr , m ỵ tj , m ỵ tj , m, ịịẩm ỵ tj , m, ị   Lẩm ỵ tr , m ỵ tj , m ỵ tj , m, ịị,   ẩm þ tr , m, Þ X f0, , rg , ’b ð, m, r, :Þ ! b L     ẩm ỵ tr , m, ị X f0, , rg K!F b L   ẩm ỵ tr , m, Þ K !b : L Þ bð2L 1904 P.V Hai This is enough to show that   ẩm ỵ tr , m, ị : Applying Lemma 2.7, we obtain the uniform exponential stability of  COROLLARY 3.6 equivalent: g Let U be an evolution family The following statements are (1) U is uniformly exponentially stable   Pn  (2) There is b N such that supn N P j ÞÞ 1:  j¼0 bð Uðn,  n   (3) There is b N such that supn, m N jẳ0 b Un ỵ m, j ỵ mị ị 1: Downloaded by [Colorado College] at 16:12 29 October 2014 Proof (1) ¼4(2) It is a simple exercise, for b(t) ¼ t (2) ¼4(3) Indeed, for n, m N, we have the inequality n X   bðUðn ỵ m, j ỵ mịị jẳ0 nX ỵm   bUn ỵ m, j ịị, jẳ0 Pn   bUn þ m, j þ mÞÞ this yields supn, m N j¼0 (3) ¼4(1) Putting È(t, s, ) ¼ U(t, s) and (t, s, ) ¼ , for every  and t ! s, one can easily check that  ¼ (È, ) is a linear skew-evolution semiflow and  &  b Um ỵ n, m ỵ j Þ , for j f0, , ng, b , m, n, j ị ẳ 0, for j 2= f0, , ng, Thus we have X ’b ð, m, n, j Þ ẳ n ỵn  X X  m   b Um ỵ n, m ỵ j ị ẳ b Um ỵ n, j ị jẳ0 jẳm jẳ0 m ỵn X   b Um ỵ n, j ị : j¼0 P1 It follows supm, n,  j¼0 ’b ð, m, n, j Þ 5P Applying Theorem 3.5 for Fsị ẳ nẳ0 snị, the proof is complete COROLLARY 3.7 g An evolution family U is uniformly exponentially stable if and only if sup n À Y  Á ỵ Un ỵ m, j ỵ mị 1: n, m, N j¼0 Proof The necessity can be followed from the inequality n À Y j¼0  Á ỵ Un ỵ m, j ỵ mị n Y ekUnỵm,jỵmịk ẳ e Pn jẳ0 kUnỵm, jỵmịk : jẳ0 From Theorem 3.5, we can follow the sufficiency, using Fsị ẳ Q1 jẳ0 ỵ s j ịị g 1905 Applicable Analysis Now, we give a characterization of the uniform exponential stability of linear skew-evolution semiflows, which generalizes the well-known theorem of Barbashin THEOREM 3.8 The linear skew-evolution semiflow  is uniformly exponentially stable if and only if there exist G H(Rỵ), b N , K40 such that sup Gð b ð, m, n, :ÞÞ K, m, n,  where Downloaded by [Colorado College] at 16:12 29 October 2014  &  b ẩm ỵ n, m ỵ , m ỵ , m, ịị , b , m, n, ị ẳ 0, for  ẵ0, n, for  2= ẵ0, n: Proof RNecessity Let G : Mỵ ! [0, 1] be the mapping defined by Gð f Þ ¼ f ðÞ d and b(t) ¼ t From Definition 2.6, there are K, 40 such that kÈ(t, s, )k KeÀ(tÀs) Hence we get inequalities Z Gð b , m, n, :ịị ẳ n  ẩm ỵ n, m ỵ , m ỵ , m, ịịd Z0n Zn Z1 Kenị d ẳ Ke d Ke d 1: 0 Sufficiency For t Rỵ, we put tị :ẳ b Met ! , then N Also for s S ỵ let fs : Rỵ ! Rỵ be the mapping given by fs() ẳ s([]) and FG : S ỵ ! [0, 1] the functional defined by FG(s) :¼ G( fs) Using the fact that G H(Rỵ), one can easily verify that FG H(N) and by observing that   Èðm ỵ n, m ỵ ẵ, m ỵ ẵ, m, ịị    ẩm ỵ n, m ỵ , m ỵ , m, ịịẩm ỵ , m ỵ ẵ, m ỵ ẵ, m, ịị   Me!ẵị ẩm ỵ n, m ỵ , m ỵ , m, ịị   Me! ẩm ỵ n, m ỵ , m ỵ , m, ÞÞ for every  [0, n] It follows that   ẩm ỵ n, m ỵ ẵ, m ỵ ẵ, m, ịị , m, n, ị ! b b Me!   ẳ ẩm ỵ n, m ỵ ẵ, m ỵ ẵ, m, ịịị ẳ f , m, n, :Þ ðÞ, which implies that b ð, m, n, :Þ K ! Gð b ð, m, n, :ÞÞ ! f’ ð, m, n, :Þ ðÁÞ, À Á ! G f , m, n, :ị ị ẳ FG , m, n, :ÞÞ: This proves the uniform boundedness of FG(’ (, m, n, )) Applying Theorem 3.5 we obtain the uniform exponential stability of  g COROLLARY 3.9 equivalent Let U be an evolution family The following statements are (1) U is uniformly exponentially stable 1906 P.V Hai Rt (2) There is b N such that supt RỵR bðkUðt, ÞkÞ d n (3) There is b N such that supn N 0R bðkUðn, ÞkÞ d n (4) There is b N such that supn, m N bkUm ỵ n, m ỵ ịkị d Proof (1) ẳ4(2) One can easily check that b(t) ¼ t (2) ¼4(3) Zn Zt     bðUðn, ÞÞ d sup bUt, ịị d 1: t Rỵ Downloaded by [Colorado College] at 16:12 29 October 2014 Hence we get (3) (3) ẳ4(4) From Zn   bUm ỵ n, m ỵ ịị d Z mỵn   bUm ỵ n, ịị d, we obtain Z n sup n, m N   bUn ỵ m,  ỵ mịị d 1: (4) ẳ4(1) By a similar argument to Corollary 3.6, we have Á &  b Um ỵ n, m ỵ ị for  ẵ0, n, b , m, n, ị ẳ for  2= ½0, nŠ, and Z sup n, m N Z b ð, m, n, Þ d ¼ sup Applying Theorem 3.7 for Gð f Þ ¼ n, m N R1 n Á  b Um ỵ n, m ỵ ị d 1: f ðÞ d, the proof is complete g Acknowledgements The author is grateful to the referees for carefully reading this article and for their valuable comments The author was partially supported by the grant TN-10-08 of College of Science, Vietnam National University, Hanoi (Dai hoc Khoa hoc TU Nhien, Dai hoc Quoc Gia Ha Noi) References [1] O Perron, Die stabilitatsfrage bei differentialgleichungen, Math Z 32 (1930), pp 703728 [2] N.V Minh, F.R Raăbiger, and R Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equ Oper Theory 32 (1998), pp 332–353 [3] N.T Huy and N.V Minh, Exponential Dichotomy of Difference Equations and Applications to Evolution Equations on the Half-Line, Comp Math Appl 42 (2001), pp 301–311 Downloaded by [Colorado College] at 16:12 29 October 2014 Applicable Analysis 1907 [4] Y Latushkin and R Schnaubelt, Evolution semigroups, translation algebras and exponential dichotomy of cocycles, J Diff 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Barbashin-type theorem of linear skew-evolution semiflows Pham Viet Hai* Department of Mathematics, College of Science, Vietnam National University,... differential equations in the frame work of finitedimensional spaces The aim of this article is to give discrete and continuous versions of Barbashin-type theorem for the case linear skewevolution semiflows. .. concepts and results on linear skew-evolution semiflows Section is devoted to the proof of main results First, the discrete version of Barbashin’s theorem is proved And then we prove Barbashin’s theorem