On Uniform Exponential Trichotomy of Evolution Operators in Banach Spaces

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1 On Uniform Exponential Trichotomy of Evolution Operator in Banach Space Mihail Megan, Codruta Stoica To cite thi verion: Mihail Megan, Codruta Stoica. On Uniform Exponential Trichotomy of Evolution Operator in Banach Space. Integral Equation and Operator Theory, Springer Verlag, 2008, 60 (4), pp < / z>. <hal > HAL Id: hal Submitted on 29 May 2008 HAL i a multi-diciplinary open acce archive for the depoit and diemination of cientific reearch document, whether they are publihed or not. The document may come from teaching and reearch intitution in France or abroad, or from public or private reearch center. L archive ouverte pluridiciplinaire HAL, et detinée au dépôt et à la diffuion de document cientifique de niveau recherche, publié ou non, émanant de établiement d eneignement et de recherche françai ou étranger, de laboratoire public ou privé.

2 On Uniform Exponential Trichotomy of Evolution Operator in Banach Space Mihail Megan Codruţa Stoica Abtract. Thi paper preent neceary and ufficient condition for uniform exponential trichotomy of nonlinear evolution operator in Banach pace. Thu are obtained reult which extend well-known reult for uniform exponential tability in the linear cae. Mathematic Subject Claification: Primary 34D05, 34D09, 93D20 hal , verion 1-29 May 2008 Keyword: Nonlinear evolution operator, exponential tability, exponential dichotomy, exponential trichotomy 1 Introduction One of the mot notable reult in the theory of tability of linear evolution operator ha been proved by Datko in [2]. Generalization of thi reult were obtained in [1], [5], [7] and [14] for exponential tability, in [9] and [11] for exponential intability and in [8], [10] and [15] for the cae of exponential dichotomy. In thi paper we hall extend thee reult in two direction. Firt, we hall conider the cae of uniform exponential trichotomy property ([3], [4], [6], [13]) and econd, we hall not aume the linearity of evolution operator. A unified treatment for uniform aymptotic behavior (exponential decay, exponential growth, exponential tability, exponential intability, exponential dichotomy, exponential trichotomy) of nonlinear evolution operator i given. Example that motivate the extenion of the aymptotic behavior for the nonlinear cae are given in [5]. In our paper we obtain ome theorem which extend well-known reult for uniform exponential tability etablihed in the linear cae. Let X be a real or complex Banach pace. The norm on X will be denoted by. The et of all mapping from X into itelf i denoted by F(X). Let T be the et of all pair (t,t 0 ) of real number with the property t t

3 2 Evolution operator Definition 2.1 A mapping E : T F(X) i called evolution operator on X if it ha the property E(t,)E(,t 0 ) = E(t,t 0 ), (t,),(,t 0 ) T. (2.1) In order to emphaize the neceity of extending the tudy of evolution operator in the nonlinear etting, we will conider the next Example 2.1 Let u conider the Cauchy problem { v(t) = Av(t), t > 0 v(0) = v 0 on a Banach pace X with nonlinear operator A. If A generate a nonlinear trongly continuou emigroup (S(t)) t 0, then E(t,) = S(t ), where t 0, define an evolution operator on X. Definition 2.2 The evolution operator E : T F(X) i aid to be with (i) uniform exponential decay if there exit M > 1 and ω > 0 uch that E(,t 0 )x Me ω(t ) E(t,t 0 )x (2.2) for all t t 0 0 and all x X; (ii) uniform exponential growth if there are M > 1 and ω > 0 uch that E(t,t 0 )x Me ω(t ) E(,t 0 )x (2.3) Lemma 2.1 The evolution operator E : T F(X) ha uniform exponential decay if and only if there exit a nondecreaing function f : [0, ) (1, ) with the property lim t f(t) = uch that E(,t 0 )x f(t ) E(t,t 0 )x Proof. Neceity. It follow from Definition 2.2 (i) for f(t) = Me ωt. Sufficiency. If t t 0 0 then there exit a natural number n uch that n t < n + 1. If we denote M = f(1) and ω = lnm, then by hypothei we have E(,t 0 )x M E( + 1,t 0 )x M 2 E( + 2,t 0 )x 2

4 M n E( + n,t 0 )x M n+1 E(t,t 0 )x = = Me nω E(t,t 0 )x Me ω(t ) E(t,t 0 )x Finally, we conclude that E ha exponential decay. Lemma 2.2 The evolution operator E : T F(X) ha uniform exponential growth if and only if there exit a nondecreaing function g : [0, ) (1, ) with the property lim t g(t) = uch that E(t,t 0 )x g(t ) E(,t 0 )x Proof. It i imilar with the proof of Lemma Uniform exponential trichotomy of evolution operator Let E be an evolution operator on the Banach pace X. Definition 3.1 An application P : R + F(X) i aid to be a projection family on X if P(t) 2 = P(t), t R +. (3.1) Definition 3.2 Three projection familie P 0,P 1,P 2 : R + F(X) are aid to be compatible with the evolution operator E : T F(X) if (ct 1 ) P 0 (t) + P 1 (t) + P 2 (t) = I, t 0 (ct 2 ) P i (t)p j (t) = 0, i,j {0,1,2}, i j, t 0 (ct 3 ) E(t,t 0 )P k (t 0 ) = P k (t)e(t,t 0 ), (t,t 0 ) T and k {0,1,2}. In what follow we will denote for all (t,t 0 ) T and k {0,1,2}. E k (t,t 0 ) = E(t,t 0 )P k (t 0 ) = P k (t)e(t,t 0 ) Remark 3.1 If E i an evolution operator on X, then E 0, E 1 and E 2 are alo evolution operator on X, fact proved by the following relation E k (t,)e k (,t 0 ) = E(t,)P k ()E(,t 0 )P k (t 0 ) = = P k (t)e(t,t 0 )P k (t 0 ) = E k (t,t 0 ), for all t t 0 0 and k {0,1,2}. 3

5 Definition 3.3 An evolution operator E : T F(X) on a Banach pace X i aid to be uniformly exponentially trichotomic if there exit N 0,N 1,N 2 > 1, ν 0,ν 1,ν 2 > 0 and three projection familie P 0,P 1 and P 2 compatible with E uch that (uet 0 ) E 0 (,t 0 )x N 0 e ν 0(t ) E 0 (t,t 0 )x N 2 0 e2ν 0(t ) E 0 (,t 0 )x (uet 1 ) e ν 1(t ) E 1 (t,t 0 )x N 1 E 1 (,t 0 )x (uet 2 ) e ν 2(t ) E 2 (,t 0 )x N 2 E 2 (t,t 0 )x Remark 3.2 For P 0 = 0 we obtain the property of uniform exponential dichotomy for evolution operator tudied in [8], [10] and [15]. It i obviou that if the evolution operator E i uniformly exponentially dichotomic then it i uniformly exponentially trichotomic. Remark 3.3 If P 0 = P 2 = 0 the property of uniform exponential tability i obtained, a in [1], [2], [7] and [12]. It follow that a uniformly exponentially table evolution operator i uniformly exponentially dichotomic and, further, uniformly exponentially trichotomic. Remark 3.4 Without any lo of generality, in Definition 3.4 we can uppoe that N 0 = N 1 = N 2 = N and ν 1 = ν 2 = ν becaue otherwie we can conider N = max{n 0,N 1,N 2 } and ν = min{ν 1,ν 2 }. Example 3.1 Let u conider X = R 3 with the norm (x 1,x 2,x 3 ) = x 1 + x 2 + x 3, x = (x 1,x 2,x 3 ) X. Let ϕ : R + (0, ) be a decreaing continuou function with the property that there exit lim t ϕ(t) = l > 0. Then the mapping E : T F(X) defined by E(t,t 0 )x = (e ϕ(τ)dτ t ϕ(τ)dτ t 0 x1, e t 0 x2, e (t t 0)ϕ(0)+ ϕ(τ)dτ t 0 x3 ) i an evolution operator on X. Let u conider the projection defined by P 1 (t)(x 1,x 2,x 3 ) = (x 1,0,0) P 2 (t)(x 1,x 2,x 3 ) = (0,x 2,0) P 3 (t)(x 1,x 2,x 3 ) = (0,0,x 3 ). for all t 0 and all x = (x 1,x 2,x 3 ) X. 4

6 Following relation hold E(t,t 0 )P 1 (t 0 )x) e l(t ) E(,t 0 )P 1 (t 0 )x) E(t,t 0 )P 2 (t 0 )x) e l(t ) E(,t 0 )P 2 (t 0 )x) E(t,t 0 )P 3 (t 0 )x) e ϕ(0)(t ) E(,t 0 )P 3 (t 0 )x) E(t,t 0 )P 3 (t 0 )x) e ϕ(0)(t ) E(,t 0 )P 3 (t 0 )x) We conclude that E i uniformly exponentially trichotomic. Theorem 3.1 Let E : T F(X) be an evolution operator on the Banach pace X with the property that there exit three projection familie P 0,P 1 and P 2 compatible with E. Then E i uniformly exponentially trichotomic if and only if there exit two nondecreaing function f, g : [0, ) (1, ) with the property lim f(t) = lim g(t) = t t uch that (uet 0 ) E 0(,t 0 )x f(t ) E 0 (t,t 0 )x f 2 (t ) E 0 (,t 0 )x (uet 1 ) g(t ) E 1(t,t 0 )x E 1 (,t 0 )x (uet 2 ) g(t ) E 2(,t 0 )x E 2 (t,t 0 )x Proof. Neceity. A the evolution operator E : T F(X) i uniformly exponentially trichotomic it follow from Definition 3.3 that there exit three projection familie P 0,P 1 and P 2 compatible with E uch that E 0 ha uniform exponential growth and uniform exponential decay, E 1 i uniformly exponentially table and E 2 i uniformly exponentially intable. According to Lemma 2.1 and Lemma 2.2 there exit a nondecreaing function f : [0, ) (1, ) with the property uch that and lim f(t) = t E 0 (,t 0 )x f(t ) E 0 (t,t 0 )x E 0 (t,t 0 )x f(t ) E 0 (,t 0 )x Hence (uet 0 ) i proved. By a imilar proof a in Lemma 2.1 one can characterize the propertie of uniform exponential tability for E 1 and uniform exponential intability 5

7 for E 2 (ee [12]) by mean of a nondecreaing function g : [0, ) (1, ) with the property lim t g(t) = uch that repectively g(t ) E 1 (t,t 0 )x E 1 (,t 0 )x g(t ) E 2 (,t 0 )x E 2 (t,t 0 )x for all t t 0 0 and all x X, which complete the proof of (uet 1 ) and (uet 2 ). Sufficiency. According to Lemma 2.1 and Lemma 2.2, the two inequalitie of tatement (uet 0 ) imply that E 0 ha exponential decay and exponential growth. The inequality (uet 1 ) characterize the property of uniform exponential tability for E 1 and (uet 2 ) how that E 2 i uniformly exponentially untable, a in [12]. Thu, according to Definition 3.3, E i uniformly exponentially trichotomic. Definition 3.4 The evolution operator E : T F(X) i aid to be trongly meaurable if for every (t 0,x) R + X the mapping t E(t,t 0 )x i meaurable. Theorem 3.2 Let E : T F(X) be an evolution operator on the Banach pace X with the property that there exit three projection familie P 0,P 1 and P 2 compatible with E uch that the evolution operator E 1 and E 2 are trongly meaurable. Then E i uniformly exponentially trichotomic if and only if (i) E 0 and E 1 have uniform exponential growth; (ii) E 0 and E 2 have uniform exponential decay; (iii) there exit M 1 uch that following inequalitie hold and E 1 (τ,t 0 )x dτ M E 1 (,t 0 )x (3.2) E 2 (τ,t 0 )x dτ M E 2 (t,t 0 )x (3.3) Proof. The property of uniform exponential trichotomy i equivalent with the exitence of three projection familie P 0,P 1 and P 2 compatible with E uch that E 0 i with uniform exponential growth and uniform exponential decay, E 1 i uniformly exponentially table and E 2 i uniformly exponentially intable. 6

8 It i ufficient to prove that if the evolution operator E 1 ha uniform exponential growth and atifie (3.2) than it i uniformly table. Indeed, if we denote by 1 1 N = du 0 g(u) where function g i given by Lemma 2.2, then E 1 (t,t 0 )x N = t 1 E 1 (t,t 0 )x dτ g(t τ) and hence E 1 (τ,t 0 )x dτ M E 1 (,t 0 )x E 1 (t,t 0 )x MN E(,t 0 )x for all t + 1, t 0 0 and all x X. If t [, + 1] then E 1 (t,t 0 )x g(t ) E 1 (,t 0 )x g(1) E 1 (,t 0 )x for all t 0 0 and all x X. Finally, we deduce that E 1 i uniformly exponentially table. Similarly, it i ufficient to prove that if the evolution operator E 2 ha uniform exponential decay and atifie relation (3.3), then it i uniformly intable. Indeed, if we denote by 1 1 N = du 0 f(u) where function f i given by Lemma 2.1, then and hence E 2 (,t 0 )x N = +1 E 2 (,t 0 )x dv f(v ) +1 t 0 E 2 (v,t 0 )x dv M E 2 (t,t 0 )x E 2 (,t 0 )x MN E 2 (t,t 0 )x E 2 (v,t 0 )x dv for all t t 0 0 and all x X and o E 2 i uniformly exponentially intable. Remark 3.5 Theorem 3.2 can be conidered a generalization of a wellknown reult due to Datko (Theorem 11 from [2]). We remark that our proof are not generalization of Datko proof for the characterization of the uniform exponential tability property. 7

9 Reference [1] C. Buşe, On nonuniform exponential tability of evolutionary procee. Rend. Sem. Mat. Univ. Polit. Torino 52 (1994), [2] R. Datko, Uniform aymptotic tability of evolutionary procee in Banach pace. SIAM J. Math. Anal. 3 (1972), [3] S. Elaydi, O. Hajek, Exponential trichotomy of differential ytem. J. Math. Anal. Appl. 129 (1988), [4] S. Elaydi, O. Hajek, Exponential dichotomy and trichotomy of nonlinear differential equation. Diff. Integral. Eq. 3 (1990), [5] A. Ichikawa, Equivalence of L p tability for a cla of nonlinear emigroup. Nonlinear Analyi 8 (1984) No. 7, [6] H. Jianlin, Exponential trichotomie and Fredholm operator. Ann. of Diff. Equation 9 (1993), [7] M. Megan, On (h,k)-tability of evolution operator in Banach pace. Dynamic Sytem and Application 5 (1996), [8] M. Megan, D.R. Laţcu, Exponential dichotomy of evolution operator in Banach pace. Internat. Ser. Num. Math, Birkhäuer Verlag 107 (1992), [9] M. Megan, A.L. Sau, B. Sau, Banach function pace and exponential intability of evolution operator. Archivum Mathematicum 39 (2003), [10] M. Megan, A.L. Sau, B. Sau, On nonuniform exponential dichotomy of evolution operator in Banach pace. Integral Equation Operator Theory 44 (2002), [11] M. Megan, A. Pogan, On exponential h-expanivene of emigroup of operator in Banach pace. Nonlinear Analyi 52 (2003), [12] M. Megan, C. Stoica, On aymptotic behavior of evolution operator in Banach pace. Seminar on Mathematical Analyi and Application in Control Theory, Wet Univerity of Timioara (2006), [13] M. Megan, C. Stoica, On null uniform exponential trichotomy of evolution operator in Hilbert pace. Annal of the Tiberiu Popoviciu Seminar on Functional Equation, Approximation and Convexity 3 (2005), [14] J.M.A.M. van Neerven, The aymptotic behavior of linear operator. Birkhäuer Verlag, [15] P. Preda, M. Megan, Exponential dichotomy of evolutionary procee in Banach pace. Czecholovak Mathematical Journal 35 (1985) No. 110, Mihail Megan Faculty of Mathematic Wet Univerity of Timişoara Bd. Pârvan, No Timişoara Romania megan@math.uvt.ro Codruţa Stoica Department of Mathematic Aurel Vlaicu Univerity of Arad Bd. Revoluţiei, No Arad Romania toicad@rdlink.ro 8