Some Characterizations for the Uniform Exponential Expansiveness of Linear Skew-evolution Semiflows
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1 Ξ45ffΞ3 μ fl Ω $ Vol.45, No " ADVANCES IN MATHEMATICS CHINA May, 206 doi: 0.845/sxjz.20473b Some Characterizations for the Uniform Exponential Expansiveness of Linear Skew-evolution Semiflows YUE Tian,, LEI Guoliang, SONG Xiaoqiu 2. School of Science, Hubei University of Automotive Technology, Shiyan, Hubei, , P. R. China; 2. College of Science, China University of Mining and Technology, Xuzhou, Jiangsu, 226, P. R. China Abstract: We give necessary and sufficient conditions for uniform exponential expansiveness of linear skew-evolution semiflows in terms of Banach sequence spaces and Banach function spaces, respectively. Thus we obtain generalizations of some well-known results due to Rolewicz, Hai, Megan, Sasu in the case of uniform exponential expansiveness of linear skew-evolution semiflows in Banach spaces. Keywords: linear skew-evolution semiflow; uniform exponential expansiveness; Banach sequence spaces; Banach function spaces MR200 Subject Classification: 37L5; 46B45 / CLC number: O77.2 Document code: A Article ID: Introduction In recent years, an impressive progress has been made in the field of the asymptotic behaviors of solutions of evolution equations in finite and infinite dimensional spaces see [3 4, 3, 5] and the references therein. Besides stability and dichotomy, a special attention was devoted to the study of expansiveness of evolution equations see [5 0, 4, 7 8]. Since the existence problem of exponential expansiveness of evolution equations is distinct compared to the studies devoted to stability and respectively to dichotomy, exponential expansiveness is a powerful tool when people analyze the asymptotic behavior of dynamical systems. In the last few years, new concepts of exponential expansiveness and in particular, of exponential instability, have been introduced and characterized. or instance, in [7] Megan et al. obtained some necessary and sufficient conditions for uniform exponential instability of linear skew-product semiflows in terms of Banach sequence spaces and Banach function spaces. In [5] and [8], the cases of uniform exponential instability has been considered for evolution families and linear skew-product flows, respectively. In [6], Megan et al. gave discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs c 0 N,X,c 0 N,X and C 0 R +,X,C 0 R +,X respectively; they also generalized Received date: Revised date: oundation item: The work is supported by NSC No and the undamental Research unds for the Central Universities No. 202LWB53. ytcumt@63.com
2 434 fi Ψ # 45ff an expansiveness theorem due to Minh et al. [] for the case of linear skew-product flows. The concept of skew-evolution semiflows, introduced and characterized by Stoica and Megan in [0] by means of evolution semiflows and cocycles, seems to be more appropriate for the study of asymptotic behaviors of evolution equations. They depend on three variables, contrary to a skew-product semiflow or an evolution operator, for which they are generalizations and which depend only on two. The exponential instability and uniform exponential stability for skewevolution semiflows are studied by Stoica and Megan in [0] and [6], respectively. In [], Hai characterized the uniform exponential stability of linear skew-evolution semiflows in terms of Banach function spaces. In [2], some discrete and continuous versions of Barbashin-type theorem for the case of linear skew-evolution semiflows were obtained. Additionally, in [8] we considered a weaker notion of instability for skew-evolution semiflows, and obtained some necessary and sufficient characterizations for weak exponential expansiveness of skew-evolution semiflows. In the spirit of the recent works of Megan et al. [5, 7], in the present paper, our main objective is to give some characterizations for uniform exponential expansiveness of linear skew-evolution semiflows in terms of Banach sequence spaces and Banach function spaces, respectively, and variants for uniform exponential expansiveness of some well-known results in uniform exponential stability theory see Zabczyk [9], Rolewicz [2] and Hai [] and exponential instability theory see Megan et al. [5, 7] are obtained. Preliminaries Let Θ,dbeametricspaceandX be a Banach space. The norm on X andonthespace LX of all bounded linear operators on X is denoted by.denotet = {t, t 0 R 2 + : t t 0 0}. Let be the set of all non-decreasing functions : R + R + with the properties 0 = 0 and t > 0 for every t>0.. Banach unction Spaces and Sequence Spaces Let Ω, Σ,μbeapositiveσ-finite measure space. By M we denote the linear space of μ-measurable functions f :Ω C, identifying the functions which are equal μ-a.e. [, 5, 7] Definition. A following properties: Banach function norm is a function N : M [0, ] withthe i Nf = 0 if and only if f =0μ-a.e.; ii if f g μ-a.e., then Nf Ng; iii Naf = a Nf for all a C and all f Mwith Nf < ; iv Nf + g Nf+Ng for all f,g M. Let B = B N be the set defined by B := {f M: f B := Nf < }. It is easy to see that B, B is a normed linear space. If B is complete, then B is called Banach function space over Ω. [, 5, 7] Definition.2 If Ω, Σ,μ=R +, L,m, where L is the σ-algebra of all Lebesgue measurable sets, and m is the Lebesgue measure, then
3 3!ß, fl±ffl, νρ : Uniform Exponential Expansiveness of Skew-evolution Semiflows 435 i for each Banach function space over R +, we define B : R + R + { }by { χ B t := [0,t B, χ [0,t B;, χ [0,t / B, where χ A denotes the characteristic function of A, and B is called the fundamental function of the Banach space B; ii BR + is the set of all Banach function space: lim Bt =, t inf χ[t,t+ B > 0. t R + [, 5, 7] Definition.3 If Ω, Σ,μ=N, PN,μ c, where μ c is the countable measure, then i for each Banach function space B over N in this case B is called Banach sequence space, we define B : N R + { }by { χ {0,,,n } B, χ {0,,,n } B; B n :=, χ {0,,,n } / B, which is called the fundamental function of B; ii BN is the set of all Banach sequence spaces B: lim Bn =, n inf χ{n} B > 0. n N Remark. [, 5, 7] If B BR +, then S B := {α n n : α nχ [n,n+ B} with respect to the norm α n n SB := α n χ [n,n+ B is a Banach sequence space which belongs to BN..2 Linear Skew-evolution Semiflow Definition.4 [] A continuous mapping σ : T Θ Θ is called an evolution semiflow on Θ, if it has the properties σt, t, θ =θ and σt, s, σs, r, θ = σt, r, θ for all t s r 0 and θ Θ. Definition.5 [] Apairπ =Φ,σ is called a linear skew-evolution semiflow on E = X Θ if σ is an evolution semiflow on Θ and Φ : T Θ LX satisfies the following conditions: i Φt, t, θ =I, the identity operator on X for all t, θ R + Θ; ii Φt, r, θ =Φt, s, σs, r, θφs, r, θ for all t s r 0andθ Θ; iii There are M, ω>0, such that Φt, s, θx Me ωt s x for all t, s,θ,x T Θ X. Remark.2 i The mapping Φ given by the definition above is called the evolution cocycle associated with the linear skew-evolution semiflow π.
4 436 fi Ψ # 45ff ii The linear skew-evolution semiflows are generalizations of the C 0 -semigroups, of the evolution operators and of the skew-product semiflows. iii If π = Φ,σ is a linear skew-evolution semiflow and α R is a parameter, then π α =Φ α,σ, where Φ α : T Θ LX, Φ α t, s, θ =e αt s Φt, s, θ, is also a linear skewevolution semiflow. Example. Let Θ be a compact metric space, σ an evolution semiflow on Θ, X a Banach space and A :Θ LX a continuous map. If Φt, s, θx is a solution of the Cauchy problem u t =Aσt, s, θut, t s, then the pair π =Φ,σ is a linear skew-evolution semiflow. Definition.6 A linear skew-evolution semiflow π =Φ,σ is said to be uniformly exponentially expansive, if there are K>0andv>0 such that Φt, s, θx Ke vt s x for all t, s,θ,x T Θ X. Example.2 We consider Θ = R +, X = R. Let a e 4, e 3. It is obvious that the mapping σ : T R + R + defined by σt, r, θ 0 =t r + θ 0, t, r, θ 0 T R + is an evolution semiflow on R + and the mapping Φ : T R + LR givenby Φt, r, θ 0 =a t r, t, r, θ 0 T R + is an evolution cocycle on X. Then the skew-evolution semiflow π =Φ,σ is uniformly exponentially expansive on E. Definition.7 A linear skew-evolution semiflow π =Φ,σ is called i injective if for every t, s,θ T Θ, the operator Φt, s, θ is injective; ii strongly mensurable if the mapping t Φt, s, θx is measurable on [s, for all s, x, θ R + E. Throughout this paper we denote U = {x X : x =}. 2MainResults Theorem 2. If there are two constants r 0 > 0andc> such that Φs + r 0,s,θx c x, s, x, θ R + E, 2 then the linear skew-evolution semiflow π =Φ,σ is uniformly exponentially expansive. Proof Let M, ω>0begiven by Definition.5 and v>0such that c =e vr0. Let t 0. There exist n N and l [0,r 0 such that t = nr 0 + l. Ifx, θ Eand s 0, then using
5 3!ß, fl±ffl, νρ : Uniform Exponential Expansiveness of Skew-evolution Semiflows 437 the hypothesis, it follows that Me ωr0 Φs + t, s, θx Me ωr0 l Φs + t, s, θx Φs +n +r 0,s,θx = Φs +n +r 0,s+ nr 0,σs + nr 0,s,θΦs + nr 0,s,θx c Φs + nr 0,s,θx c n+ x. Hence Φs + t, s, θx Ke n+vr0 x Ke vt x, s, t, x, θ R 2 + E, wherewehavedenotedk = Me.Soπis uniformly exponentially expansive. ωr 0 Theorem 2.2 Let π =Φ,σ be an injective linear skew-evolution semiflow on E = X Θ. Then π is uniformly exponentially expansive if and only if there are a Banach sequence space B BN and a constant L>0 such that i the mapping ϕx, θ, s, :N R +, ϕx, θ, s, n = Φs + n, s, θx belongs to B for all x, θ, s U Θ R + ; ii ϕx, θ, s, B L for all x, θ, s U Θ R +. Moreover, Condition ii holds for L = K e v. Proof Necessity. If π is uniformly exponentially expansive, then from Definition.6, it follows that there are K, v > 0 such that for all s, n, θ, x R + N Θ U. Thus Φs + n, s, θx Ke vn Φs + n, s, θx K e vn = So B := l N, C andϕx, θ, s, B. Sufficiency. Let x, θ U ΘandB BN, lim Bn =, n λ =inf χ {n} B > 0. n N Since ϕx, θ, s, nχ {n} ϕx, θ, s,, it follows that K e v. L ϕx, θ, s, B ϕx, θ, s, n χ {n} B λϕx, θ, s, n. Hence ϕx, θ, s, n L λ, x, θ, s, n U Θ R + N. 3
6 438 fi Ψ # 45ff Let m N such that B m > 3L2 2λ. Using the relation 3 and Condition ii of Definition.5, we have Φs + m, s, θx L λ Φs + i, s, θx for every i {0,,,m } and every x, θ, s X Θ R +. Then we deduce that Φs + m, s, θx χ {0,,,m } L ϕx, θ, s,, λ which yields 3L 2 2λ Φs + m, s, θx Φs + m, s, θx Bm L λ ϕx, θ, s, B urthermore, we obtain L2 λ. Φs + m, s, θx 3 2, x, θ, s X Θ R +. According to Theorem 2., π is uniformly exponentially expansive, which ends the proof. Theorem 2.3 Let π =Φ,σ be an injective and strongly measurable linear skew-evolution semiflow on E = X Θ. Then π is uniformly exponentially expansive if and only if there are a Banach function space B BR + and a constant L>0 such that i the mapping ψx, θ, s, :R + R +, ψx, θ, s, t = Φs + t, s, θx belongs to B for all x, θ, s U Θ R + ; ii ψx, θ, s, B L for all x, θ, s U Θ R +. Proof Necessity. If π is uniformly exponentially expansive, then by Definition.6, there are K, v > 0 such that Φs + τ,s,θx dτ K 0 0 e vτ dτ = Kv = L for all s, θ, x R + Θ U. Thus it is immediate for B = L R +, C. Sufficiency. We consider the Banach sequence space S B, associated with B by Remark.. Let x, θ, s U Θ R + and ϕx, θ, s, :N R +, ϕx, θ, s, n = Φs + n, s, θx. or every t R +,thereexistsn N such that t [n, n +,andwehave Φs + t, s, θx Me ωt n Φs + n, s, θx Me ω Φs + n, s, θx,
7 3!ß, fl±ffl, νρ : Uniform Exponential Expansiveness of Skew-evolution Semiflows 439 where M, ω>0 are given by Definition.5. Hence ϕx, θ, s, n Me ω ψx, θ, s, t for all n N and all t [n, n +. urthermore, we obtain ϕx, θ, s, nχ [n,n+ Me ω ψx, θ, s,. So ϕx, θ, s, SB = ϕx, θ, s, nχ [n,n+ B Me ω ψx, θ, s, B LMe ω, x, θ, s U Θ R +. inally, by Theorem 2.2 we conclude that π is uniformly exponentially expansive. Corollary 2. Let π = Φ,σ be an injective and strongly measurable linear skewevolution semiflow on E = X Θ. Then π is uniformly exponentially expansive if and only if there exists p such that sup θ Θ,s R + 0 Φs + τ,s,θx p dτ< 4 for all x U. Proof Necessity. It is trivial. Sufficiency. It is immediate by Theorem 2.3 for B := L p R +, C. Remark 2. Theorems 2.2 and 2.3 are some versions of the classical instability theorems due to Megan et al. [5, 7] for uniform exponential expansiveness of linear skew-evolution semiflows. They can also be considered as the variants for stability of theorems proved by Hai in [] for the case of uniform exponential stability of linear skew-evolution semiflows. Theorem 2.4 Let π =Φ,σ be an injective linear skew-evolution semiflow on E = X Θ. Then π is uniformly exponentially expansive if and only if there are a Banach sequence space B BN, a function and a constant L>0 such that i the mapping ϕx, θ, s, :N R +, ϕx, θ, s, n = Φs + n, s, θx belongs to B for all x, θ, s U Θ R + ; ii ϕx, θ, s, B L for all x, θ, s U Θ R +. Proof Necessity. It is a simple verification for t =t, t 0andB = l N, C. Sufficiency. Let x, θ U Θ. Since B BN, there exists m N such that B m +> L. 5
8 440 fi Ψ # 45ff Let s R + and n N. By Condition iii in Definition.5 and since is a non-decreasing function, we have Me ωm Φs + n, s, θx Φs + n + i, s, θx for every i {0,,,m}, wherem,ω are given by Definition.5. urthermore, we obtain Me ωm χ {0,,,m} ϕx, θ, s + n,. Φs + n, s, θx Using the relation 5 we have L Me ωm Φs + n, s, θx which yields < Me ωm Φs + n, s, θx ϕx, θ, s + n, B L, B m + Φs + n, s, θx Meωm, m N, x, θ, s U Θ R +. 6 Let m 0 N such that L B m 0 > 2 3Me. 7 ωm Using the inequality 6 and the condition ii of Definition.5, we have Φs + m 0,s,θx Me ωm Φs + i, s, θx for every i {0,,,m 0 } and every x, θ, s X Θ R +.Thus Me ωm, i {0,,,m 0 }. Φs + m 0,s,θx Φs + i, s, θx It follows that Me ωm Φs + m 0,s,θx and hence from 7 we deduce that χ {0,,,m0 } ϕx, θ, s,, Φs + m 0,s,θx 3 2, x, θ, s X Θ R +. By Theorem 2. we conclude that π is uniformly exponentially expansive. Corollary 2.2 Let π =Φ,σ be an injective linear skew-evolution semiflow on E = X Θ. Then π is uniformly exponentially expansive if and only if there are a function and a constant L>0 such that L, x, θ, s U Θ R +. 8 Φs + n, s, θx
9 3!ß, fl±ffl, νρ : Uniform Exponential Expansiveness of Skew-evolution Semiflows 44 Remark 2.2 Corollary 2.2 is a version of the classical stability theorem due to Zabczyk [9] for uniform exponential expansiveness of linear skew-evolution semiflows. It can also be considered as a variant for instability of some theorems proved by Megan et al. in [5, 7] for the case of uniform exponential instability of linear skew-product semiflows and of evolution families. Theorem 2.5 Let π =Φ,σ be an injective and strongly measurable linear skew-evolution semiflow on E = X Θ. Then π is uniformly exponentially expansive if and only if there are a Banach function space B BR +, a function and a constant L>0 such that i the mapping ψx, θ, s, :R + R +, ψx, θ, s, t = Φs + t, s, θx belongs to B for all x, θ, s U Θ R + ; ii ψx, θ, s, B L for all x, θ, s U Θ R +. Proof Necessity. It is a simple verification for t =t, t 0andB = L R +, C. Sufficiency. Let x, θ, s U Θ R +,andm,ω be given by Definition.5. We put γt = t Me and define ω ϕx, θ, s, :N R +, ϕx, θ, s, n =γ. Φs + n, s, θx rom the following relation for all n N and all t [n, n +,wehave γ Φs + n, s, θx It follows that Φs + t, s, θx Me ω Φs + n, s, θx, Φs + t, s, θx, n N, t [n, n +. ϕx, θ, s, nχ [n,n+ ψx, θ, s,. 9 We consider the Banach sequence space S B, associated with B by Remark.. Then from the relation 9 we deduce that ϕx, θ, s, SB ψx, θ, s, B L, x, θ, s U Θ R +. inally, by Theorem 2.4 we conclude that π is uniformly exponentially expansive. Corollary 2.3 Let π = Φ,σ be an injective and strongly measurable linear skewevolution semiflow on E = X Θ. Then π is uniformly exponentially expansive if and only if there exist a function and a constant L>0 such that dτ L, x, θ, s U Θ R Φs + τ,s,θx Remark 2.3 Corollary 2.3 is a version of the classical stability theorem due to Rolewicz [2] for the case of uniform exponential expansiveness of linear skew-evolution semiflows. It can
10 442 fi Ψ # 45ff also be considered as a variant for instability of some theorems proved by Megan et al. in [5, 7] for the case of uniform exponential instability of linear skew-product semiflows and of evolution families. Acknowledgements The authors thank the referees for carefully reading and very important suggestions and comments, which led to the improvement of the paper. References [] Hai, P.V., Continuous and discrete characterizations for the uniform exponential stability of linear skewevolution semiflows, Nonlinear Anal., 200, 722: [2] Hai, P.V., Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows, Appl. Anal., 20, 902: [3] Hai, P.V., On two theorems regarding exponential stability, Appl. Anal. Discrete Math., 20, 52: [4] Hai, P.V., Two new approaches to Barbashin theorem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 202, 96: [5] Megan, M., Sasu, A.L. and Sasu, B., Banach function spaces and exponential instability of evolution families, Arch. Math. Brno, 2003, 394: [6] Megan, M., Sasu, A.L. and Sasu, B., Perron conditions for uniform exponential expansiveness of linear skew-product flows, Monatsh. Math., 2003, 382: [7] Megan, M., Sasu, A.L. and Sasu, B., Exponential instability of linear skew-product semiflows in terms of Banach function spaces, Results Math., 2004, 453: [8] Megan, M., Sasu, A.L. and Sasu, B., Exponential stability and exponential instability for linear skew-product flows, Math. Bohem., 2004, 293: [9] Megan, M., Sasu, B. and Sasu, A.L., Exponential expansiveness and complete admissibility for evolution families, Czechoslovak Math. J., 2004, 543: [0] Megan, M. and Stoica, C., Exponential instability of skew-evolution semiflows in Banach spaces, Stud. Univ. Babeş-Bolyai Math., 2008, 53: [] Minh, N.V., Räbiger,. and Schnaubelt, R., Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 998, 323: [2] Rolewicz, S., On uniform N-equistability, J. Math. Anal. Appl., 986, 52: [3] Sasu, A.L. and Sasu, B., A Zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems, Appl. Math. Comput., 204, 245: [4] Sasu, B., New criteria for exponential expansiveness of variational difference equations, J. Math. Anal. Appl., 2007, 327: [5] Song, X.Q., Yue, T. and Li, D.Q., Nonuniform exponential trichotomy for linear discrete-time systems in Banach spaces, J. unct. Spaces Appl., 203, 203: Article ID , 6 pages. [6] Stoica, C. and Megan, M., On uniform exponential stability for skew-evolution semiflows on Banach spaces, Nonlinear Anal., 200, 723/4: [7] Yue, T., Song, X.Q. and Li, D.Q., On weak exponential expansiveness of evolution families in Banach spaces, Sci. World J., 203, 203: Article ID , 6 pages. [8] Yue, T., Song, X.Q. and Li, D.Q., On weak exponential expansiveness of skew-evolution semiflows in Banach spaces, J. Inequal. Appl., 204, 204: 65, pages. [9] Zabczyk, J., Remarks on the control of discrete-time distributed parameter systems, SIAM J. Control Optim., 974, 24: '09=<3;62*/, E C,?>@, BDA 2. Πχfiψffifiψ,»ffi,, ; 2. %±fiχλfiffifiψ, ff&, Φο, 226 :8 HhQ[aVl^ XUaVTd XOIPK]ikjmWG_otsdb rmjnfy, elz`gr]pqzs c Rolewicz, Hai, Megan, Sasu N. +. ikjmwg_; otsdbrk; avl^ X; avtd X
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