Nonlinear Analysis 72 (2010) 4390–4396 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows Pham Viet Hai ∗ Department of Mathematics, Viet Nam National University, Ha Noi, College of Science, Viet Nam article abstract info Article history: Received 25 October 2009 Accepted 25 January 2010 In this paper, we will consider the concept ‘‘linear skew-evolution semiflows’’ and extend theorems of R Datko, S Rolewicz, Zabczyk and J.M.A.M van Neerven for this case [15] © 2010 Elsevier Ltd All rights reserved MSC: 93D20 34D05 46E30 47D06 Keywords: Exponential stability Linear skew-evolution semiflows Function spaces Introduction One of the most important results in the theory of stability for a strongly continuous semigroup of linear operators has been obtained by Datko [1] in 1970; it states that the semigroup (T (t ))t ≥0 is uniformly exponentially stable if and only if, for each x ∈ X , the map t → T (t )x lies in L2 (R+ ) Later, in [2] Pazy proved that the result remains true even if we replace L2 (R+ ) with Lp (R+ ), where p ∈ [1, ∞) In 1972, Datko generalized the results above as follows, [3] Theorem 1.1 An evolution family (U (t , s))t ≥s≥0 with exponential growth is uniformly exponentially stable if and only if there exists p ∈ [1, ∞) such that: ∞ U (τ , s)x sup s≥0 p dτ < ∞ s for all x ∈ X The result provided by Theorem 1.1 was extended to dichotomy by Preda and Megan [4] in 1985 The same result was generalized in 1986 by Rolewicz [5] in the following way Theorem 1.2 Let φ : R+ → R+ be a continuous, non-decreasing function with φ(0) = and φ(t ) > for each positive t, and (U (t , s))t ≥s≥0 an evolution family, with exponential growth If ∞ φ ( U (τ , s)x ) dτ < ∞, sup s≥0 (x ∈ X ) s ∗ Corresponding address: Faculty of Mathematics, Mechanics and Informatics, College of Science, Viet Nam National University, Ha Noi, 334, Nguyen Trai Road, Thanh Xuan Dist., Ha Noi, Viet Nam E-mail address: phamviethai86@gmail.com 0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved doi:10.1016/j.na.2010.01.046 P.V Hai / Nonlinear Analysis 72 (2010) 4390–4396 4391 then (U (t , s))t ≥s≥0 is uniformly exponentially stable W Litman gave another proof of Theorem 1.2 for strongly continuous semigroup of linear operators, [6] In [7], the author generalize some results due to S Rolewicz, Z Zabczyk A unified treatment of the Datko–Pazy and Rolewicz theorem is presented by Neerven [8] In fact, Neerven presented in [8] a more general result Theorem 1.3 Let C0 -semigroup (T (t ))t ≥0 on a Banach space X and E be a Banach function space over R+ with limt →∞ ϕE (t ) = ∞ where ϕE (t ) := X[0,t ) E If for all x ∈ X , the map t → T (t )x belongs to E, then (T (t ))t ≥0 is uniformly exponentially stable This method can be generalized for the study of uniform exponential stability of linear skew-product semiflows In [9], Megan, A.L Sasu, B Sasu proved a more general result than Neerven’s result The same direction is given in [10] Theorem 1.4 The linear skew-product semiflow π0 = (Φ0 , σ0 ) is uniformly exponentially stable if and only if there are B ∈ B (N) and a sequence (tn ) of positive real numbers with the following properties (1) Sup | tn+1 − tn |< ∞ n∈N (2) The function: ϕx,θ (.) : N → R+ ϕx,θ (n) = Φ0 (θ , tn )x belongs to B (3) There exists K : X → (0, ∞) such that: | ϕx,θ |B ≤ K (x) for all (x, θ ) ∈ E In [11], the author proved: Theorem 1.5 The linear skew-evolution semiflow π = (Φ , σ ) is uniformly exponentially stable if and only if there are B ∈ B (N) and a constant L > such that: The mapping: ϕ(x, θ , m, ) : N → R+ ϕ(x, θ , m, n) := Φ (m + n, m, θ )x ϕ(x, θ , m, ) ∈ B for all (x, θ , m) ∈ U × × N ϕ(x, θ , m, ) B ≤ L for all (x, θ, m) ∈ U × × N Theorem 1.6 The linear skew-evolution semiflow π = (Φ , σ ) is uniformly exponentially stable if and only if there are N ∈ N and a constant L such that: ∞ N ( Φ (m + j, m, θ)x ) ≤ L j=0 for all (m, θ , x) ∈ N × × U By extending techniques in [11], this paper will study continuous characterizations for the uniform exponential stability of linear skew-evolution semiflows Preliminaries 2.1 Linear skew-evolution semiflow Let us recall basic notions of linear skew-evolution semiflows, X a Banach space, ( , d) a metric space We denote by L(X ) be the Banach algebra of all bounded linear operators acting on X ; T = {(t , s) ∈ R2+ | t ≥ s} and ∆ = {(m, n) ∈ N2 | m ≥ n} Definition 2.1 A continuous mapping σ : T × is called a evolution semiflow on (1) σ (t , t , θ ) = θ (2) σ (t , s, σ (s, r , θ )) = σ (t , r , θ ), for all t ≥ s ≥ r ≥ 0; θ ∈ if: Definition 2.2 A pair π = (Φ , σ ) is called a linear skew-evolution semiflow on E = X × on and Φ : T × → L(X ) satisfies the following conditions: if σ is an evolution semiflow 4392 P.V Hai / Nonlinear Analysis 72 (2010) 4390–4396 (1) Φ (t , t , θ ) = I the identity operator on X , for all (t , θ ) ∈ R+ × (2) Φ (t , r , θ ) = Φ (t , s, σ (s, r , θ ))Φ (s, r , θ ) for all t ≥ s ≥ r ≥ 0; θ ∈ (3) there are M , ω > such that Φ (t , s, θ )x ≤ Meω(t −s) x for all ((t , s), θ , x) ∈ T × × X Example 2.1 It is easy to see that C0 -semigroups, evolution families and linear skew-product semiflows are particular cases of linear skew-evolution semiflows Example 2.2 Let {U (t , s)}t ≥s≥0 be an evolution family, with uniform exponential growth If there exists a bounded strongly continuous family of idempotent operators {P (θ )}θ∈ , with the property that P (θ )U (t , s) = U (t , s)P (θ ), t ≥ s ≥ 0, θ ∈ then the pair π = (Φ , σ ) defined by σ (t , s, θ ) = θ , Φ (t , s, θ ) = P (θ )U (t , s) is a linear skew-evolution semiflow Example 2.3 Let be a compact metric space, σ an evolution semiflow on continuous map If Φ (t , t0 , θ )x is the solution of the Cauchy problem u (t ) = A(σ (t , t0 , θ ))u(t ), , X a Banach space and A : → L(X ) a t ≥ t0 (2.1) then the pair π = (Φ , σ ) is a linear skew-evolution semiflow Eq (2.1) is the starting point of our paper Definition 2.3 A linear skew-evolution semiflow π = (Φ , σ ) is said to be uniformly exponentially stable if there are K > and ν > such that: Φ (t , s, θ )x ≤ K e−ν(t −s) x for all ((t , s), θ , x) ∈ T × × X Throughout this paper we shall denote: U = {x ∈ X : x = 1} 2.2 Function spaces, sequence spaces Let (Ω , Σ , µ) be a positive σ -finite measure space By M we denote the linear space of µ-measure functions f : Ω → C, identifying the functions which are equal to µ-a.e Definition 2.4 A Banach function norm is a function N : M → [0; ∞] with the following properties: (1) (2) (3) (4) N (f ) = if and only if f = µ-a.e If |f | ≤ |g | µ-a.e then N (f ) ≤ N (g ) N (af ) = |a|N (f ) for all a ∈ C and f ∈ M with N (f ) < ∞ N (f + g ) ≤ N (f ) + N (g ) for all f , g ∈ M Let B = BN be the set defined by: B := {f ∈ M : f B := N (f ) < ∞} 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 For more details about function spaces, we can see [12–14] Definition 2.5 If (Ω , Σ , µ) = (R+ , L, m) where L is the σ -algebra of all Lebesgue measurable sets and m the Lebesgue measure then (1) For each Banach function space over R+ we define: FB : R+ → R+ ∪ {∞} FB (t ) := X[0;t ) B ; X[0;t ) ∈ B ∞; X[0;t ) ∈ B where XA denotes the characteristic function of A FB is called the fundamental function of the Banach space B (2) B (R+ ) is the set of all Banach function space: lim FB (t ) = ∞ t →∞ inf t ∈R+ X[t ;t +1) B > P.V Hai / Nonlinear Analysis 72 (2010) 4390–4396 Example 2.4 A trivial example of Banach function space over R+ which belongs B (R+ ) is Lp (R+ , C) with p ∈ [1, ∞) Definition 2.6 If (Ω , Σ , µ) = (N, P (N), µc ) where µc is the countable measure then (1) For each Banach function space over N (or Banach sequence space), B, we define: FB : N∗ → R+ ∪ {∞} FB (n) = X{0; ;n−1} B ; X{0; ;n−1} ∈ B ∞; X{0; ;n−1} ∈ B called the fundamental function of B (2) B (N) the set of all Banach sequence spaces B: lim FB (n) = ∞ n→∞ inf X{n} n∈N B > Example 2.5 If p ∈ [1, ∞) then B = lp with: p ∞ s p |s(n)| = p n =0 is a Banach sequence space which belongs to B (N) Remark 2.6 If B is a Banach function space over R+ which belongs to B (R+ ) then: ∞ SB := (αn )n : αn X[n,n+1) ∈ B n =0 with respect to the norm: ∞ (αn )n SB αn X[n,n+1) := n =0 B is a Banach sequence space which belongs to B (N) Definition 2.7 Let N be the set of all non-decreasing continuous functions: N : [0; ∞) → [0; ∞) with properties: N (0) = N (t ) > 0; t > Main results 3.1 Discrete characterization Lemma 3.1 If there are two constants p ∈ N∗ and c ∈ (0; 1) (does not belong to (x, θ , m)) such that: Φ (p + m, m, θ )x ≤ c x for all (x, θ , m) ∈ X × × N, then the linear skew-evolution semiflow π = (Φ , σ ) is uniformly exponentially stable Proof For each (t , s) ∈ R+ × R+ : If t ∈ [0; 1]: Φ (t + s, s, θ )x ≤ Meωt x ≤ Meω x If t ≥ 1: t + s ≥ [t ] + [s] ≥ + [s] > s 4393 4394 P.V Hai / Nonlinear Analysis 72 (2010) 4390–4396 There exist k ∈ N and r ∈ [0, p) such that: [t ] − = kp + r [t ] − t −2 k> −1> − p p It is clear that: Φ (t + s, s, θ )x ≤ ≤ ≤ ≤ Meω(t +s−[t ]−[s]) Φ (kp + r + + [s], s, θ )x Meω(t +s−[t ]−[s]) Meωr Φ (kp + + [s], s, θ )x M eω(t +s−[t ]−[s]) eωp c k Φ (1 + [s], s, θ )x M eω(t +s−[t ]−[s]) eωp c k eω(1+[s]−s) x ≤ M eωp+2ω c t −2 p −1 x where M , ω in Definition 2.2 So Lemma is proved Theorem 3.2 π = (Φ , σ ) is uniformly exponentially stable if and only if there are d ∈ N; B ∈ B (N) and a constant L > such that: (1) The mapping: ϕd (x, θ , m, ) : N → R+ ϕd (x, θ , m, n) := Φ (m + n + d, m, θ )x (2) ϕd (x, θ , m, ) ∈ B for all (x, θ , m) ∈ U × × N ϕd (x, θ , m, ) B ≤ L for all (x, θ , m) ∈ U × × N Proof Necessity: π = (Φ , σ ) is uniformly exponentially stable There are K , ν > such that: Φ (m + n, m, θ )x ≤ K e−ν n for all (m, n, x) ∈ N × N × U ∞ ∞ Φ (m + n, m, θ )x ≤ K n =0 e−ν n n =0 = K − e−ν So d = 0; B := l1 and ϕ(x, θ , m, ) ∈ B Sufficiency: Let x ∈ U and B ∈ B (N) lim FB (n) = ∞ n→∞ c = inf n∈N X{n} B > We see that ϕd (x, θ , m, n)X{n} ≤ ϕd (x, θ , m, ) L ≥ ϕd (x, θ , m, ) B ≥ ϕd (x, θ , m, n) X{n} B ≥ ϕd (x, θ , m, n)c L ϕd (x, θ , m, n) ≤ c For j ∈ {0; ; n} and θ1 := σ (m + j + d, m, θ ), we see that Φ (m + n + 2d, m, θ )x ≤ L c Φ (m + j + d, m, θ)x Φ (m + n + 2d, m, θ )x X{0; ;n} ≤ L c Φ (m + n + 2d, m, θ )x FB (n + 1) ≤ ≤ lim n→∞ Φ (m + n + 2d, m, θ )x = ϕd (x, θ , m, ) L |ϕd (x, θ , m, )|B c L2 c P.V Hai / Nonlinear Analysis 72 (2010) 4390–4396 4395 There exists n0 ∈ N such that: Φ (m + n0 + 2d, m, θ )x ≤ By Lemma 3.1, Theorem is proved 3.2 Continuous characterization Theorem 3.3 The linear skew-evolution semiflow π = (Φ , σ ) is uniformly exponentially stable if and only if there are B ∈ B (R+ ) and L > such that: (1) The mapping: ψ(x, θ , m, ) : R+ → R+ ψ(x, θ , m, t ) := Φ (m + t , m, θ )x ψ(x, θ , m, ) ∈ B for all (x, θ , m) ∈ U × (2) L ≥ ψ(x, θ , m, ) B × N Proof Necessity: B := L1 (R+ , C) and L := Kν Indeed, we have: ∞ ∞ Φ (m + τ , m, θ )x dτ ≤ K e−ντ dτ = K ν = L Sufficiency For each t ∈ R+ , there exists n ∈ N such that t ∈ [n, n + 1) ≤ Meω(n+1−t ) Φ (m + t , m, θ )x ≤ Meω Φ (m + t , m, θ )x ω ϕ1 (x, θ , m, n) ≤ Me ψ(x, θ , m, t ) Φ (m + n + 1, m, θ )x ∞ ϕ1 (x, θ , m, n)X[n,n+1) ≤ Meω ψ(x, θ , m, ) n =0 So ∞ ϕ1 (x, θ , m, ) SB ϕ1 (x, θ , m, n)X[n,n+1) = n =0 ≤ Meω ψ(x, θ , m, ) ≤ Meω L B B where M , ω in Definition 2.2 By using Theorem 3.2, Theorem is proved Corollary 3.4 The linear skew-evolution semiflow π = (Φ , σ ) is uniformly exponentially stable if and only if there exists p ∈ [1, ∞) such that: ∞ Φ (τ + m, m, θ )x sup θ∈ m∈N p dτ < ∞ for all x ∈ X Proof Necessity It is trivial Sufficiency It results by Theorem 3.3 for B := Lp (R+ , C) Theorem 3.5 The linear skew-evolution semiflow π = (Φ , σ ) is uniformly exponentially stable if and only if there are N ∈ N and a constant L such that: ∞ N ( Φ (τ + m, m, θ )x ) dτ ≤ L for all (m, θ , x) ∈ N × × U 4396 P.V Hai / Nonlinear Analysis 72 (2010) 4390–4396 Proof The necessity is obvious for N (t ) = t The sufficiency: M ; ω in Definition 2.2 We put γ (t ) = N ∞ t Meω ∞ γ ( Φ (n + m, m, θ )x ) = n =0 γ ( Φ (n + m, m, θ )x ) + γ (1) n=1 ∞ γ ( Φ (n + + m, m, θ )x ) + N ≤ n=0 ∞ Meω n +1 N ( Φ (τ + m, m, θ )x ) dτ + N ≤ n=0 n ≤ L+N Meω Meω By Theorem 3.4 in [11], Theorem is proved References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] R Datko, Extending a theorem of Lyapunov to Hilbert spaces, J Math Anal Appl 32 (1970) 610–616 A Pazy, Semigroups of Linear Operators and Applications to Partial Differnetial Equations, Springer, Berlin, 1983 R Datko, Uniform asymtotic stability of evolutionary processes in Banach spaces, SIAM J Math Anal (1972) 428–445 P Preda, M Megan, Exponential dichotomy of evolutionary processes in Banach spaces, Czech Math J 35 (1985) 312–323 S Rolewicz, On uniform N-equistability, J Math Anal Appl 115 (1986) 434–441 W Litman, A generalization of the theorem Datko–Pazy, Lect Notes Control Inform Sci 130 (1983) 318–323 A.L Sasu, B Sasu, Exponential stability for linear skew-product flows, Bull Sci Math 128 (2004) 727–738 J.M.A.M Van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, in: Theory, Advances and Applications, Vol 88, Birkhauser, Boston, 1996 M Megan, A.L Sasu, B Sasu, On uniform exponential stability of linear skew-product semiflows in Banach spaces, Bull Belg Math Soc Simon Stevin (2002) 143–154 M Megan, A.L Sasu, B Sasu, Exponential stability and exponential instability for linear skew-product flows, Math Bohem 129 (2004) 225–243 Pham Viet Hai, Some results about uniform exponential stability of linear skew-evolution semiflows, Int J Evol Equ 04 (03) (2009) C Bennett, S Sharpley, Interpolation of Operators, Academic Press, Boston, 1988 S.G Krein, Yu.I Petunin, E.M Semeonov, Interpolation of Linear Operators, in: Transl Math Monogr., vol 54, Amer Math Soc., Providence, 1982 P Meyer-Nieberg, Banach Lattices, Springer Verlag, Berlin, Heidelberg, New York, 1991 Z Zabczyk, Remarks on the control of discrete-time distributed parameter systems, SIAM J Control Optim 12 (1971) 721–735 ... Nonlinear Analysis 72 (2010) 4390–4396 4391 then (U (t , s))t ≥s≥0 is uniformly exponentially stable W Litman gave another proof of Theorem 1.2 for strongly continuous semigroup of linear operators,... (t )x belongs to E, then (T (t ))t ≥0 is uniformly exponentially stable This method can be generalized for the study of uniform exponential stability of linear skew-product semiflows In [9], Megan,... [11], the author proved: Theorem 1.5 The linear skew-evolution semiflow π = (Φ , σ ) is uniformly exponentially stable if and only if there are B ∈ B (N) and a constant L > such that: The mapping: