arXiv:1606.05159v1 [math.DS] 16 Jun 2016
ADMISSIBLE BANACH FUNCTION SPACES FOR LINEAR DYNAMICS WITH NONUNIFORM BEHAVIOR ON THE HALF-LINE NICOLAE LUPA AND LIVIU HORIA POPESCU Abstract. For nonuniform exponentially bounded evolution families on the half-line we introduce a class of Banach function spaces on which we define nonuniform evolution semigroups. We completely characterize nonuniform exponential stability in terms of invertibility of the corresponding generators. We emphasize that in particular our results apply to all linear differential equations with bounded operator and finite Lyapunov exponent.
1. Introduction A linear dynamics is called well-posed if we assume the existence, uniqueness and continuous dependence of solutions on initial data. In the case of the nonautonomous equation dx/dt = A (t) x, well-posedness is equivalent to the existence of an evolution family that solves the equation (see, for instance, Proposition 9.3 in [7], p. 478), that is a family of bounded linear operators U = {U (t, s)}t≥s≥0 acting on the underlying Banach space X, with properties: (e1 ) U (t, t) = Id, t ≥ 0; (e2 ) U (t, τ )U (τ, s) = U (t, s), t ≥ τ ≥ s ≥ 0; (e3 ) The map (t, s) 7→ U (t, s)x is continuous for each x ∈ X. If the linear operators A (t) are bounded, then well-posedness is guaranteed [6, Chapter 3]. In the autonomous case, the equation dx/dt = Ax is well-posed if and only if the linear operator A generates a C0 -semigroup (Corollary 6.9 in [7], p. 151). We recall that a family of bounded linear operators T = {T (t)}t≥0 acting on X is said to be a C0 -semigroup if (s1 ) T (0) = Id; (s2 ) T (t)T (s) = T (t + s) for t, s ≥ 0; (s3 ) lim T (t)x = x for every x ∈ X. t→0+
The (closed and densely defined) linear operator G : D(G) ⊂ X → X defined by Gx = lim
t→0+
T (t)x − x t
is called the (infinitesimal) generator of the C0 -semigroup T . For more details on the theory of C0 -semigroups we refer the reader to [7, 10]. 2010 Mathematics Subject Classification. 34D20,47D06. Key words and phrases. Evolution families, nonuniform exponential stability, evolution semigroups. 1
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N. LUPA AND L.H. POPESCU
In the particular case of linear dynamics, the concept of uniform asymptotic stability (in the sense of Lyapunov) translates as uniform exponential stability. The evolution family U is called uniform exponentially stable if kU (t, s)k ≤ M e−α(t−s) , t ≥ s ≥ 0, for some constants M , α > 0. Under such hypothesis it is possible to introduce a C0 -semigroup on some Banach function spaces, basically defined as ( U (s, s − t)u(s − t), if s > t, (T (t)u)(s) = (1) U (s, 0)u(0), if 0 ≤ s ≤ t, and it is called the (uniform) evolution semigroup. This construction reduces the study of a nonautonomous equation to the analysis of an autonomous one in the form dy/dt = Gy, where G is the generator of the evolution semigroup. Thus, asymptotic behavior of an evolution family can be essentially characterized in terms of spectral properties of the generator of the corresponding evolution semigroup. The clue is connecting the generator to a special integral equation. For more details on the theory of (classical) evolution semigroups we mention in particular the complete and self-contained monograph of C. Chicone and Yu. Latushkin [5] and the paper of Nguyen Van Minh, F. R¨ abiger and R. Schnaubelt [9]. The theory of uniform behavior is too restrictive, and it is important to consider a more general view, such as nonuniform asymptotic stability. A serious motivation for weakening the notion of uniform exponential behavior lies in the ergodic theory. In this regard, a consistent contribution is due to Ya. Pesin, L. Barreira and C. Valls (we refer the reader to monographs [1, 3] and the references therein). Roughly speaking, while uniformity relates to the finiteness of the Bohl exponents [6, Chapter 3], nonuniformity analyses situations when the Lyapunov exponent is finite [3]. This type of argumentation is not at all of a formal type, as illustrated in our examples. The orbits of the evolution family in our Example 4.4 are all asymptotically stable, even if the evolution family is not uniform exponentially bounded. In this case it is impossible to construct the evolution semigroup. From another hand, Example 4.5 (Perron) delivers a uniform exponentially bounded evolution family which is not uniform exponentially stable, with all the orbits being asymptotically stable. In this case the evolution semigroup exists, but it does not furnish any kind of information about the (nonuniform) asymptotic behavior of the orbits. To sum up, in many situation the classical tool either does not exist or it is completely useless. Our main goal is to emphasize an important difference between uniform and nonuniform behavior: while in the uniform case all the evolution families reflect into a unique output function space (of functions vanishing at 0 and infinity, for instance), in the nonuniform case there are infinitely many output function spaces, which are the admissible ones. They depend on each evolution family, and on each particular admissible exponent. Let us point out that the theory of evolution semigroups and that of admissibility are quite similar. In fact, admissibility methods deal with couples of Banach function spaces, while in the case of evolution semigroups the spaces into each couple coincide. In this view, we prefer to use the term “admissible” throughout our paper. For a better understanding, we refer the reader to [8, 11] for uniform behavior, and to [4, 13] for the nonuniform setting. We emphasize that the idea of connecting the nonuniform behavior to C0 semigroups is already present in the recent paper [2], in the case of discrete families. This paper is organized as follows.
ADMISSIBLE BANACH FUNCTION SPACES FOR LINEAR DYNAMICS
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In Section 2 we introduce the set of admissible exponents associated to a nonuniform exponentially bounded evolution family, and the class of the corresponding Banach function spaces that we also call admissible. We emphasize the connections with the conditions on the finiteness of the corresponding Bohl and Lyapunov exponents. In Section 3 we generalize the notion of evolution semigroup for linear dynamics with nonuniform behavior, by shrinking the output space. We call this mathematical object a nonuniform evolution semigroup. Moreover, we point out the connection with the uniform case. The last section presents an introspective study of nonuniformity via evolution semigroups. We characterize nonuniform exponential stability in terms of invertibility of the corresponding infinitesimal generators. In this regard, we introduce a new notion that we call quasi-negative exponent, specific to nonuniform behavior. As a consequence, we prove a spectral mapping theorem for nonuniform evolution semigroups. We consider as our main result the statement in Theorem 4.6. 2. Admissible Banach function spaces Throughout our paper X is a Banach space. We denote C(R+ , X) the space of all continuous, X-valued functions defined on the half-line, and Cc,0 (R+ , X) is the space of all functions in C(R+ , X) with compact support vanishing at 0. We also make use of the following notation: n o C00 (R+ , X) = u ∈ C(R+ , X) : lim u(t) = u(0) = 0 . t→∞
Definition 2.1. For any fixed α ∈ R, the evolution family U is called α-nonuniform exponentially bounded, if there exists a continuous map Mα : R+ → (0, ∞) such that k U (t, s) k≤ Mα (s)eα(t−s) , t ≥ s ≥ 0. (2)
If the above estimation holds for some α < 0, then U is called α-nonuniform exponentially stable. Each α satisfying (2) is called an admissible exponent, and the set of all admissible exponents is denoted A (U). Evidently for each evolution family U, the set A (U) is either a (semi) infinite interval, or empty. If A (U) 6= ∅, then the evolution family U is called nonuniform exponentially bounded, and if A (U) contains negative admissible exponents, we say that U is nonuniform exponentially stable. We emphasize that our paper is devoted to the study of the nonuniform exponentially bounded evolution families, that is we only consider the case A (U) 6= ∅. In the above terminology, whenever there exists a bounded map Mα (s) satisfying (2) (which is equivalent to the existence of a constant one), we just replace the term “nonuniform” with “uniform”. Also, in such cases we call α a strict (admissible) exponent, and we denote As (U) the set of all strict exponents. Remark 2.2. Assume that the evolution family U is reversible (i.e. U (t, s) is invertible for all t ≥ s ≥ 0, U (s, t) denotes its inverse). If the Lyapunov exponent KL is finite and not attained, then A (U) = (KL , ∞) (in particular KL = −∞ whenever A (U) = R). Also if the Bohl exponent KB is finite and not attained, then As (U) = (KB , ∞). The intervals of admissibility are closed at their left endpoints whenever the Lyapunov or the Bohl exponents are attained.
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Indeed, as
KL = inf α ∈ R : there exists Mα > 0 with kU (t, 0)k ≤ Mα eαt , t ≥ 0
and is not attained, if α ∈ A (U), replacing s = 0 in (2), one has k U (t, 0) k≤ Mα (0)eαt ,
thus α ∈ (KL , ∞). For α ∈ (KL , ∞), assuming that U is reversible, we get
kU (t, s)k = kU (t, 0)U (0, s)k ≤ Mα eαt kU (0, s)k = Mα eαs kU (0, s)k eα(t−s) ,
that implies α ∈ A (U). The second statement can be proved similarly if we notice that n o KB = inf α ∈ R : there exists Mα > 0 with kU (t, s)k ≤ Mα eα(t−s) , t ≥ s ≥ 0 . Suppose that A (U) 6= ∅, and let α ∈ A (U). For each t ≥ 0 and u ∈ C(R+ , X) we set ϕU ,α (t, u) = sup e−α(τ −t) k U (τ, t)u(t) k .
(3)
τ ≥t
Inequality (2) implies
k u(t) k≤ ϕU ,α (t, u) ≤ Mα (t) k u(t) k, t ≥ 0.
(4)
If in particular u(t) ≡ x for some x ∈ X, we step over the norm on X defined in [4, Eq. (5)], precisely kxkt = sup e−α(τ −t) k U (τ, t)x k. In this regard, we notice τ ≥t
that the map ϕU ,α in (3) can be (indirectly) defined as ϕU ,α (t, u) = ku(t)kt . The following result is essential in the sequel. Proposition 2.3. The map R+ ∋ t 7→ ϕU ,α (t, u) ∈ R+ is continuous for each fixed α ∈ A (U) and u ∈ C(R+ , X). In addition, for each u ∈ C(R+ , X) for which lim ϕU ,α (t, u) = 0, there exists (possibly not unique) tu ≥ 0 such that
t→∞
sup ϕU ,α (t, u) = ϕU ,α (tu , u). t≥0
Proof. To prove the first statement we set V (t, s) = e−α(t−s) U (t, s). It follows that V = {V (t, s)}t≥s≥0 is also an evolution family with k V (t, s) k≤ Mα (s), t ≥ s ≥ 0. For fixed u ∈ C(R+ , X), t0 ≥ 0 and ε > 0, there exists δ1 , δ2 > 0 such that |t − t0 | < δ1 ⇒ Mα (t) ku (t) − u (t0 )k < ε/3,
t0 − δ2 < t0 ≤ t < t0 + δ2 ⇒ Mα (t) ku (t0 ) − V (t, t0 ) u (t0 )k < ε/3. Let δ = max {δ1 , δ2 } and choose t ≥ 0 with |t − t0 | < δ. We only analyze the case t ≥ t0 . For any τ ≥ t we have kV (τ, t)u (t)k ≤ kV (τ, t) (u (t) − u (t0 ))k + kV (τ, t)u (t0 ) − V (τ, t0 )u (t0 )k + kV (τ, t0 )u (t0 )k
≤ kV (τ, t)k ku (t) − u (t0 )k + kV (τ, t)k ku (t0 ) − V (t, t0 ) u (t0 )k + kV (τ, t0 )u (t0 )k
≤ Mα (t) ku (t) − u (t0 )k + Mα (t) ku (t0 ) − V (t, t0 ) u (t0 )k + kV (τ, t0 )u (t0 )k
≤ (2ε)/3 + ϕU ,α (t0 , u).
ADMISSIBLE BANACH FUNCTION SPACES FOR LINEAR DYNAMICS
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When taking the supremum with respect to τ ≥ t, we get ϕU ,α (t, u) − ϕU ,α (t0 , u) < ε. Similarly one can prove that ϕU ,α (t0 , u) − ϕU ,α (t, u) < ε, that is the map t 7→ ϕU ,α (t, u) is continuous at t0 . The second statement follows from the continuity of the map in question, together with condition lim ϕU ,α (t, u) = 0. t→∞
For each α ∈ A (U) we set n o C(U, α) = u ∈ C(R+ , X) : lim ϕU ,α (t, u) = ϕU ,α (0, u) = 0 . t→∞
It is a kind of straightforward argument to verify that C(U, α) is a Banach (function) space equipped with the norm k u kU ,α = sup ϕU ,α (t, u), t≥0
called the admissible Banach function space corresponding to the evolution family U and the admissible exponent α ∈ A(U). Eq. (4) also implies Cc,0 (R+ , X) ⊂ C(U, α) ⊂ C00 (R+ , X).
(5)
Let us remark that if α ∈ A (U), u ∈ C(R+ , X) and β ≥ α, then β ∈ A (U), ϕU ,β (t, u) ≤ ϕU ,α (t, u). Moreover, C(U, α) ⊂ C (U, β) and k u kU ,β ≤k u kU ,α . 3. Nonuniform evolution semigroups In this section we introduce the concept of nonuniform evolution semigroup, emphasizing the connections with the uniform case. Theorem 3.1. Each α ∈ A (U) defines a C0 -semigroup Tα = {Tα (t)}t≥0 on C(U, α) by setting ( U (s, s − t)u(s − t), if s > t, (Tα (t)u)(s) = (6) 0, if 0 ≤ s ≤ t.
Moreover, the following estimation holds
k Tα (t)u kU ,α ≤ eαt k u kU ,α , u ∈ C(U, α).
(7)
Proof. Evidently Tα (0) = Id and Tα (t)Tα (s) = Tα (t + s), for all t, s ≥ 0. It remains to prove that the map Tα (t) in (6) is well defined on C(U, α), and the norm k Tα (t)u − u kU ,α → 0 as t → 0+ , for each fixed u ∈ C(U, α). Pick u ∈ C(U, α) and t ≥ 0. For arbitrary s ≥ t we have ϕU ,α (s, Tα (t)u) = sup e−α(τ −s) k U (τ, s − t)u(s − t) k τ ≥s
= eαt sup e−α[τ −(s−t)] k U (τ, s − t)u(s − t) k τ ≥s
≤ eαt ϕU ,α (s − t, u), therefore ϕU ,α (s, Tα (t)u) → 0 as s → ∞, which leads to Tα (t)u ∈ C(U, α). Notice that the above estimation also proves inequality (7). We now show that the space Cc,0 (R+ , X) is dense in C(U, α) with respect to the norm k · kU ,α . For any fixed
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u ∈ C(U, α) and any non-negative integer n ∈ N, let us consider a continuous function αn : R+ → [0, 1] such that αn (t) = 1, for all t ∈ [0, n], and αn (t) = 0, for all t ≥ n + 1. Putting un = αn u, we notice that un ∈ Cc,0 (R+ , X). We claim that the limit lim k un − u kU ,α = 0. Indeed, as lim ϕU ,α (t, u) = 0, it follows that for each ε > 0 n→∞
t→∞
there exists δ > 0 such that for t > δ we have ϕU ,α (t, u) < ε/2. Set n0 = [δ] + 1 and choose n ≥ n0 . The definition of map αn readily implies k un − u kU ,α = sup ϕU ,α (t, un − u) ≤ sup ϕU ,α (t, u) < ε, t≥n
t≥n
which concludes the claim. For the second statement, pick u ∈ Cc,0 (R+ , X). There exist a, b ≥ 0, a < b such that supp(Tα (t)u − u) ⊂ [a, b], for sufficiently small t ≥ 0. For such t we have k Tα (t)u − u kU ,α = sup ϕU ,α (s, Tα (t)u − u) s≥0
≤ sup Mα (s) k (Tα (t)u)(s) − u(s) k s≥0
=
sup s∈supp(Tα (t)u−u)
≤ Kα
Mα (s) k (Tα (t)u)(s) − u(s) k
sup s∈supp(Tα (t)u−u)
k (Tα (t)u)(s) − u(s) k,
where Kα = max Mα (s). Using standard arguments (ex. [12]), one can easily s∈[a,b]
prove that k Tα (t)u − u kU ,α → 0 as t → 0+ , and this completes the proof.
The C0 -semigroup Tα defined above is called the nonuniform evolution semigroup associated to the evolution family U and the admissible exponent α. We denote GU ,α its generator. Remark 3.2. If the Banach function spaces C(U, α) and C(U, β) coincide for some admissible exponents α, β ∈ A(U), then Tα = Tβ and GU ,α = GU ,β . While in the uniform case an evolution family defines a unique evolution semigroup T on C00 (R+ , X), in the nonuniform case there are infinitely many nonuniform evolution semigroups Tα defined on C(U, α), α ∈ A(U). The next proposition emphasizes the connection between the concepts of uniform and nonuniform evolution semigroup. Proposition 3.3. For α ∈ A (U), the Banach spaces C(U, α) and C00 (R+ , X) coincide if and only if α is a strict exponent. In this case, the nonuniform evolution semigroup Tα coincides with the (uniform) evolution semigroup T on C00 (R+ , X). Proof. Necessity. Let us assume that C(U, α) = C00 (R+ , X), for some α ∈ A (U). If α is not a strict exponent, then for each positive integer n ∈ N∗ there exist tn ≥ sn ≥ 0 and xn ∈ X with k xn k= 1, such that k U (tn , sn )xn k> neα(tn −sn ) .
If the sequence {sn }n∈N∗ is bounded, say sn ≤ k, then inequality n < e−α(tn −sn ) k U (tn , sn )xn k≤ Mα (sn ) ≤ sup Mα (s) 0≤s≤k
ADMISSIBLE BANACH FUNCTION SPACES FOR LINEAR DYNAMICS
7
leads to a contradiction, n < sup Mα (s) for all n ∈ N∗ . Thus, without loss of 0≤s≤k
generality one can always assume that the sequence {sn }n∈N∗ is strictly increasing, xn unbounded and let us put t0 = s0 = 0, x0 = 0. Setting yn = √ , n ≥ 1 and y0 = 0, n one gets √ e−α(tn −sn ) k U (tn , sn )yn k≥ n, n ∈ N. Consider uy : R+ → X by uy (s) =
s (yn+1 − yn ) sn+1 yn − sn yn+1 + , if s ∈ [sn , sn+1 ] , n ∈ N. sn+1 − sn sn+1 − sn
We notice that uy (sn ) = yn for each n ∈ N, that results in uy ∈ C00 (R+ , X). From estimation √ ϕU ,α (sn , uy ) ≥ e−α(tn −sn ) k U (tn , sn )yn k≥ n,
we deduce uy ∈ / C(U, α), that is C(U, α) 6= C00 (R+ , X), which is a contradiction. Sufficiency. If α is a strict exponent, then there exists Mα > 0 such that k U (t, s) k≤ Mα eα(t−s) for t ≥ s ≥ 0. Assumption u ∈ C00 (R+ , X) implies ϕU ,α (t, u) ≤ Mα k u(t) k→ 0 as t → ∞, hence u ∈ C(U, α).
The following result is taken from the theory of C0 -semigroups (see [10, Theorem 2.4] or [7, Lemma 1.3]): Lemma 3.4. Let T = {T (t)}t≥0 be a C0 -semigroup on a Banach space E, and G its infinitesimal generator. If x, y ∈ E, then x ∈ D(G) and Gx = y if and only if Z t T (t)x − x = T (ξ)ydξ, t ≥ 0. 0
Let us substitute E = C(U, α), x = u, y = −f and G = GU ,α . This rewrites as follows: if u, f ∈ C(U, α), then u ∈ D(GU ,α ) and GU ,α u = −f if and only if Z t u(t) = U (t, ξ)f (ξ)dξ, t ≥ 0. (8) 0
We notice that Lemma 1.1 in [9] uses the same source (Lemma 3.4) to present its similar conclusions. 4. Criteria for nonuniform exponential stability Lemma 4.1. If there exists α ∈ A(U), α < 0 (i.e. U is α-nonuniform exponentially stable), then the generator GU ,α is invertible and
1
−1 ≤ − kf kU ,α . (9)
GU ,α f α U ,α Proof. For each f ∈ C(U, α) we define Z t uf (t) = U (t, ξ)f (ξ)dξ, t ≥ 0. 0
From f = 0 ⇒ uf = 0 we deduce that GU ,α is a one-to-one map on C(U, α). To prove that GU ,α is invertible, one needs to show first that uf ∈ C(U, α) for each
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f ∈ C(U, α). From estimation ϕU ,α (t, uf ) ≤ sup e
−α(τ −t)
τ ≥t
τ ≥t
=
0
t
k U (τ, ξ)f (ξ) k dξ
0
≤ sup e−α(τ −t) Z
Z
t
Z
0
t
eα(τ −ξ) ϕU ,α (ξ, f )dξ
eα(t−ξ) ϕU ,α (ξ, f )dξ,
as lim ϕU ,α (t, f ) = 0, one gets lim ϕU ,α (t, uf ) = 0, therefore uf ∈ C(U, α). We t→∞
t→∞
conclude that GU ,α is algebraically invertible and −uf = G−1 U ,α f . Furthermore, inequality Z t ϕU ,α (t, uf ) ≤ eα(t−ξ) ϕU ,α (ξ, f )dξ 0
yields
kuf kU ,α ≤ kf kU ,α
Z
t 0
eα(t−ξ) dξ ≤ −
1 kf kU ,α , α
that is G−1 U ,α is bounded and estimation (9) holds.
Next Lemma is crucial in the sequel, and generalizes the implication (ii) ⇒ (i) from Theorem 2.2 in [9], for nonuniform exponentially bounded families. Obviously our techniques, based on induction, are of a completely different type (otherwise the methods and constructions used in the bibliographic source we refer to cannot apply in the nonuniform setting). Lemma 4.2. If the generator GU ,α is invertible for some α ∈ A (U), then U is nonuniform exponentially stable. Proof. Without loss of generality we may assume that α ≥ 0. Suppose that the operator GU ,α : D(GU ,α ) ⊂ C(U, α) → C(U, α) is invertible and put c = c(α) =k G−1 U ,α k .
n
For each positive integer n ∈ N∗ we denote θn = ln ene−1 → 0. For fixed t > s ≥ 0 and n large enough such that s + θn ≤ t ≤ n, let us consider a continuous function αn : R+ → [0, 1] with 0, if 0 ≤ ξ ≤ s, αn (ξ) = 1, if s + θn ≤ ξ ≤ n, 0, if ξ ≥ n + θn . Step 1. We prove that
k U (t, s) k≤ (cα + 1)Mα (s), for t ≥ s ≥ 0.
(10)
Evidently (10) holds for α = 0. Assume now that α > 0 and fix t > s ≥ 0, and x ∈ X. For n large enough we define ( αn (ξ)e−α(ξ−s) U (ξ, s)x, if ξ > s, fn (ξ) = 0, if 0 ≤ ξ ≤ s. We have fn ∈ Cc,0 (R+ , X) and thus fn ∈ C(U, α). We claim that k fn kU ,α ≤ Mα (s) k x k .
ADMISSIBLE BANACH FUNCTION SPACES FOR LINEAR DYNAMICS
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Indeed, if ξ ≤ s, as fn (ξ) = 0, one has ϕU ,α (ξ, fn ) = 0, meanwhile for ξ > s the following estimation holds: ϕU ,α (ξ, fn ) = sup e−α(τ −ξ) k U (τ, s)x k αn (ξ)e−α(ξ−s) τ ≥ξ
≤ sup e−α(τ −s) k U (τ, s)x k≤ Mα (s) k x k . τ ≥ξ
Putting un = G−1 U ,α (−fn ) one gets Z t un (t) = U (t, ξ)fn (ξ)dξ 0
=
Z
s+θn
αn (ξ)e−α(ξ−s) dξ U (t, s)x +
Z
t
αn (ξ)e−α(ξ−s) dξ U (t, s)x
s+θn
s
i 1 h −αθn = In U (t, s)x + − e−α(t−s) U (t, s)x. e α R s+θn −α(ξ−s) Here In = s αn (ξ)e dξ. Inequality 0 ≤ In ≤ α1 1 − e−αθn leads to 1 −αθn k U (t, s)x k e α
1 1 1 − e−αθn k U (t, s)x k + e−α(t−s) k U (t, s)x k α α 1 1 ≤k un kU ,α + 1 − e−αθn k U (t, s)x k + Mα (s) k x k α α 1 1 1 − e−αθn k U (t, s)x k + Mα (s) k x k ≤ c k fn kU ,α + α α cα + 1 1 1 − e−αθn k U (t, s)x k . ≤ Mα (s) k x k + α α Now inequality (10) results immediately when letting n → ∞. Step 2. For all k ∈ N the following holds ≤k un (t) k +
k U (t, s) k≤
ck k! (cα + 1)Mα (s), t > s ≥ 0. (t − s)k
(11)
Step 1 implies that inequality (11) holds for k = 0. Assume that (11) holds for some k ∈ N. For fixed t > s ≥ 0, x ∈ X and sufficiently large n we consider ( αn (ξ)(ξ − s)k U (ξ, s)x, if ξ > s, gn,k (ξ) = 0, if 0 ≤ ξ ≤ s. Since gn,k ∈ Cc,0 (R+ , X), it follows that gn,k ∈ C(U, α). For 0 ≤ ξ ≤ s we notice that ϕU ,α (ξ, gn,k ) = 0, and if ξ > s we have ϕU ,α (ξ, gn,k ) = sup e−α(τ −ξ) k U (τ, ξ)gn,k (ξ) k τ ≥ξ
= sup e−α(τ −ξ) αn (ξ)(ξ − s)k k U (τ, s)x k τ ≥ξ
≤ sup(τ − s)k k U (τ, s)x k τ ≥ξ
that results in
≤ ck k!(cα + 1)Mα (s) k x k , k gn,k kU ,α ≤ ck k!(cα + 1)Mα (s) k x k .
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N. LUPA AND L.H. POPESCU
If un,k = G−1 (−gn,k ), then Z t un,k (t) = U (t, ξ)gn,k (ξ)dξ 0 Z t (ξ − s)k dξ U (t, s)x = In,k U (t, s)x + s+θn
= In,k U (t, s)x + where In,k =
R s+θn s
1 (t − s)k+1 − θnk+1 U (t, s)x, k+1
αn (ξ)(ξ − s)k dξ ≤
1 k+1 k+1 θn .
Let us estimate
(t − s)k+1 k U (t, s)x k k+1 1 k+1 θ k U (t, s)x k +In,k k U (t, s)x k k+1 n 2 ≤ ck+1 k!(cα + 1)Mα (s) k x k + θk+1 k U (t, s)x k . k+1 n Letting n → ∞ we deduce that (11) works for k + 1. Step 3. Pick δ ∈ (0, 1). Multiplying (11) by δ k and summing with respect to k ∈ N one easily gets δ cα + 1 Mα (s)e− c (t−s) , t ≥ s ≥ 0, k U (t, s) k≤ 1−δ that ends our proof. ≤k un,k (t) k +
Definition 4.3. We say that the admissible exponent α ∈A (U) is quasi-negative if C(U, α) = C(U, −ν), for some admissible exponent −ν < 0. It follows that each negative admissible exponent is quasi-negative. We also notice that the set of all quasi-negative exponents is an interval. According to Proposition 3.3, the evolution family U is uniform exponentially stable if and only if it has strict quasi-negative exponents. In the nonuniform case the situation is more complicate, as illustrated in the below examples. Example 4.4. For t ≥ s ≥ 0 we set
E (t, s) = s (2 + sin s) − t (2 + sin t) ,
E(t,s)
and let U (t, s) x = e x, x ∈ X. We claim that the evolution family U is nonuniform exponentially stable, but not uniform exponentially bounded. In fact, A (U) = [−1, ∞) and C(U, α) = C (U, −1) for all α ∈ A (U) ,
that is all admissible exponents are quasi-negative. Indeed, let us notice that
E1 (t, s) = E (t, s) + t − s = s(1 + sin s) − t(1 + sin t) ≤ s + s sin s = f1 (s) , and as E1 (2nπ + 3π/2, s) = f1 (s) for n sufficiently large, one obtains supE1 (t, s) = f1 (s) . t≥s
It follows that −1 ∈ A (U). If ε > 0, then for
Eε (t, s) = E (t, s) + (1 + ε) (t − s) = s (1 − ε + sin s) − t (1 − ε + sin t) ,
ADMISSIBLE BANACH FUNCTION SPACES FOR LINEAR DYNAMICS
11
we get Eε (2nπ + 3π/2, s) = s (1 − ε + sin s) + ε (2nπ + 3π/2) ,
which implies that supEε (t, s) = ∞, and consequently −1 − ε ∈ / A (U). Thus, t≥s
A (U) = [−1, ∞). Since the map f1 is unbounded, the evolution family U is not uniform exponentially bounded. For any fixed α ∈ A (U) let us define Eα (t, s) = E (t, s) − α (t − s) = s (2 + α + sin s) − t (2 + α + sin t) , and put f2 (s) = supEα (t, s). For u ∈ C(R+ , X) we have t≥s
ϕU ,α (s, u) = supe−α(t−s) kU (t, s) u (s)k = supeEα (t,s) k u(s) k= ef2 (s) k u (s) k , t≥s
t≥s
f1 (s)
and similarly k u (s) k . For any fixed s ≥ 0 we denote s ϕU ,−1 (s, u) = e ns = 2π − 34 , that implies ts = 2(ns + 1)π + 3π/2 ∈ (s, s + 2π]. Let us remark that f1 (s) − f2 (s) ≤ f1 (s) − Eα (ts , s) = (1 + α) (ts − s) ≤ 2π (1 + α) . Gathering the above identities and estimation, one gets ϕU ,−1 (s, u) = ef1 (s) k u (s) k= ef1 (s)−f2 (s) ϕU ,α (s, u) ≤ e2π(1+α) ϕU ,α (s, u) . We conclude that C(U, α) = C(U, −1). For the evolution family in the next example the situation is completely different. It has no admissible exponents both positive and quasi-negative at the same time. Besides, it is uniform exponentially bounded. Example 4.5. For t ≥ s ≥ 0 we set √ √ E (t, s) = s 2 + sin ln s − t 2 + sin ln t ,
and let
√ √ Eα (t, s) = E (t, s) − α (t − s) = s α + 2 + sin ln s − t α + 2 + sin ln t ,
for fixed α ∈ R. We consider the evolution family U (t, s) x = eE(t,s) x, t ≥ s ≥ 0, x ∈ X. √ We claim that A (U) = 1 − 2, ∞ , the quasi-negative admissible exponents are only the negative ones, and U is uniform exponentially bounded. √ √ If α < 1 − 2, then α = 1 − 2 − ε, for some ε > 0. Let us notice that E1−√2−ε (t, s) = s (1 − ε + sin ln s) − t (1 − ε + sin ln t) . 3π
For tn = e2nπ+ 2 , n ∈ N, we have E1−√2−ε (tn , 0) = εtn → ∞, which implies that α∈ / A (U). √ √ √ If α ∈ 1 − 2, 0 , then there exists ε ∈ 0, 2 − 1 such that α = 1 − 2 + ε. In this case we have E1−√2+ε (t, s) = s (1 + ε + sin ln s) − t (1 + ε + sin ln t)
≤ s (1 + sin ln s) = f1 (s) . √ Thus, α ∈ A (U). We conclude that A (U) = 1 − 2, ∞ .
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N. LUPA AND L.H. POPESCU
√ For α ≥ 0 let us put fα (t) = t α + 2 + sin ln t , t ≥ 0. The derivative is π i √ h + ln t ≥ 0. fα′ (t) = α + 2 1 + sin 4
Thus,
Eα (t, s) = fα (s) − fα (t) = fα′ (θt,s ) (s − t) ≤ 0, for t > s ≥ 0, α(t−s) and so kU(t, s)k . This means that U is uniform exponentially bounded. √ ≤ e If α ∈ 1 − 2, 0 , as f1 (s) is unbounded, the inclusion C(U, α) ⊂ C00 (R+ , X) is strict. In this case the admissible exponent α is not quasi-negative.
We consider the next theorem as the main result of the paper. Theorem 4.6. Let α ∈ A (U). The generator GU ,α is invertible if and only if α is a quasi-negative admissible exponent. Proof. The sufficiency follows directly from Lemma 4.1 and Remark 3.2. For the necessity, assume that GU ,α is invertible. According to Lemma 4.2, U is nonuniform exponentially stable. Choose ν > 0 with −ν ∈ A (U), −ν 6= inf A (U). It suffices to consider the case α ≥ 0. For each fixed s ≥ 0 and x ∈ X r {0} we construct the map u es,x ∈ C(R+ , X) given by ( eν(ξ−s) U (ξ, s) x, if ξ > s; u es,x (ξ) = (12) x, if 0 ≤ ξ ≤ s. Pick ε > 0 such that −ν − ε ∈ A (U). For t > s we have
ϕU ,α (t, u es,x ) = sup e−α(τ −t) eν(t−s) kU (τ, s) xk τ ≥t
≤ sup eν(t−s) e−(ν+ε)(τ −s) M−ν−ε (s) kxk τ ≥t
≤ e−ε(t−s) M−ν−ε (s) kxk , therefore lim ϕU ,α (t, u es,x ) = 0. Proposition 2.3 implies that the below set is t→∞ nonempty: ( ) Λs,x =
t ≥ 0 : sup ϕU ,α (ξ, u es,x ) = ϕU ,α (t, u es,x ) . ξ≥0
As the map ξ 7→ ϕU ,α (ξ, u es,x ) is continuous, it follows that ts,x = inf Λs,x ∈ Λs,x . We have the alternative: (A1) There exists ν > 0, −ν ∈ A (U), −ν 6= inf A (U) such that for each s ≥ 0 and x ∈ X r {0}, ts,x ∈ s, s + ν1 . (A2) For any sufficiently small ν > 0, −ν ∈ A (U), −ν 6= inf A (U), there exists s ≥ 0 and x ∈ X r {0} with ts,x > s + ν1 . Assume that (A1) holds. In this case, for all t ≥ s and x ∈ X r {0} we have ku es,x (t) k ≤ ϕU ,α (t, u es,x ) ≤ ϕU ,α (ts,x , u es,x )
= sup e−α(τ −ts,x ) eν(ts,x −s) kU (τ, s) xk . τ ≥ts,x
ADMISSIBLE BANACH FUNCTION SPACES FOR LINEAR DYNAMICS
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The above estimation yields sup eν(t−s) kU (t, s) xk ≤ sup e−α(τ −ts,x ) eν(ts,x −s) kU (τ, s) xk τ ≥ts,x
t≥s
= sup e(α+ν)(ts,x −s) e−α(τ −s) kU (τ, s) xk τ ≥ts,x 1
≤ e ν (α+ν) sup e−α(τ −s) kU (τ, s) xk . τ ≥s
Therefore, for all s ≥ 0 and x ∈ X we have
sup e−α(t−s) kU (t, s) xk ≤ sup eν(t−s) kU (t, s) xk ≤ Ksup e−α(t−s) kU (t, s) xk , t≥s
t≥s
where K = e
1 ν (α+ν)
t≥s
. It turns out that
ϕU ,α (s, u) ≤ ϕU ,−ν (s, u) ≤ KϕU ,α (s, u) , s ≥ 0, u ∈ C(R+ , X).
We conclude that C(U, α) = C(U, −ν), as Banach spaces, thus α is quasi-negative. Assume that (A2) holds. For sufficiently large n ∈ N∗ such that − n1 ∈ A (U), 1 − n 6= inf A (U), we put sn ≥ 0, xn ∈ X r {0} and tn = tsn ,xn > sn + n, as in the statement (A2). Let us define the C 1 -map ψn : R+ → R+ by 1 (t−s ) n , if t > tn ; en t e n1 (tn −sn ) + e n1 (tn −sn ) 1 − tn , if s + δ < t ≤ t ; n n n n ψn (t) = n 2 a t + b t + c , if s < t ≤ s + δ n n n n n n; 0, if 0 ≤ t ≤ sn .
The constants an , bn , cn ∈ R and δn ∈ (sn , tn ) are determined by condition ψn ∈ C 1 (R+ ). We also define un , fn : R+ → X by ( ψn (t) U (t, sn ) xn , if t > sn ; un (t) = 0, if 0 ≤ t ≤ sn , and fn (t) =
(
ψn′ (t) U (t, sn ) xn , if t > sn ; 0, if 0 ≤ t ≤ sn .
Evidently un , fn ∈ C(U, α) and GU ,α un = −fn . Notice that for t ∈ [0, tn ] we have ϕU ,α (t, un ) = sup e−α(τ −t) ψn (t) kU (τ, sn ) xn k τ ≥t
1
≤ sup e−α(τ −t) e n (t−sn ) kU (τ, sn ) xn k τ ≥t
= ϕU ,α (t, u esn ,xn ) ,
esn ,xn ), where u esn ,xn is defined in (12). If t ≥ tn , as ϕU ,α (t, un ) = ϕU ,α (t, u it follows that k un kU ,α = ϕU ,α (tn , u esn ,xn ), and similarly one gets k fn kU ,α = 1 esn ,xn ). These identities imply n ϕU ,α (tn , u which leads to
sup f ∈C(U ,α)
kG−1 U,α f kU,α kf kU,α
k un kU ,α = n, k fn kU ,α
≥ n. Thus, G−1 U ,α is not bounded, which is false,
and eventually only alternative (A1) holds, that proves the claim.
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N. LUPA AND L.H. POPESCU
As in [9], the spectral mapping theorem is to be deduced from exponential stability. Corollary 4.7 (The Spectral Mapping Theorem). For each α ∈ A (U), the nonuniform evolution semigroup Tα = {Tα (t)}t≥0 satisfies the identity etσ(GU,α ) = σ (Tα (t)) r {0} , t ≥ 0. Moreover, σ (GU ,α ) is a left half-plane and σ (Tα (t)), t ≥ 0, is a disc. Proof. If λ ∈ C, then we consider the rescaled evolution family Uλ (t, s) = e−λ(t−s) U (t, s). For arbitrary α ∈ A (U), t ≥ 0 and u ∈ C(R+ , X) one has ϕUλ ,α−Re λ (t, u) = sup e−(α−Re λ)(τ −t) kUλ (τ, t) u (t)k τ ≥t
= sup e−α(τ −t) kU (τ, t) u (t)k = ϕU ,α (t, u) . τ ≥t
We deduce that the admissible sets A (Uλ ) = A (U) − Re λ, and the admissible Banach spaces C(Uλ , α − Re λ) = C(U, α). From another hand, the definition of the generator implies that D (GU ,α ) = D (GUλ ,α−Re λ ) and GUλ ,α−Re λ = GU ,α − λId.
(13)
Let λ ∈ ρ (GU ,α ), that is GUλ ,α−Re λ is invertible. Choose µ ∈ C with Re µ ≥ Re λ. We only have two options: (B1) Re µ > α; (B2) Re λ ≤ Re µ ≤ α. In case (B1), Theorem 4.6 and formula (13) imply that GUµ ,α−Re µ is invertible, consequently µ ∈ ρ (GU ,α ). Assume that (B2) holds. According to the same Theorem 4.6, as GUλ ,α−Re λ is invertible, there exists ν > 0 with C(Uλ , α − Re λ) = C(Uλ , −ν), and GUλ ,α−Re λ = GUλ ,−ν . Using Eq. (13), we successively have GUµ ,α−Re µ = GUλ ,α−Re λ + (λ − µ) Id = GUλ ,−ν + (λ − µ) Id = GUµ ,−ν+Re λ−Re µ .
Since −ν + Re λ − Re µ < 0, then GUµ ,−ν+Re λ−Re µ is invertible, hence GUµ ,α−Re µ is also invertible. This shows that µ ∈ ρ (GU ,α ), which implies that the spectrum σ (GU ,α ) is a left half-plane. For the rest of the proof we refer the reader to [9]. We now recover the result from Theorem 2.2 in [9], valid in the particular case of uniform exponential stability. Corollary 4.8. Let U be a uniform exponentially bounded evolution family. Then U is uniform exponentially stable if and only if the generator of the corresponding evolution semigroup is invertible.
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[email protected] L. H. Popescu, Department of Mathematics and Informatics, Faculty of Sciences, ˘t University of Oradea, Universita ¸ ii St. 1, 410087 Oradea, Romania E-mail address:
[email protected]
