Admissible Banach Function Spaces and Nonuniform Stabilities
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Admissible Banach Function Spaces and Nonuniform Stabilities Nicolae Lupa
and Liviu Horia Popescu
Abstract. For nonuniform exponentially bounded evolution families defined on Banach spaces, we introduce a class of Banach function spaces, whose norms are completely determined by the nonuniform behaviour of the corresponding evolution family. We generalize a classical theorem of Datko on these spaces. In addition, we obtain new criteria for the existence of nonuniform stability. Mathematics Subject Classification. 34D20, 37D25. Keywords. Nonuniform stabilities, admissible exponents, Datko’s theorem.
1. Introduction In a recent paper, the authors introduced a special class of Banach function spaces to characterize the concept of nonuniform exponential stability for evolution families in terms of invertibility of the infinitesimal generators of certain C0 -semigroups [12]. In fact, the results obtained in [12] prove the necessity of the study of nonuniform behaviour of an evolution family for each individual admissible exponent, requiring substantial departure from well-known ideas. The aim of this paper is twofold. On the one hand, we try to give some new criteria for the existence of nonuniform stability, which is the case when 0 is an admissible exponent. So, we define some appropriate projections and analyse their invariance in respect to a class of admissible Banach function spaces. On the other hand, motivated by a recent work of Dragiˇcevi´c [8], we intend to generalize a classical result of Datko on these spaces. The reader will surely notice that our techniques are of a completely different type from those in [8], where the results in the case of continuous time are obtained by reducing the dynamics to the case of discrete time (see also Remark 3.9). 0123456789().: V,-vol
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N. Lupa and L. H. Popescu
MJOM
A notable result in the stability theory of ODEs lies in Datko’s theorem. To extend a classical theorem of Lyapunov to strongly continuous semigroups of operators in Hilbert spaces, Datko proved that ∞ if {T (t)}t≥0 is a C0 semigroup on a complex Hilbert space X such that 0 T (t)x 2 dt < ∞ for some x ∈ X, then limt→∞ T (t)x = 0 (Lemma 3 in [5]). Later, using different techniques, Pazy showed that if for some p ∈ [1, ∞), the integral ∞ T (t)x p dt is finite for every x ∈ X, then {T (t)}t≥0 is uniform expo0 nentially stable, i.e. there exist M, α > 0 such that T (t) ≤ M e−αt , t ≥ 0 (see [14] and [15, Theorem 4.1, pp. 116]). In 1972 Datko generalized this result for evolution families on Banach spaces [6]. It is shown that a uniform exponentially bounded evolution family {U (t, s)}t≥s≥0 on a Banach space X is uniform exponentially stable if and only if there exists p ∈ [1, ∞) such that ∞ U (ξ, t)x p dξ < ∞, x ∈ X. sup t≥0
t
Ichikawa obtained in [9] a Datko-type theorem for families of nonlinear operators U (t, s) : Xs → Xt , t ≥ s ≥ 0, which satisfy the following conditions: (e1 ) U (t, t)x = x, x ∈ Xt ; (e2 ) U (t, τ )U (τ, s) = U (t, s) on Xs , t ≥ τ ≥ s; (e3 ) the map t →
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