Polynomial Expansiveness and Admissibility of Weighted Lebesgue Spaces
- PDF / 209,857 Bytes
- 26 Pages / 499 x 709 pts Page_size
- 88 Downloads / 199 Views
Czechoslovak Mathematical Journal
26 pp
Online first
POLYNOMIAL EXPANSIVENESS AND ADMISSIBILITY OF WEIGHTED LEBESGUE SPACES Pham Viet Hai, Hanoi Received April 29, 2019. Published online August 18, 2020.
Abstract. The paper investigates the interaction between the notions of expansiveness and admissibility. We consider a polynomially bounded discrete evolution family and define an admissibility notion via the solvability of an associated difference equation. Using the admissibility of weighted Lebesgue spaces of sequences, we give a characterization of discrete evolution families which are polynomially expansive and also those which are expansive in the ordinary sense. Thereafter, we apply the main results in order to infer continuous-time characterizations for the notions of expansiveness through the solvability of an associated integral equation. Keywords: polynomial expansiveness; evolution family MSC 2020 : 34E05, 34D05
1. Introduction Throughout this paper, we denote by X a complex Banach space, and by L(X) the algebra of all bounded linear operators on X. The norm on X and L(X) will be denoted as k·k. Let Z, R denote the sets of integer, real numbers, respectively. For A ⊆ R we write A>δ = {x ∈ A : x > δ}. Denote ∆ = {(t, s) : t > s > 0}. Consider the homogeneous equation (1.1)
x′ (t) = A(t)y(t),
t>0
and the inhomogeneous equation (1.2)
y ′ (t) = A(t)y(t) + h(t),
t > 0,
where A(t) is in general an unbounded linear operator on X for all fixed t > 0. Equation (1.1) is called well-posed if we assume the existence, uniqueness and continuous DOI: 10.21136/CMJ.2020.0195-19
1
dependence of solutions on initial condition. Note that if A(t) is a bounded linear operator for all fixed t > 0, then the well-posedness is always ensured. In the qualitative theory of differential equations and dynamical systems, a central problem is to study the interaction between the solvability of equation (1.2) and the asymptotic behaviors of solutions of equation (1.1). Historically, this problem has the origin in the pioneering work of Perron in 1930, see [13]. Theorem 1.1 ([13]). Let X = Rd . If equation (1.2) admits at least one bounded solution for each bounded continuous function h(·), then each bounded solution of equation (1.1) goes to zero as t → ∞. Many results related to differential equations can carry over quite easily to corresponding results for difference equations. Similarities between differential equations and difference equations have been recognized and exploited very regularly. Thus, shortly after the publication of the work [13], a corresponding study for the case of discrete time was undertaken by Ta Li (see [8]) in which a similar method was done to obtain analogous results for difference equations. In both papers, a central interest is the relationship, for linear equations, between the solvability of the inhomogeneous equation for every bounded perturbation and a certain form of behavior of the solutions of the homogeneous equation. The assumption in Perron’s theorem is known as the admissibility
Data Loading...
