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In this paper, we present two quite general approximation theorems for the propagators of higher order (in time) abstract Cauchy problems, which extend largely the classical Trotter-Kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions. Then, we apply the approximation theorems to deal with the second… (More)
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For a strongly continuous semigroup (T (t)) t0 with generator A we introduce its critical spectrum crit (T (t)). This yields in an optimal way the spectral mapping theorem (T(t)) = e tt(A) crit (T (t)) and improves classical stability results.
In this article we survey results concerning asymptotic properties of C0-semigroups on Banach spaces with respect to the weak operator topology. The property " no eigenvalues of the generator on the imaginary axis " is equivalent to weak stability for most time values; a phenomenon called " almost weak stability ". Further, sufficient conditions implying… (More)
We propose an abstract framework for the computation of the spectrum (A) of a linear operator A : D(A) X ! X on a Banach space X through a condition in a smaller Banach space X 1. If this space is nite dimensional this yields a characteristic equation for (A). The method is tested for delay, integro-diierential and population equations and is applicable to… (More)
We consider a strongly continuous semigroup (T(t)) t0 with generator A on a Banach space X, an Abounded perturbation B, and the semigroup (S(t)) t0 generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t)) t0. The results are applied to a cell equation with age… (More)
For a generator A of a C 0-semigroup T (·) on a Banach space X we consider the semi-norm M k x := lim sup t→0+ t −1 (T (t) − I)A k−1 x on the Favard space F k of order k associated with A. The use of this semi-norm is motivated by the functional analytic treatment of time-discretization methods of linear evolution equations. We show that sharp inequalities… (More)
We study linear neutral PDEs of the form (∂/∂t)F u t = BFu t + Φu t , t ≥ 0; u 0 (t) = ϕ(t), t ≤ 0, where the function u(·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ, we construct a C 0-semigroup on C 0 (R − ,X) yielding the solutions of the equation.
Introduction Partial differential equations on bounded domains of R n have traditionally been equipped with homogeneous boundary conditions (usually Dirichlet, Neumann, or Robin). However, other kinds of boundary conditions can also be considered, and for a number of concrete application it seems that dynamic (i.e., time-dependent) boundary conditions are… (More)