Sharp estimates for solutions of systems with aftereffect

Abstract

Abstract. Sharp estimates are established for strong solutions of systems of differential-difference equations of both neutral and retarded type.The approach is based on the study of the resolvent corresponding to the generator of the semigroup of shifts along the trajectories of a dynamical system. In the case of neutral type equations, the Riesz basis property of the subsystem of exponential solutions is used. §1. Introduction. Statement of the problem and main resultsIn this paper, we consider the initial value problem for a differential-difference equation of the formHere, A kj is a matrix of size r × r with constant complex entries, and the numbers h k are shifts, 0We denote by L(λ) the characteristic matrix of equation (1.1), The following statement about quasipolynomials is well known. Proposition 1.2. If det A 0m = 0, then the quantity κ + := sup λ q ∈Λ Re λ q is finite. Now we state the main result of the paper.2000 Mathematics Subject Classification. Primary 34K12.

Correction of the Mathematical Method for Studying a Cathode Lens with Two Planes of Symmetry

“…Classical methods for studying electron lenses [4][5][6] are unsuitable for developing the theory of cathode lenses because they do not take into account the specific nature of their initial conditions. For example, in single and immersion lenses, it is accurately assumed that the trajectories of charged particles throughout have a small inclination to the main optical axis.…”