Controllability of semilinear boundary problems with nonlocal initial conditions

The main purpose of this paper is the existence of solutions and controllability for semilinear boundary problems with nonlocal initial conditions. We show that the solutions are given by a variation of constants formula which allows us to study the exact controllability for this kind of problems with control and nonlinear terms at the boundary. The included application to a size structured population equation provides a motivation for abstract results.     

Exact boundary observability for nonautonomous first-order quasilinear hyperbolic systems

By means of the theory on the semiglobal C¹ solution to the mixed initial-boundary value problem for first-order quasilinear hyperbolic systems, we establish the local exact boundary observability for general nonautonomous first-order quasilinear hyperbolic systems without zero eigenvalues and reveal the essential difference between nonautonomous hyperbolic systems and autonomous hyperbolic systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Attractor of a Nonautonomous Hyperbolic Equation with a Small Parameter

Mathematical Notes -     

非自治系统的周期解

本文在齐次线性系统ｘ＝Ａ（ｔ）ｘ的解均为ω周期解的条件下，给出了非自治系统ｘ＝Ａ（ｔ）ｘ＋ｇ（ｔ，ｘ）。     

A note on the dichotomy spectrum

In many ways, exponential dichotomies are an appropriate hyperbolicity notion for nonautonomous linear differential or difference equations. The corresponding dichotomy spectrum generalizes the classical set of eigenvalues or Floquet multipliers and is therefore of eminent importance in a stability theory for explicitly time-dependent systems, as well as to establish a geometric theory of nonautonomous problems with ingredients like invariant manifolds and normal forms, or to deduce continuation and bifurcation techniques.

In this note, we derive some invariance and perturbation properties of the dichotomy spectrum for nonautonomous linear difference equations in Banach spaces. They easily follow from the observation that the dichotomy spectrum is strongly related to a weighted shift operator on an ambient sequence space.     

非自治多种群时滞Kolmogorov系统的持续性和灭绝性

研究非自治多种群时滞Kolmogorov系统,给出了该系统中种群持续和灭绝的充分条件,并把得到的结果应用于非自治时滞Lotka-Volterra系统.把相关文献的有关结果推广到时滞系统.     

A Smoothness Theorem for Invariant Fiber Bundles

Invariant fiber bundles are the generalization of invariant manifolds from discrete dynamical systems (mappings) to non-autonomous difference equations. In this paper we present a self-contained proof of their existence and smoothness. Our main result generalizes the so-called Hadamard–Perron-Theorem for time-dependent families of pseudo-hyperbolic mappings from the finite-dimensional invertible to the infinite-dimensional non-invertible case.     

Bounded and Almost-Periodic Solutions of Nonlocal Perturbations of Nonautonomous Logistic-Type Equations

This paper is concerned with the study of dynamics of nonlinear nonlocal perturbations of nonautonomous logistic-type equations. More specifically, we prove the existence and uniqueness of a bounded solution that is positive, and that does not approach the zero solution in the past and in the future. We also show that this solution is an attractor for all other positive solutions. Since the zero solution is shown to be a repeller for all solutions that remain below the aforementioned one, we obtain an attractor-repeller pair. Almost-periodic attractor is also discussed.