Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and an integral equation . We characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations.     

Existence and robustness of exponential dichotomy of linear skew-product semiflows over semiflows

In this paper we investigate the exponential dichotomy of linear skew-product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on Banach bundles. We characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy.     

Exponential trichotomy and p-admissibility for evolution families on the real line

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1,∞) and the pair (C_b(R,X),C_c(R,X)) is uniformly p-admissible for an evolution family ={U(t,s)}_t≥_s then is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C_b(R, X), C_c(R, X)) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C_b(R,X),L^p(R,X)) for an evolution family ={U(t,s)}_t≥_s is equivalent with its uniform exponential dichotomy.     

Discretized characterizations of exponential dichotomy of linear skew-product semiflows over semiflows

Using the method of discretization, we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew-product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear-skew product semiflows. We then apply the obtained results to consider the roughness of exponential dichotomy of linear-skew product semiflows.     

A discrete Perron-Ta Li type theorem for the dichotomy of evolution operators

Following the Perron-Ta Li line of results, we give a characterization of the uniform exponential dichotomy property for the abstract continuous-time evolution families using a discrete-time approach.     

On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations

For a delay difference equation

     

Exponential dichotomy and dichotomy radius for difference equations

The aim of this paper is to provide a new approach concerning the characterization of exponential dichotomy of difference equations by means of admissible pair of sequence spaces. We classify the classes of input and output spaces, respectively, and deduce necessary and sufficient conditions for exponential dichotomy applicable for a large variety of systems. By an example we show that the obtained results are the most general in this topic. As an application we deduce a general lower bound for the dichotomy radius of difference equations in terms of input-output operators acting on sequence spaces which are invariant under translations.     

Almost automorphic solutions of nonautonomous evolution equations

In this paper, we establish some new theorems about the existence of almost automorphic solutions to nonautonomous evolution equations u^′(t)=A(t)u(t)+f(t) and u^′(t)=A(t)u(t)+f(t,u(t)) in Banach spaces. As we will see, our results allow for a more general A(t) to some extent. An example is also given to illustrate our results. In addition, by means of an example, we show that one cannot ensure the existence of almost automorphic solutions to u^′(t)=A(t)u(t)+f(t) even if the evolution family U(t,s) generated by A(t) is exponentially stable and fAA(X).     

Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and a semi-linear integral equation . We prove the existence of stable manifolds of solutions to this equation in the case that (U(t,s))_t?s?0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x₁)−f(t,x₂)‖?φ(t)‖x₁−x₂‖ for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.     

Uniform exponential dichotomy of parabolic operators with rapidly oscillating coefficients

The paper deals with averaging problems for parabolic equations. We prove that exponential dichotomy is preserved without any assumption concerning the almost-periodicity of the coefficients.     

On admissible perturbations for exponential dichotomy

In this work we obtain an optimal upper bound for exponential dichotomy roughness in infinite-dimensional Banach spaces. Unlike some previous works, we do not assume bounded growth. We consider linear, non-autonomous ordinary differential equations with bounded and unbounded coefficients.     

On the proof of characterizations of the exponential dichotomy

We describe explicitly the generator of the evolutionary semigroup associated with the evolutionary operator generated by the linear differential equation . From this we give a short proof of some known characterizations of the exponential dichotomy of the above mentioned equation.

     

指数型二分性与不变流形

该文证明了非线性微分方程组在对应的齐次方程具有指数型二分性、非线性部分满足适当的条件下存在稳定流形和不稳定流形；并且对所得的结果给出一个应用．     

Pseudo-almost periodicity of some nonautonomous evolution equations with delay

This paper is concerned with pseudo-almost periodicity of the solutions to the nonautonomous evolution equation with delay u^′(t)=A(t)u(t)+f(t,u(t−h))$u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f(t,u(t-h))$. Some sufficient conditions which ensure the existence and uniqueness of pseudo-almost periodic mild solutions to the evolution equation with delay are given. An example is shown to illustrate our results.     

Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

LetU=(U(t, s)) _tsO be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsG _O,G _X andI _X on certain spaces ofX-valued continuous functions connected with the integral equation , and we characterize exponential stability, exponential expansiveness and exponential dichotomy ofU by properties ofG _O,G _X andI _X , respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.This work was done while the first author was visiting the Department of Mathematics of the University of Tübingen. The support of the Alexander von Humboldt Foundation is gratefully acknowledged. The author wishes to thank R. Nagel and the Functional Analysis group in Tübingen for their kind hospitality and constant encouragement.Support by Deutsche Forschungsgemeinschaft DFG is gratefully acknowledged.     

Exponential dichotomy roughness on Banach spaces

In the present paper we extend existing results on exponential dichotomy roughness for linear ODE systems to infinite dimensional Banach space. We give new conditions for the existence of exponential dichotomy roughness in infinite dimensional space and in the finite interval case. We also improve previous results by indicating the exact values of the dichotomic constants of the perturbed equation.     

The Fredholm alternative for parabolic evolution equations with inhomogeneous boundary conditions

We study the Fredholm properties of parabolic evolution equations on R with inhomogeneous boundary values. These problems are transformed into evolution equations with inhomogeneities taking values in certain extrapolation spaces. Assuming that the underlying homogeneous problem is asymptotically hyperbolic, we show the Fredholm alternative for these equations. The results are applied to parabolic partial differential equations.     

Input-output admissibility and exponential trichotomy of difference equations

We propose a new method for the study of the asymptotic behavior of difference equations in infinite-dimensional spaces, providing characterizations for the property of uniform exponential trichotomy. We deduce the structure of the stable, unstable and bounded subspace and prove the uniqueness of the projection families. We introduce a new admissibility concept with respect to a discrete input-output system and prove that this is a necessary and sufficient condition for the existence of uniform exponential trichotomy. Throughout the paper, there is no assumption on the coefficients and the obtained results are applicable to any class of difference equations.     

Mean-square exponential dichotomy of numerical solutions to stochastic differential equations

We present the ability of numerical simulations to reproduce the mean-square exponential dichotomy of stochastic differential equations. Under some conditions, we show that the mean-square exponential dichotomy of stochastic differential equations is equivalent to that of the numerical method for sufficient small step sizes