Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy

We establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that the linear equation v^′=A(t)v admits a nonuniform exponential dichotomy. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of Ck functions with α-Hölder continuous kth derivative. This is a closed subset of the space of continuous functions with the supremum norm, by an apparently not so well-known lemma of Henry (see Proposition 3). The estimates showing that the functions maintain the original bounds when transformed under the fixed-point operator are obtained through a careful application of the Faà di Bruno formula for the higher derivatives of the compositions (see (31) and (35)). As a consequence, we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class Ck.     

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.     

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 characterization of generalized exponential dichotomy

This paper studies some important properties of the notion generalized exponential dichotomy. A new notion called generalized bounded growth is introduced to describe the characterization of generalized exponential dichotomy. The relations between generalized bounded growth and generalized exponential dichotomy are established.     

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.     

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.

     

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     

Generalized exponential dichotomy and global linearization

Hartman's linearization theorem says that if all eigenvalues of matrix A have no zero real part and f(x) is small Lipschitzian, then nonlinear system x^′=Ax+f(x) and its linear system x^′=Ax are topologically equivalent. In 1970s Palmer extended the theorem to nonautonomous systems. In this paper we extend Hartman's theorem to the systems with generalized exponential dichotomy.     

Dichotomy spectrum for difference equations in Banach spaces

In this paper, we study the structure and properties of the dichotomy spectrum for linear difference equations in Banach spaces avoiding the commonly made assumption of having a compact flow. In this situation, any compact subset of the positive-half line may occur as a dichotomy spectrum. To overcome this difficulty, we introduce the upper and lower dichotomy index by means of a measure of noncompactness. It allows to decompose the dichotomy spectrum into the upper, essential and lower dichotomy spectrum. The main result is a Spectral Theorem which describes all possible cases of the upper and lower dichotomy spectrum and yields a “nonautonomous linear algebra” appropriate also for systems in the absence of Lyapunov regularity. Furthermore, we give explicit examples of difference equations to illustrate the different cases of the Spectral Theorem and to underline that a new form of the spectrum not present in the literature can occur. Finally, we indicate applications to nonlinear difference equations and the continuous time situation.     

Robustness of nonuniform exponential trichotomies in Banach spaces

We establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x^′=A(t)x in Banach spaces. We also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy. We consider both trichotomies in semi-infinite intervals and trichotomies in R.     

A topological equivalence result for a family of nonlinear difference systems having generalized exponential dichotomy

We obtain sufficient conditions ensuring the topological and strong topological equivalence of two perturbed difference systems whose linear part has a property of generalized exponential dichotomy. When the exponential dichotomy is verified, we obtain a strongly and Hölder topological equivalence.     

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C ₀,C ₀) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.     

Strongly topological linearization with generalized exponential dichotomy

It has been proved that if x^′=A(t)x$x^{\prime }=A\left(t\right)x$ has a generalized exponential dichotomy and f(t,x)$f(t,x)$ satisfies certain conditions, then the nonlinear system x^′=A(t)x+f(t,x)$x^{\prime }=A\left(t\right)x+f(t,x)$ is topologically equivalent to its linear system x^′=A(t)x$x^{\prime }=A\left(t\right)x$. In this paper, we prove that if the condition |A(t)|≤M⋅a(t)$\left|A\left(t\right)\right|\le M\cdot a\left(t\right)$ is added, then x^′=A(t)x+f(t,x)$x^{\prime }=A\left(t\right)x+f(t,x)$ is strongly topologically equivalent to x^′=A(t)x$x^{\prime }=A\left(t\right)x$, where M$M$ is some positive number and a(t)$a\left(t\right)$ is the eigenfunction of the generalized exponential dichotomy, and therefore the corresponding solutions of x^′=A(t)x+f(t,x)$x^{\prime }=A\left(t\right)x+f(t,x)$ and x^′=A(t)x$x^{\prime }=A\left(t\right)x$ have the same stability.     

Uniform exponential stabilization of nonlinear systems in Banach spaces

This paper presents necessary and sufficient conditions for uniform exponential stabilization of a class of nonlinear systems in Banach state spaces. The stabilization assumptions are formulated in terms of integral estimates involving the control operator and the state of the uncontrolled version of the system at hand. An explicit estimate of the convergence speed is given. Applications to feedback stabilization of affine control systems are given. Illustrative examples are further provided.     

广义指数二分性与微分方程的不变流形

用 Hadamar和 Bogoliubov的方法,在常微分方程、泛函微分方程以及半线性抛物型方程所能满足的条件下对Banach或Hilbert空间上的非自治抽象微分方程建立了不变流形理论。首先,对相应的线性方程提出了“广义指数二分性”概念并讨论了它和线性方程谱的关系,然后,我们给出了不变流形存在性结论以及强稳定(不稳定)流形、弱稳定(不稳定)流形和弱双曲流形的分类。进而,我们对这些不变流形给出了C~k光滑性、周期性、概周期性和吸斥性的结论。     

Uniform dichotomy and exponential dichotomy of evolution families on the half-line

The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family Φ={Φ(m,n)}_(m,n)∈Δ the subspace . Supposing that X₁ is closed and complemented, we prove that the admissibility of the pair implies the uniform dichotomy of Φ. Under the same hypothesis on X₁, we obtain that the admissibility of the pair with p∈(1,∞] is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.     

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.