Stability of dichotomies in difference equations with infinite delay

For delay difference equations with infinite delay we consider the notion of nonuniform exponential dichotomy. This includes the notion of uniform exponential dichotomy as a very special case. Our main aim is to establish a stable manifold theorem under sufficiently small nonlinear perturbations. We also establish the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. Finally, we characterize the nonuniform exponential dichotomies in terms of strict Lyapunov sequences. In particular, we construct explicitly a strict Lyapunov sequence for each exponential dichotomy.     

Smooth robustness of parameterized perturbations of exponential dichotomies

We establish the robustness of nonuniform exponential dichotomies in Banach spaces, under sufficiently small C¹-parameterized perturbations. Moreover, we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class C¹ on the parameter, thus yielding an optimal smoothness.     

Stability in delay difference equations with nonuniform exponential behavior

We study the stability under perturbations for delay difference equations in Banach spaces. Namely, we establish the (nonuniform) stability of linear nonuniform exponential contractions under sufficiently small perturbations. We also obtain a stable manifold theorem for perturbations of linear delay difference equations admitting a nonuniform exponential dichotomy, and show that the stable manifolds are Lipschitz in the perturbation.     

Hyperbolicity of linear partial differential equations with delay

Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.The author thanks W. Desch (Graz), I. Gyri (Veszprém) and R. Schnaubelt (Halle) for helpful discussions.     

Robustness of nonuniform exponential dichotomies in Banach spaces

We give conditions for the robustness of nonuniform exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation x^′=A(t)x persists under a sufficiently small linear perturbation. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the “angles” between the stable and unstable subspaces. Our proofs exhibit (implicitly) the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions. We emphasize that we do not need the notion of admissibility (of bounded nonlinear perturbations). We also obtain related robustness results in the case of nonuniform exponential contractions. In addition, we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time.     

Stable manifolds for nonautonomous equations without exponential dichotomy

We consider nonautonomous ordinary differential equations v^′=A(t)v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v), which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v^′=A(t)v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.     

On small solutions of delay equations in infinite dimensions

LetX be a Banach space and 1p<. LetL be a bounded linear operator fromL ^p ([–1,0],X) intoX. Consider the delay differential equationu(t)=Lu _t ,u(0)=x,u ₀=f on the state spaceL ^p ([–1,0],X). We prove that a mild solutionu(t)=u(t;x,f) is a small solution if and only if the Laplace transform ofu(t;x,f) extends to an entire function. The same result holds for the state spaceC([–1,0],X).This paper was written while the authors were affiliated with the University of Tübingen. It is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG. The authors warmly thank Professor Rainer Nagel and the AG Funktionalanalysis for the stimulating and enjoyable working environment.Support by DAAD is gratefully acknowledged.Support by an Individual Fellowship from the Human Capital and Mobility programme of the European Community is gratefully acknowledged.     

Stable manifolds for delay equations and parameter dependence

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x^′=L(t)_xt+f(t,_xt,λ)$x^{\prime }=L\left(t\right)x_t+f(t,x_t,\lambda )$, assuming that the linear equation x^′=L(t)_xt$x^{\prime }=L\left(t\right)x_t$ admits a nonuniform exponential dichotomy and that f$f$ is a sufficiently small Lipschitz perturbation. We also show that the stable invariant manifolds are Lipschitz in the parameter λ$\lambda$.     

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.     

Nonlinear perturbations of nonuniform exponential dichotomy on measure chains

This paper focuses on nonlinear perturbations of flows in Banach spaces, corresponding to a nonautonomous dynamical system on measure chains admitting a nonuniform exponential dichotomy. We first define the nonuniform exponential dichotomy of linear nonuniformly hyperbolic systems on measure chains, then establish a new version of the Grobman-Hartman theorem for nonuniformly hyperbolic dynamics on measure chains with the help of nonuniform exponential dichotomies. Moreover, we also construct stable invariant manifolds for sufficiently small nonlinear perturbations of a nonuniform exponential dichotomy. In particular, it is shown that the stable invariant manifolds are Lipschitz in the initial values provided that the nonlinear perturbation is a sufficiently small Lipschitz perturbation.     

Admissibility for nonuniform exponential contractions

We study the relation between the notions of nonuniform exponential stability and admissibility. In particular, using appropriate adapted norms (which can be seen as Lyapunov norms), we show that if any of their associated Lp spaces, with p∈(1,∞], is admissible for a given evolution process, then this process is a nonuniform exponential contraction. We also provide a collection of admissible Banach spaces for any given nonuniform exponential contraction.     

On dynamic iteration for delay differential equations

In this paper we study dynamic iteration techniques for systems of nonlinear delay differential equations. After pointing out a close connection to the truncated infinite embedding, as proposed by Feldstein, Iserles, and Levin, we give a proof of the superlinear convergence of the simple dynamic iteration scheme. Then we propose a more general scheme that in addition allows for a decoupling of the equations into disjoint subsystems, just like what we are used to from dynamic iteration schemes for ODEs. This scheme is also shown to converge superlinearly.     

Moment boundedness of linear stochastic delay differential equations with distributed delay

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the first moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of the Laplace transform. From the characteristic equation, sufficient conditions for the second moment to be bounded or unbounded are proposed.     

On weak exponential stability of evolution operators in Banach spaces

The aim of this paper is to give some characterizations for weak exponential stability properties of evolution operators in Banach spaces. Variants for weak exponential stability of some well-known results in uniform stability theory (Bu?e and Niculescu (2009) [1], Daleckij and Krein (1974) [2], Datko (1973) [3], Rolewicz (1986) [7], Stoica and Megan (2009) [8]) are obtained.     

Lyapunov sequences for exponential trichotomies

For a linear cocycle with discrete time, we give a complete characterization of nonuniform exponential trichotomies in terms of strict Lyapunov sequences. We also obtain inverse theorems by constructing explicitly strict Lyapunov sequences for each nonuniform exponential trichotomy. These are constructed in terms of Lyapunov norms, with respect to which the nonuniform behavior of the trichotomies becomes uniform. We also obtain a corresponding version of the results for cocyles over measure-preserving transformations.     

Maximal regularity of second-order equations with delay

We characterize existence and uniqueness of solutions of an inhomogeneous abstract delay equation in Hölder spaces. The method is based on the theory of operator-valued Fourier multipliers.     

Lyapunov sequences for exponential dichotomies

For a linear nonautonomous dynamics with discrete time, we study the relation between nonuniform exponential dichotomies and strict Lyapunov sequences. Given such a sequence, we obtain the stable and unstable subspaces from the intersection of the images and preimages of the cones defined by each element of the sequence. The main difficulty is to extract some information about the angles between the stable and unstable subspaces (or some appropriate notion in the case of Banach spaces) from the Lyapunov sequence. In particular, for a large class of nonuniform exponential dichotomies we give a complete characterization in terms of strict quadratic Lyapunov sequences, that is, strict Lyapunov sequences defined by quadratic forms. We also construct explicitly families of strict Lyapunov sequences for each nonuniform exponential dichotomy, in terms of Lyapunov norms.     

Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations

Recently, the existence of Morse decompositions for nonautonomous dynamical systems was shown for three different time domains: the past, the future and—in the linear case—the entire time. In this article, notions of exponential dichotomy are discussed with respect to the three time domains. It is shown that an exponential dichotomy gives rise to an attractor-repeller pair in the projective space, which is a building block of a Morse decomposition. Moreover, based on the notions of exponential dichotomy, dichotomy spectra are introduced, and it is proved that the corresponding spectral manifolds lead to Morse decompositions in the projective space.     

Admissibility and nonuniform polynomial dichotomies

For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a polynomial dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility of bounded solutions. As a nontrivial application, we establish the robustness of the notion of a nonuniform polynomial dichotomy.     

Pseudo-almost periodic solutions of neutral integral and differential equations with applications

The existence and uniqueness of pseudo-almost periodic solutions to general neutral integral equations with deviations are obtained. For this, pseudo-almost periodic functions in two variables are considered. The results extend the corresponding ones to the convolution type integral equations. They are used to study pseudo-almost periodic solutions of general neutral differential equations and to the so-called scalar neutral logistic equation version.