共查询到20条相似文献,搜索用时 31 毫秒
1.
本文给出了巴拿赫空间中线性差分方程的两个多项式二分性概念, 使其在相应空间中的范数的增长速度不快于指数型增长. 并用实例阐释了相关概念之间的关系. 借助于指数二分性的研究方法讨论了多项式二分性的特征, 所得结论推广了指数稳定性及指数二分性中的一些已有结果. 相似文献
2.
For the flow determined by a nonautonomous linear differential equation, we characterize the existence of a strong nonuniform exponential dichotomy in terms of the Fredholm property of a certain linear operator. We consider both cases of one-sided and two-sided exponential dichotomies. Moreover, we use the characterizations to establish the robustness of the notion of a strong nonuniform exponential dichotomy in a simple manner. 相似文献
3.
Álvaro Castañeda 《Journal of Difference Equations and Applications》2016,22(9):1271-1291
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. 相似文献
4.
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. 相似文献
5.
Bogdan Sasu 《Bulletin des Sciences Mathématiques》2010,134(3):235
We give new necessary and sufficient integral conditions for the existence of exponential dichotomy of skew-product flows. Our methods are based on the structure of the associated stable subspace and unstable subspace. We propose a nonlinear approach, extending the study of exponential stability in terms of integral conditions to the case of uniform exponential dichotomy. We apply the main results to the study of uniform exponential dichotomy of non-autonomous systems. 相似文献
6.
Martin Rasmussen 《Journal of Differential Equations》2009,246(6):2242-2263
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. 相似文献
7.
Fernando Cardoso 《Journal of Difference Equations and Applications》2013,19(3):261-290
We characterize the exponential dichotomy of difference equations with infinite delay. We apply the results to study the robustness of exponential dichotomy. This kind of dichotomy gives us relevant information about boundedness of solutions for several perturbed quasi linear systems with infinite delay. Applications to Volterra difference equations are shown. 相似文献
8.
In this paper we introduce a concept of exponential dichotomy for linear skew-product semiflows (LSPS) in infinite dimensional Banach spaces, which is an extension of the classical concept of exponential dichotomy for time dependent linear differential equations in Banach spaces. We prove that the concept of exponential dichotomy used by Sacker-Sell and Magalhães in recent years is stronger than this one, but they are equivalent under suitable conditions. Using this concept we where able to find a formula for all the bounded negative continuations. After that, we characterize the stable and unstable subbundles in terms of the boundedness of the corresponding projector along (forward/backward) the LSPS and in terms of the exponential decay of the semiflow. The linear theory presented here provides a foundation for studying the nonlinear theory. Also, this concept can be used to study the existence of exponential dichotomy and the roughness property for LSPS.
9.
Jimin Zhang Meng Fan Xiaoyuan Chang 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(2):670-683
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. 相似文献
10.
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. 相似文献
11.
Dennis D. Berkey 《Journal of Mathematical Analysis and Applications》1976,55(1):140-149
It is shown that the dichotomy obtained by Lazer for diagonally dominant linear systems is a weaker condition than that of exponential dichotomy but that the two conditions are equivalent in the case of bounded coefficients. Thus Fink's result that exponential decay of all solutions implies exponential dichotomy does not extend to the case of mixed growth and decay. It is also shown that column dominant systems of mixed sign admit exponential dichotomies when the coefficients are bounded. 相似文献
12.
Nguyen Thieu Huy 《Journal of Mathematical Analysis and Applications》2007,333(2):731-752
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. 相似文献
13.
We study the problem of exponential dichotomy for the systems of linear difference equations with periodic coefficients. Some criterion is established for exponential dichotomy in terms of solvability of a special boundary value problem for a system of discrete Lyapunov equations. We also give estimates for dichotomy parameters. 相似文献
14.
For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. 相似文献
15.
We establish the robustness of linear cocycles in Banach spaces admitting a nonuniform exponential dichotomy. We first obtain robustness results for positive and negative time, by establishing exponential behavior along certain subspaces, and showing that the associated sequences of projections have bounded exponential growth. We then establish a robustness result in Z by constructing explicitly appropriate projections on the stable and unstable subspaces. We emphasize that in general these projections may be different from those obtained separately from the robustness for positive and negative time. We also consider the case of strong nonuniform exponential dichotomies. 相似文献
16.
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. 相似文献
17.
The purpose of this paper is to provide a new, unified and complete study for uniform dichotomy and exponential dichotomy
on the half-line. First we deduce conditions for the existence of uniform dichotomy, using classes of function spaces over
_+{\mathbb {R}_+} which are invariant under translations. After that, we obtain a classification of the main classes of function spaces over
\mathbb R+{\mathbb {R}_+}, in order to deduce necessary and sufficient conditions for the existence of exponential dichotomy, emphasizing on the main
technical qualitative properties of the underlying spaces. We motivate our approach by illustrative examples and show that
the main hypotheses cannot be dropped. We provide optimal methods regarding the input space in the study of dichotomy and
deduce as particular cases some interesting situations as well as several dichotomy results published in the past few years. 相似文献
18.
19.
Hugo Leiva 《Applicable analysis》2013,92(1-2):9-28
In this paper we compute the dynamical spectrum for time dependent scalar parabolic equations with both Neumann and Dirichlet boundary conditions. In order to do that, first, we put forward the concepts of negative continuation, exponential dichotomy and Dynamical Spectrum for linear skew product semiflows. Second, we set the problem in the skew product semiflow frame work and compute explicitly the dynamical spectrum for this semiflow. Finally, we compute the dynamical spectrum for a time dependent system of ordinary differential equations that is obtained by spatially discretizing of the parabolic equation 相似文献
20.
A relationship between the existence of well-posed boundary-value problems and exponential dichotomy for functional equations and linear differential-operator equations on a half-line is considered. It is shown that well-posed boundary-value problems can exist for equations without the exponential dichotomy property. 相似文献