Dichotomies with No Invariant Unstable Manifolds for Autonomous Equations

Răzvan O Moşincat,Ciprian Preda,Petre Preda

doi:10.1155/2012/527647

Răzvan O Moşincat, Ciprian Preda + Show 1 more

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https://doi.org/10.1155/2012/527647

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Abstract

We analyze the existence of (no past) exponential dichotomies for a well-posed autonomous differential equation (that generates aC0-semigroup ). The novelty of our approach consists in the fact that we do not assume theT(t)-invariance of the unstable manifolds. Roughly speaking, we prove that if the solution of the corresponding inhomogeneous difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the autonomous homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the aforementioned condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical spaces, sequence Orlicz spaces, etc.) and that from discrete-time conditions we get information about the continuous-time behavior of the solutions.

Highlights

The exponential dichotomy is one of the most basic concepts arising in the theory of dynamical systems
The notion was introduced by Perron in 1, who was concerned with the problem of conditional stability of a system x A t x and its connection with the existence of bounded solutions of the equation xAtxft, x, where the state space X is a finite-dimensional Banach space and the operator-valued function A · is bounded and continuous in the strong operator topology
There were established connections between the condition that the inhomogeneous equation has some bounded solution for every bounded “second member” on the one hand and a certain form of conditional stability of the solutions of the homogeneous equation on the other

Summary

Introduction

The exponential dichotomy is one of the most basic concepts arising in the theory of dynamical systems. There were established connections between the condition that the inhomogeneous equation has some bounded solution for every bounded “second member” on the one hand and a certain form of conditional stability of the solutions of the homogeneous equation on the other This idea was later extensively developed for discrete-time systems in the infinite-dimensional case by Coffman and Schaffer 8 and Henry 9. We prove that if the solution of the corresponding inhomogeneous difference equation belongs to any sequence space on which the right shift is an isometry for every inhomogeneity from the same class of sequence spaces, the continuoustime solutions of the autonomous homogeneous differential equation will exhibit a no past exponential dichotomic behavior This approach has many advantages among which we emphasize on the facts that the previous condition is very general since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical p spaces, sequence Orlicz spaces, etc. It is worth to mention that from discrete-time conditions we get information about the continuous-time behavior of the solutions

Sequence Schaffer Spaces

Main Results

Consider γ as in