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Continuous-time dichotomies without unstable invariant manifolds for autonomous system

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Abstract

It is known that a differential equation ẋ(t) = Ax(t) on the Banach space X (we assume the well-posedness, i.e., A generates a C0-semigroup {T(t)}t ≥ 0) is hyperbolic if X can be decomposed as X = X1 ⊕ X2 so that solutions ẋ(∙) starting from X1 (respectively, from X2) decay exponentially in forward time (respectively, in backward time). Hyperbolicity forces the solutions that start from X2 to exist for negative time (or, equivalently, the semigroup generated by A to extend to a C0-group on X2). We generalize this notion by replacing the exponential decay in negative time for the solutions starting in X2 with an exponential blow-up in positive time (we call this an exponential dichotomy). It is obvious that hyperbolicity implies the existence of an exponential dichotomy, but the converse is not valid (we point out an example in this context). We obtain a unified treatment of admissibility-type conditions guaranteeing the existence of an exponential dichotomy and complete characterizations of the hyperbolicity of autonomous differential equations.

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Authors and Affiliations

  1. West University of Timişoara, Bd. V. Parvan, no. 4, 300223, Timişoara, Romania

    Ciprian Preda & Florin Bǎtǎran

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  1. Ciprian Preda
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  2. Florin Bǎtǎran
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Correspondence to Ciprian Preda.

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Preda, C., Bǎtǎran, F. Continuous-time dichotomies without unstable invariant manifolds for autonomous system. Lith Math J 58, 167–184 (2018). https://doi.org/10.1007/s10986-018-9388-1

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  • DOI: https://doi.org/10.1007/s10986-018-9388-1

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