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Relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems. It was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy. In this paper we find an appropriate admissible pair of classes of Lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half-lines R ± separately, on both half-lines R ± simultaneously, and on the whole line R . Additionally, the maximal admissibility is proved in the case on both half-lines R ± simultaneously.

In this paper we study the roughness of tempered exponential dichotomies for linear random dynamical systems in Banach spaces. Such a dichotomy has a tempered bound and describes nonuniform hyperbolicity. We prove the roughness without assuming their invertibility and the integrability condition of the Multiplicative Ergodic Theorem. We give an explicit bound for the linear perturbation such that the dichotomy is persistent. We also obtain explicit forms for the exponent and the bound of tempered exponential dichotomy of the perturbed random system in terms of the original ones and the perturbations.

Given a piecewise continuous function A : R ¯ + → L ( C N ) and a projection P 1 onto a subspace X 1 of C N , we investigate the injectivity, surjectivity and, more generally, the Fredholm properties of the ordinary differential operator with boundary condition ( u ˙ + A u , P 1 u ( 0 ) ) . This operator acts from the “natural” space W A 1 , 2 = { u : u ˙ ∈ L 2 , A u ∈ L 2 } into L 2 × X 1 . A main novelty is that it is not assumed that A is bounded or that u ˙ + A u = 0 has any dichotomy, except to discuss the impact of the results on this special case. We show that all the functional properties of interest, including the characterization of the Fredholm index, can be related to the existence of a selfadjoint solution H of the Riccati differential inequality H A + A ∗ H − H ˙ ⩾ ν ( A ∗ A + H 2 ) . Special attention is given to the simple case when H = A + A ∗ satisfies this inequality. When H is known, all the other hypotheses and criteria are easily verifiable in most concrete problems.

It is shown that there cannot exist a uniform exponential dichotomy for any linear delay equation with a positive finite delay.

Intermediate spaces are useful for obtaining maximal regularity when studying reaction-diffusion equations. We give a definition of exponential dichotomies in these spaces. Roughness of the exponential dichotomies and other basic properties are also proved. We then use these notions to investigate the stability of a periodic orbit in a predator-prey model studied by Conway and Smoller when diffusion terms are added.

This paper is written in two parts. Part I deals with the asymptotic integra- tions of functional differential equations, and Part II deals with the invariance of the essential spectra of ?zth order differential operators under certain classes of perturbations. Both parts depend on a result of Schgffer [14] (cf. [ll], 72.A and 72.C) to the effect that if there is an exponential dichotomy for the linear first order system y’ = A(t)y on t > 0, then the same is true for the system y’ = A(t)y + B(t)y providedthat lim+tup I”’ [ B(s)1

In this paper we introduce a cone invariance property, which ensures dichotomy for a linear differential system under suitable technical conditions. This property consists in the existence of two disjoint cones, such that the forward (backward) flow originating from a plane that lies in the first (second) cone remains in the cone itself. This property can be verified even if the system matrix is not diagonal dominant. As a consequence, the resulting conditions for dichotomy are not related to diagonal dominance and can be applied to cases in which some of the classical hypotheses appearing in literature do not hold.