Weak Topologies for Carathéodory Differential Equations: Continuous Dependence, Exponential Dichotomy and Attractors

Abstract

We introduce new weak topologies and spaces of Carathéodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Carathéodory vector field f provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limit set or the whole hull of f. Conditions for the existence of a pullback or a global attractor for the skew-product semiflow, as well as application examples are also given. Key words and phrases. Non-autonomous Carathéodory differential equations, linearized skewproduct flow, exponential dichotomy, pullback and forward attractors. Partly supported by MINECO/FEDER under project MTM2015-66330-P and EU Marie-Sk lodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-ITN-2014 643073 CRITICS).

Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics

“…Nevertheless, playing with the functions in the compact sets K J j and K J j , we are still able to achieve the properties we need, as shown in the technical lemma below. Such result is the analogous of Lemma 2.13(ii) in [16] for σ Θ , and since the two proofs differ only on minor details, we skip the proof.…”

“…Therefore we have that (g n ) n∈N converges to g in (LC(R N , R N ), σ Θ ), and since x(t, f n , φ n ) and x(t, f, φ) are respectively the solutions of the systems in (3.5), then, applying Theorem 3.8 in [16], we have that x(·, f n , φ n ) The proof of (ii) is a consequence of (i) and the fact that, due to Theorem 2.20(ii), σ Θ Θ coincides with σ D under the given assumptions.…”

“…As regards the topology σ D , the idea is, once again, to use Theorem 2.20(ii) so that for any specific converging sequence of initial data, one selects the appropriate suitable set of moduli of continuity as shown in the following result which also permits to simplify the proof of Theorem 3.8 in [16].…”

“…Nevertheless, the same problems for Carathéodory delay differential equations remained still untreated. Relying on some of the structural and topological results in [15,16], we fill such gap and pose new questions. It is worth noticing, however, that, as the problem becomes essentially infinitedimensional, this is not a trivial extension of the previous results.…”

“…In Section 2, we set some preliminary notation and introduce the topological spaces which will be used throughout the rest of the work. Particularly, we recall the definitions of the spaces of Lipschitz Carathéodory (LC) and Strong Carathéodory (SC) functions, and endow them with several possible topologies: besides the already established topologies (which are collectively analyzed in [15,16]), such as the classic T B , T D (firstly presented in [17,18]) and σ D (also treated in [2,3,4,12,19]), and the more recent T Θ and σ Θ (firstly presented in [15] and [16], respectively), we introduce the new hybrid topologies T ΘD , σ ΘD , T Θ Θ , σ Θ Θ and T ΘB , where by hybrid we mean that they are, somehow, derived from the previous ones in a way that it is possible to treat the first N components of the spatial variable (representing the current state in a delay differential equation) in a different way from the last N ones (representing the history of the state). The symbols Θ and Θ stand for sets of moduli of continuity which identify numerable quantities of compact sets of continuous functions on which the sequences of elements in LC are asked to converge.…”