Abstract
A type of nonautonomous n-dimensional state-dependent delay differential equation (SDDE) is studied. The evolution law is supposed to satisfy standard conditions ensuring that it can be imbedded, via the Bebutov hull construction, in a new map which determines a family of SDDEs when it is evaluated along the orbits of a flow on a compact metric space. Additional conditions on the initial equation, inherited by those of the family, ensure the existence and uniqueness of the maximal solution of each initial value problem. The solutions give rise to a skew-product semiflow which may be noncontinuous, but which satisfies strong continuity properties. In addition, the solutions of the variational equation associated to the SDDE determine the Frechet differential with respect to the initial state of the orbits of the semiflow at the compatibility points. These results are key points to start using topological tools in the analysis of the long-term behavior of the solution of this type of nonautonomous SDDEs.
Highlights
Functional differential equations of state-dependent delay type (SDDEs for short) have been object of active analysis during the last years, due in part to the high theoretical interest of this study, but mainly to the increasing number of models of applied sciences which respond to this pattern: see e.g. Hartung [6], Wu [19], Hartung et al [7], Mallet-Paret and Nussbaum [13], Barbarossa and Walther [1], He and de la Llave [8], and Krisztin and Rezounenko [12], as well as the many references therein
The regularity properties required on the vector field to guarantee existence, uniqueness, and continuous variation of solutions of initial value problems are much more exigent than in the case of fixed delay or even time-dependent delay differential equations
Complex is the nonautonomous case: due to the time-dependence, the solutions do not generate a semiflow on the state space, and more sophisticated tools must be designed in order to use the methods of the topological dynamics in the analysis of the dynamical properties of the solutions
Summary
Functional differential equations of state-dependent delay type (SDDEs for short) have been object of active analysis during the last years, due in part to the high theoretical interest of this study, but mainly to the increasing number of models of applied sciences which respond to this pattern: see e.g. Hartung [6], Wu [19], Hartung et al [7], Mallet-Paret and Nussbaum [13], Barbarossa and Walther [1], He and de la Llave [8], and Krisztin and Rezounenko [12], as well as the many references therein. These properties are one of the key points required to prove that, if z(t, ω, x, v) represents the solution of (1.3) agreeing with v ∈ W 1,∞ in [−r, 0], and w(t, ω, x, v)(s) := z(t + s, ω, x, v) for s ∈ [−r, 0], the map (t, ω, x, v) → (Π(t, ω, x), w(t, ω, x, v)) defines a new pseudo-continuous semiflow on K × W 1,∞ (linear in this case), where K is any compact Π-invariant subset of Ω × W 1,∞ The importance of this result relies on the fact that w(t, ω, x, v) = ux(t, ω, x)v; that is, that w(t, ω, x, ·) represents the differential (in the Frechet sense, as a matter of fact) with respect to the state variable of the Π-semiorbit corresponding to a compatibility point. He and de la Llave use in [8] the parameterization method in order to construct quasiperiodic solutions of quasi-periodic SDDEs, which are defined as the ε-perturbation of an hyperbolic family of ordinary differential equations
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