On forced fast oscillations for delay differential equations on compact manifolds

Abstract

We prove an existence result for forced oscillations of delay differential equations on compact manifolds with nonzero EulerPoincaré characteristic. When the period is smaller than the delay we need the asymptotic fixed point index theory for C 1 maps due to Eells and Fournier, and Nussbaum.

A Continuation Result for Forced Oscillations of Constrained Motion Problems with Infinite Delay

“…x ′′ π (t) = F (t, x t ) − εx ′ (t) admits at least one forced oscillation (see Corollary 4.2 below). This consequence of Theorem 4.1 generalizes results given in [2] and [4] for equations with constant time lag (see also [11] for the undelayed case). On the opposite, in the frictionless case, the existence of an unbounded bifurcating branch is not sufficient to guarantee the existence of a forced oscillation of the equation…”

Delay Differential Equations on Manifolds and Applications to Motion Problems for Forced Constrained Systems

“…Since M \∂M is a boundaryless manifold which is not compact when ∂M = ∅, unless otherwise stated we will assume that M is boundaryless, but not necessarily compact. In this context, we prove a global bifurcation result, Theorem 3.13, whose consequence, Corollary 3.16, provides the desired extension of the results in [1] and [3].…”

“…x (t) = λf (t, x(t), x(t − 1)). In two recent papers, [1] and [3], we investigated the structure of the set of T -periodic pairs of (1.1). In the first one we tackled the case when the period T is not smaller than the delay, that, without loss of generality, we supposed to be 1.…”

“…In [3] we dealt with the case of arbitrary period T > 0, and we proved a global bifurcation result as in [1], but with the additional assumption that ∂M = ∅. This extra condition is due to the fact that, when 0 < T < 1, the Poincaré operator is not locally compact and, consequently, we applied the fixed point index theory of Eells-Fournier (see [6]) and Nussbaum (see [18]) instead of the classical one.…”

“…In fact, in this case M is not a C 1 retract of any of its neighborhoods and, consequently, the argument used in [5] to prove that C([−1, 0], M ) is a C 1 -ANR fails. Our purpose here is to overcome this difficulty, in order to give a complete extension of the results in [1] and [3]. Since M \∂M is a boundaryless manifold which is not compact when ∂M = ∅, unless otherwise stated we will assume that M is boundaryless, but not necessarily compact.…”