Periodic Perturbations With Delay of Autonomous Differential Equations on Manifolds

Abstract

We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincaré T -translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.

Multiplicity of forced oscillations for scalar retarded functional differential equations

“…We need some results taken mostly from Furi et al In what follows, we will mainly work with Equation .…”

“…The following consequence of Corollary 4.4 of Furi et al yields the existence of a Rabinowitz‐type branch of T‐ periodic pairs for . Its proof relies on the notion of degree of a tangent vector field and on some of its standard properties.…”

“…The excision property of the degree yields . Now, the assertion follows from Corollary 4.4 of Furi et al □…”

“…We regard a solution of (4) as a map x ∶ J → R, defined on an open real interval J with inf J = −∞, such that the pair (x, x ′ ) ∶ J → R 2 is a solution of (5). For a different approach to the notion of solution of a second-order RFDE, see Benevieri et al 2 We need some results taken mostly from Furi et al 1,16 In what follows, we will mainly work with Equation 5. We will denote by C T (R 2 ) the Banach space of the T-periodic continuous maps from R into R 2 with the usual supremum norm.…”

“…Although the technicalities are hidden in the proofs, to obtain our multiplicity result, we apply topological methods. In fact, a key step for our result is a Rabinowitz‐type theorem on branches of periodic solutions for first‐order parametrized RFDEs obtained by M. Furi and the last 2 authors in Furi et al (see Theorem below). The proof of this theorem is based on the notion of degree of a tangent vector field (see, eg, Furi et al).…”