On a class of differential-algebraic equations with infinite delay

Abstract

We study the set of T -periodic solutions of a class of T -periodically perturbed Differential-Algebraic Equations, allowing the perturbation to contain a distributed and possibly infinite delay. Under suitable assumptions, the perturbed equations are equivalent to Retarded Functional (Ordinary) Differential Equations on a manifold. Our study is based on known results about the latter class of equations and on a "reduction" formula for the degree of a tangent vector field to implicitly defined differentiable manifolds. AbstractWe present here a corrected version of Lemma 5.5 and Corollary 5.7 of EJQTDE 2011, No. 81.

About the notion of non‐T‐resonance and applications to topological multiplicity results for ODEs on differentiable manifolds

“…As a consequence, when the manifold M ⊆ R k is a closed subset, closed and bounded sets of T -pairs are compact. 1 The version of this software, adapted for scalar equations, is available through its author web page http://www.dma.unifi.it/∼spadini/Software/01 Utilities/StartingPoints-1D, whereas the one for two-dimensional equations is still under development. 2 In the sense that when the zeros of v are all non-degenerate, then the set v −1 (0) ∩ W is finite and…”

“…On one hand, proceeding as in (i) we see that there are no T -periodic solutions in a neighborhood of p 0 for λ > 0 small. On the other hand we know that isolated zeros with nonzero index are ejecting for (1). This yields a contradiction.…”

“…Theorem 2.3 implies that there exists an unbounded connected set Γ of nontrivial T -pairs of (1) in Ω whose closure in the space [0, ∞)× C T (M ) intersects g −1 (0)∩Ω. By assumption (ii) it follows that g −1 (0) ∩ V 0 is an ejecting set of the set of T -pairs for (1). Remark 3.3 shows that p 1 , .…”

“…In fact, the technique used in the proof of our main multiplicity result, Theorem 3.5, is shared by a number different contexts. For example differential-algebraic equations have been considered in [1,19] and constrained mechanical systems in [7].…”

“…We conclude this introduction noting that the pictures present in the paper have been produced by the popular interactive plotting package Gnuplot [8] using data generated by simple software written by one of the authors (M. Spadini) for the purpose of analyzing the set of starting points for scalar and two-dimensional parametrized equations. 1…”