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

Abstract

We prove a global bifurcation result for T -periodic solutions of the delay T -periodic differential equation x (t) = λf (t, x(t), x(t − 1)) on a smooth manifold (λ is a nonnegative parameter). The approach is based on the asymptotic fixed point index theory for C 1 maps due to Eells-Fournier and Nussbaum. As an application, we prove the existence of forced oscillations of motion problems on topologically nontrivial compact constraints. The result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.

On a class of differential-algebraic equations with infinite delay

“…For so doing we used a technical lemma from linear algebra that, unfortunately, turns out to be flawed. As briefly discussed below this affects only marginally our paper's results (just a corollary in Section 5 of [1]). The simple example below shows that there is something wrong with Lemma 5.5 in [1].…”

“…As briefly discussed below this affects only marginally our paper's results (just a corollary in Section 5 of [1]). The simple example below shows that there is something wrong with Lemma 5.5 in [1]. In the next section we provide an amended version of this result.…”

“…Which proves the assertion because P T r U r , C r , and V T r Q r are nonsingular. In view of the corrected version of the above lemma, the statement of Corollary 5.7 of [1] can be rewritten as follows: where the maps C : R → R n×n and S : BU (−∞, 0], R n → R n are continuous, E is a (constant) n × n matrix, F is locally Lipschitz and S verifies condition (K) in [1]. Suppose also that C and E satisfy (2.1) and (2.3), and that C is T -periodic.…”

“…In Section 5 of [1] we investigated examples of applications of that paper's results to a particular class of implicit differential equations. For so doing we used a technical lemma from linear algebra that, unfortunately, turns out to be flawed.…”

“…Consider the matrices , ker C T = ker E T = span{( 0 1 )} for all t ∈ R. The matrices the form expected from Lemma 5.5 in[1]. The problem, as it turns out, is that ker C = ker E.Luckily, the impact of the wrong statement of [1, Lemma 5.5] on[1] is minor: all results and examples (besides Lemma 5.5, of course) remain correct, with the exception of Corollary 5.7 where it is necessary to assume the following further hypothesis:ker C(t) = ker E, ∀ t ∈ R.(A corrected statement of Corollary 5.7 of[1] can be found in the next section, Corollary 2.2. )…”