Abstract
We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Our result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation.
Highlights
Consider, on a differentiable manifold M ⊆ Rk, the class of constrained functional differential equations of the following form:t x(t) = g x(t), γab(t − s)x(s) ds, (1) −∞where g : M × Rk → Rk is a continuous map with the properties that g(x, y) belongs to the tangent space Tx M to M at x for all (x, y) ∈ M × Rk
Where g : M × Rk → Rk is a continuous map with the properties that g(x, y) belongs to the tangent space Tx M to M at x for all (x, y) ∈ M × Rk
In this paper we are interested in the periodic response of (1) subject to periodic perturbations
Summary
On a differentiable manifold M ⊆ Rk, the class of constrained functional differential equations of the following form:. The main contribution of this paper follows by specializing the techniques of [10,13] to such ordinary differential equation This application is made possible by a formula, Theorem 2.2, for the computation of the degree (or rotation or characteristic) of the associated vector field (for λ = 0). Validity of the last statement using the properties of the degree of a tangent vector field in combination with approximation results on manifolds This somewhat technical proof is set apart in a dedicated section at the end of the paper in order not to divert the reader’s attention from the main theme. We will defer a brief discussion of these ideas to a later section on “Perspectives and further developments”
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