Multiplicity of forced oscillations for scalar retarded functional differential equations

Abstract

We find multiplicity results for forced oscillations of a periodically perturbed autonomous second‐order equation, the perturbing term possibly depending on the whole history of the system. The techniques that we use are topological in nature, but the technical details are hidden in the proofs and completely transparent to the reader only interested in the results.

Periodic perturbations of a class of scalar second order functional differential equations

“…Our results are so-to-speak dual to those of [10], where periodic perturbations containing delay terms are applied to scalar, second order ODE's. The study, by means of topological methods, of the branching and multiplicity of periodic solutions of periodically perturbed equations, is now a well-investigated subject in the case of ODE's both in Euclidean spaces and on manifolds (see, e.g., [21,24]).…”

“…Computation formula. If (f, U, q) is admissible, f is smooth, and q is a regular value for f in U , then (10) deg(f,…”

“…The study, by means of topological methods, of the branching and multiplicity of periodic solutions of periodically perturbed equations, is now a well-investigated subject in the case of ODE's both in Euclidean spaces and on manifolds (see, e.g., [21,24]). The case of ODE's perturbed with delayed forcing terms is also studied in the literature, although not so broadly (see, e.g., [10,11]). However, the presence of delay terms in the unperturbed equation ( 1) is peculiar of the problem addressed here.…”

Branches of Forced Oscillations Induced by a Delayed Periodic Force

“…The same drawbacks plague also the situation when, unlike in (1.4), the functional field F perturbs a nonzero vector field. In fact, in [10] for the spherical pendulum (i.e., when M = S 2 ), as well as in [11] for the scalar case (i.e., M = ℝ), multiplicity results for T-periodic solutions of functional perturbations of autonomous vector fields have been obtained by the authors, even in the possible presence of friction, at the cost of imposing the same limitations as above on the forcing perturbations. This does not come as a surprise as also for first order RFDEs, like in, e.g., [16], it seems necessary to impose similar restrictions on the forcing term.…”

Introduction to: Topological degree and fixed point theories in differential and difference equations

“…5 In order to solve this problem an a priori estimate on the speed of solutions depending on the initial condition was found using both the geometry of the sphere and a generalized notion of winding number (this is another topological tool, see [62] for an informative and entertaining account). Another example of a quite different situation in which topological information must be joined by another of analytic nature is the problem of the existence and multiplicity of harmonic solutions of a small periodic perturbation of an autonomous differential equation (possibly with delay), see [63][64][65][66]. In this situation, assuming the unperturbed vector field is C 1 at its zeros, what turns out to play a key role is a notion of T-resonance 6 (e.g.…”