Computational fixed-point theory for differential delay equations with multiple time lags

We introduce a general computational fixed-point method to prove existence of periodic solutions of differential delay equations with multiple time lags. The idea of such a method is to compute numerical approximations of periodic solutions using Newtonʼs method applied on a finite dimensional projection, to derive a set of analytic estimates to bound the truncation error term and finally to use this explicit information to verify computationally the hypotheses of a contraction mapping theorem in a given Banach space. The fixed point so obtained gives us the desired periodic solution. We provide two applications. The first one is a proof of coexistence of three periodic solutions for a given delay equation with two time lags, and the second one provides rigorous computations of several nontrivial periodic solutions for a delay equation with three time lags.