On the existence of periodic solutions for linear inhomogeneous and quasilinear functional differential equations (Q1188222)

The equation \(x'(t)=\int^{+\infty}_{-\infty}(dE(s))x(t+s)+f(t)\) is considered with \(T\)-periodic \(f\). Associate the characteristic equation \(\text{det} \Delta(\mu)=0\) \(\Delta(\mu)=\mu I-\int^{+\infty}_{- \infty}e^{\mu s}dE(s)\). Define \(\mu_ R={2k\pi i\over T}\) and let \(\hat f(k)\) be the \(k\)th Fourier coefficient of \(f\). An elementary proof is given for the following: The equation has a \(T\)-periodic solution iff \(a\hat f(k)=0\) for all row vectors \(a\) for which \(a\Delta(\mu_ k)=0\). For the ``if'' part a proof using Leray-Schauder arguments allows generalizations to nonlinear situations.