A First-Order Periodic Differential Equation at Resonance

Abstract

We consider the existence of a periodic solution to the first-order nonlinear problem

\begin{eqnarray*} &&x'(t) = -a(t)x(t)+ q ( t, x(t) ),\; \mbox{ a.e. on } (0, T),\\ &&x(0) = x(T), \end{eqnarray*}

where the nonlinear term $q$ is Carathéodory with respect to $L^1[0, T]$. The coefficient function $a$ is such that the differential equation is non-invertible. The technique used to establish our existence result is Mahwin's coincidence degree theory.