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A First-Order Periodic Differential Equation at Resonance

Volume 11, Number 1 (2011), 66 - 74

A First-Order Periodic Differential Equation at Resonance

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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.