Exact multiplicity for periodic solutions of a first-order differential equation (Q1827130)

The authors deal with the following periodic equation \[ x'+ f(t, x)= h(t),\tag{1} \] where \(h(t)\in C(\mathbb{R}/\mathbb{Z})\) and \(f(t,x)\) is one-periodic in \(t\). It is known that the problem to determine the number of periodic solutions of (1) is connected with the problem determining the number of limit cycles of certain polynomial systems in the plane. However, the upper and lower solution method combined with the degree method does not allow one to give the exact number of periodic solutions. Here, by means of the singularity method, the authors obtain exact multiplicity results.