Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type (Q6587693)

The author studies the Liénard equation with a singularity of repulsive type \N\[\Nx''+f(x)x'+\varphi(t)x^\mu-\frac{1}{x^\gamma}=e(t),\tag{\(\ast\)}\N\]\Nwhere \(f:(0,\infty)\to{\mathbb R}\) is continuous (with a possible singularity at \(0\)), the functions \(\varphi,e\) are allowed to change sign and \(\mu,\gamma\geq 0\) denote reals. Based on a sequence of lemmas, a continuation theorem using Mawhin's coincidence degree and under the assumptions \N\[\N\int_0^T\varphi(s)\,ds>0,\quad \int_0^Te(s)\,ds=0,\N\] \Nsufficient conditions are provided, such that the ordinary differential equation \((\ast)\) has at least one positive \(T\)-periodic solution for each \(\mu\geq 0\). Two concrete examples illustrate these criteria.