Positive Periodic Solutions of Singular Third-Order Functional Differential Equations with $p$-Laplacian-Like Operators

The aim of this paper is to consider singular third-order functional differential equations with $p$-Laplacianlike operator of the form
$$ (\varphi_p (x''(t)))' +f(t,x(t-\tau))=e(t). $$
Unlike in previous works, $f$ has a strong singularity at $x=0$ and satisfies a small force condition at $ x=\infty$. Based on a continuation theorem due to Mawhin, new results on the existence of positive periodic solutions are obtained, which makes it possible to refine and extend some related results in the literature. A typical example demonstrating the effectiveness and flexibility of the main results is given.