On the periodic solutions of the Lienard equation

Mathematical modeling of many problems of natural science leads to the need to study quasi-linear boundary value problems for functional differential equations with a linear part that is not uniquely solvable for all right-hand parts. The specificity of such problems is that the corresponding linear operator is not reversible. In the literature, such boundary value problems are usually called resonant. Since the 70s of the last century, the development of methods for studying resonant boundary value problems. considered as a single operator equation has begun. A very important area of research from the point of view of applications is the application of General statements to the study of periodic boundary value problems for functional differential equations.
The existence problem is considered $\omega$-a periodic solution of the Lienard equation with a deviating argument of the form $x^{\prime\prime}(t)+f(x(t))x^\prime(t)+g(x_p(t))=h(t)$, $t\in [0,\omega], x_p(t)=x(p(t))$.
It is assumed that the function $p(t)$ is measurable and ${p([0,\omega])\subset [0,\omega]}$. Using an approach based on the application of theoretical existence for a quasilinear operator equation, sufficient conditions can be obtained in the work, at least one $\omega$ – a periodic solution must correspond to the equations. The obtained result refines some well-known results for the Lienard equations. Execution conditions ${p([0,\omega])\subset [0,\omega]}$ decisions do not affect the existence of solutions.