Periodic solutions to the Li\'enard type equations with phase attractive singularities

Sufficient conditions are established guaranteeing the existence of a positive $\omega$-periodic solution to the equation $$ u''+f(u)u'+g(u)=h(t,u), $$ where $f,g:(0,+\infty)\to \RR$ are continuous functions with possible singularities at zero, and $h:[0,\omega]\times\RR\to\RR$ is a Carath\'eodory function. The results obtained are rewritten for the equation of the type $$ u''+\frac{cu'}{u^{\mu}}+\frac{g_1}{u^{\nu}}-\frac{g_2}{u^{\gamma}}=h_0(t)u^{\delta}, $$ where $g_1$, $g_2$, $\delta$ are non-negative constants, $c$, $\mu$, $\nu$, $\gamma$ are real numbers, and $h_0\in\loor$. The last equation covers also the so-called Rayleigh-Plesset equation, frequently used in fluid mechanics to model the bubbel dynamics in liquid. In the paper, there is studied the case when $\nu>\gamma$, i.e., the case which covers the attractive singularity of the function $g$. The results obtained assure that there exists a positive $\omega$-periodic solution to the above-mentioned equation if the power $\mu$ or $\nu$ is sufficiently large.