Periodic solutions to the Lienard 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,+infinity) -> R are continuous functions with possible singularities at zero and h : [0,omega] x R -> R is a Caratheodory function. The results obtained are rewritten for the equation of the type u '' + cu'/u(mu) + g(1)/g(nu) - g(2)/u(gamma) = h(o)(t)u(delta), where g(1), g(2), delta are non-negative constants, c, mu, nu, gamma are real numbers, and h(0) is an element of L([0,omega]; R). The last equation also covers the so-called Rayleigh-Plesset equation, frequently used in fluid mechanics to model the bubble dynamics in liquid. In the paper, the case when nu > gamma, i.e., the case which covers the attractive singularity of the function g, is studied. 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.