Periodic solutions to the Liénard type equations with phase attractive singularities
Robert Hakl, Manuel Zamora · Boundary Value Problems · 2013
Sufficient conditions are established guaranteeing the existence of a positive ω-periodic solution to the equation where are continuous functions with possible singularities at zero and is a Carathéodory function. The results obtained are rewritten for the equation of the type where , , δ are non-negative constants, c, μ, ν, γ are real numbers, and . 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 , i.e., the case which covers the attractive singularity of the function g, is studied. The results obtained assure that there exists a positive ω-periodic solution to the above-mentioned equation if the power μ or ν is sufficiently large. MSC:34C25, 34B16, 34B18, 76N15.