Open Access Research

Periodic solutions to the Liénard type equations with phase attractive singularities

Robert Hakl1* and Manuel Zamora2

Author Affiliations

1 Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, Brno, 616 62, Czech Republic

2 Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, Campus de Fuentenueva s/n, Granada, 18071, Spain

For all author emails, please log on.

Boundary Value Problems 2013, 2013:47  doi:10.1186/1687-2770-2013-47

Published: 6 March 2013

Abstract

Sufficient conditions are established guaranteeing the existence of a positive ω-periodic solution to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M2">View MathML</a> are continuous functions with possible singularities at zero and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M3">View MathML</a> is a Carathéodory function. The results obtained are rewritten for the equation of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M4">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M6">View MathML</a>, δ are non-negative constants, c, μ, ν, γ are real numbers, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M7">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M8">View MathML</a>, 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.

Keywords:
Rayleigh-Plesset equation; singular equation; periodic solution; upper and lower function