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Periodic solutions for a singular Liénard equation with indefinite weight. (English) Zbl 1472.34080

In this paper, the authors study the following singular Liénard equation \[ x''(t)+f(x(t))x'(t)+\frac{\alpha(t)}{x^\mu(t)}= h(t),\tag{1} \] where \(f\in C((0, +\infty), \mathbb{R})\) may have a singularity at \(x=0,\, \mu\in(0, +\infty)\) is a constant, \(\alpha\) and \(h\) are \(T\)-periodic functions with \(\alpha,\, h \in L^1 ([0, T], \mathbb{R}).\) The weight function \(\alpha\) may change sign on \([0, T].\) A new method for estimating a priori bounds of all possible positive \(T\)-periodic solutions is obtained. By using a continuation theorem of Mawhin’s coincidence degree theory, some new results on the existence of positive periodic solutions for the equation (1) are established.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
70K40 Forced motions for nonlinear problems in mechanics
47N20 Applications of operator theory to differential and integral equations
37C60 Nonautonomous smooth dynamical systems
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References:

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