Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. (English) Zbl 1004.34008

The authors are interested in nonnegative and nonpositive solutions to the boundary value problem \[ u''=f(t,u),\quad u(0)=u(1), u'(0)=u'(1), \tag{1} \] where \(f\) satisfies Carathéodory conditions. They provide sets of assumptions that imply existence of lower and upper solutions \(\alpha\) and \(\beta\) which can be well-ordered \(\alpha\leq\beta\) or ordered in the reversed side \(\beta\leq\alpha\). Using such techniques together with degree arguments, the authors generalize previous results on (1). In particular, they consider Duffing equations with repulsive or attracting singularity at the origin and extend existence results by A.C. Lazer and S. Solimini and other authors. Multiple solutions are obtained and weak singularities are allowed in the repulsive case. Results are illustrated by examples.


34B15 Nonlinear boundary value problems for ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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