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**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.

Reviewer: P.Habets (Louvain-La-Neuve)

### MSC:

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 |

### Keywords:

second-order nonlinear ordinary differential equation; periodic solution; lower and upper solutions; differential inequalities; nonnegative solution; nonpositive solution; attractive and repulsive singularity; Duffing equation
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\textit{I. Rachůnková} et al., J. Differ. Equations 176, No. 2, 445--469 (2001; Zbl 1004.34008)

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### References:

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