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ISSN 1088-6850(e) ISSN 0002-9947(p)

Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities

Author(s): C. Rebelo; F. Zanolin
Journal: Trans. Amer. Math. Soc. 348 (1996), 2349-2389.
MSC (1991): Primary 34C25; Secondary 34B15
MathSciNet review: 1344211
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Abstract: We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equation $x'' + g(x) = s + w(t,x)$ , as $s\to +\infty$ or $s\to - \infty ,$ where $g$ is a smooth function defined on a open interval $]a,b[\subset {\mathbb {R}}.$ The hypotheses we assume on the nonlinearity $g(x)$ allow us to cover the case $b=+\infty$ (or $a = -\infty$ ) and $g$ having superlinear growth at infinity, as well as the case $b < +\infty$ (or $a > -\infty$ ) and $g$ having a singularity in $b$ (respectively in $a$ ). Applications are given also to situations like $g'(-\infty ) \not = g'(+\infty )$ (including the so-called ``jumping nonlinearities''). Our results are connected to the periodic Ambrosetti - Prodi problem and related problems arising from the Lazer - McKenna suspension bridges model.

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