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

Rebelo, C.; Zanolin, F.

doi:10.1090/S0002-9947-96-01580-2

Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities
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by C. Rebelo and F. Zanolin PDF

Trans. Amer. Math. Soc. 348 (1996), 2349-2389 Request permission

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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