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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: C. Rebelo and 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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Additional Information

C. Rebelo
Affiliation: International School for Advanced Studies, via Beirut 2-4, 34013 Trieste, Italy
Address at time of publication: Centro de Matemática e Aplicações Fundamentais, Av. Prof. Gama Pinto 2, 1699 Lisboa Codex, Portugal

F. Zanolin
Affiliation: Dipartimento di Matematica e Informatica, Università, via delle Scienze 208 (loc. Rizzi), 33100 Udine, Italy

Keywords: Periodic solutions, subharmonics, asymmetric nonlinearities, Poincaré-Birkhoff fixed point theorem
Received by editor(s): August 4, 1994
Received by editor(s) in revised form: February 28, 1995
Additional Notes: Work performed in the frame of the EEC project “Non linear boundary value problems: existence, multiplicity and stability of solutions”, grant ERB CHRX-CT94-0555.
The first author is on leave of absence from Faculdade de Ciências da Universidade de Lisboa with a fellowship from Programa Ciência (JNICT).
The second author’s work performed under the auspices of GNAFA-CNR and supported by MURST (40% and 60% funds).
Article copyright: © Copyright 1996 American Mathematical Society