Nonautonomous nonlinear ODEs: Nonresonance conditions and rotation numbers

We deal with the T-periodic problem associated with a nonlinear scalar differential equation like(1)x″+f(t,x)=0, where, for x→0 and |x|→+∞, the nonlinearity f is assumed behave linearly, with a time-dependent coefficient. We prove that the Landesman–Lazer conditions at zero and at infinity possess a...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2019-05, Vol.473 (1), p.490-509
Main Authors: Garrione, Maurizio, Margheri, Alessandro, Rebelo, Carlota
Format: Article
Language:English
Subjects:
Landesman–Lazer conditions
Morse index
Periodic solutions
Poincaré–Birkhoff theorem
Rotation number
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Summary:We deal with the T-periodic problem associated with a nonlinear scalar differential equation like(1)x″+f(t,x)=0, where, for x→0 and |x|→+∞, the nonlinearity f is assumed behave linearly, with a time-dependent coefficient. We prove that the Landesman–Lazer conditions at zero and at infinity possess a rotational effect on “small” and “large” (in the phase-plane) solutions of (1). As a consequence, we are able to generalize previous multiplicity results in the resonant case - i.e., when the linearizations at zero and/or at infinity are resonant, through the use of the Poincaré–Birkhoff fixed point theorem.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2018.12.063