Degree computations for positively homogeneous differential equations

Christian Fabry, Patrick Habets

DOI: http://dx.doi.org/10.12775/TMNA.2004.004

Abstract


We study $2\pi$-periodic solutions of
$$
u''+f(t,u)=0
$$
using positively homogeneous asymptotic approximations of this equation
near zero and infinity.
Our main results concern the degree of $I-P$, where $P$ is
the Poincaré map associated to
these approximations. We indicate classes of problems, some with degree 1
and others with degree different from 1.
Considering results based on first order approximations,
we work out examples of equations
for which the degree is the negative of any integer.

Keywords


Periodic solutions; Brouwer degree; Poincaré operator; positively homogeneous equation; Fučik spectrum

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