### Degree computations for positively homogeneous differential equations

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.

$$

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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