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Periodic Problems for ODE's via Multivalued Poincare Operators

##
*
Lech Gorniewicz
*

** Address.**
Faculty of Mathematics and Informatics
Nicholas Copernicus University
Chopina 12/18, 87-100 Torun, Poland

** E-mail:**
gorn@mat.uni.torun.pl

**Abstract.**
We shall consider periodic problems for ordinary differential equations
of the form
\begin{equation} \label{gorrI}
\begin{cases}
x'(t)= f(t,x(t)),\\
x(0) = x(a),
\end{cases}
\end{equation}}
where $ f:[0,a] \times R^n \to R^n$ satisfies suitable assumptions.

To study the above problem we shall follow an approach based on the
topological degree theory. Roughly speaking, if on some ball of $R^n$,
the topological degree of, associated to (\ref{gorrI}), multivalued Poincar\'e
operator $P$ turns out to be different from zero, then problem (\ref{gorrI})
has solutions.

Next by using the multivalued version of the classical Liapunov-Kras\-no\-sel\-sk\v\i\
guiding potential method we calculate the topological degree of the
Poincar\'e operator~$P$. To do it we associate with $f$ a guiding potential
$V$ which is assumed to be locally Lipschitzean (instead of $C^1$) and hence, by using
Clarke generalized gradient calculus we are able to prove existence results
for (\ref{gorrI}), of the classical type, obtained earlier under the assumption that
$V$ is $C^1$.

Note that using a technique of the same type (adopting to the random case) we
are able to obtain all of above mentioned results for the following random
periodic problem:
\begin{equation} \label{gorrII}
\begin{cases}
x'(\xi, t) = f(\xi, t, x(\xi,t)),\\
x(\xi,0) = x(\xi, a),
\end{cases}
\end{equation}}
where $f:\Omega\times[0,a]\times R^n\to R^n$ is a random operator satisfying
suitable assumptions.

This paper stands a simplification of earlier works of F.~S.~De~Blasi,
G.~Pianigiani and L.~G\'orniewicz (see: \cite{gor7}, \cite{gor8}), where the case of
differential inclusions is considered.

**AMS classification.**
34C25, 55M20, 47H10

**Key words.**
Periodic processes, topological degree,
Poincare translation operator