## 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 $$\label{gorrI} \begin{cases} x'(t)= f(t,x(t)),\\ x(0) = x(a), \end{cases}$$} 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: $$\label{gorrII} \begin{cases} x'(\xi, t) = f(\xi, t, x(\xi,t)),\\ x(\xi,0) = x(\xi, a), \end{cases}$$} 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