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Periodic processes; topological degree; Poincaré translation operator
We shall consider periodic problems for ordinary differential equations of the form \[ {\left\lbrace \begin{array}{ll} x^{\prime }(t)= f(t,x(t)),\\ x(0) = x(a), \end{array}\right.} \] where $ f:[0,a] \times R^n \rightarrow 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 (), multivalued Poincaré operator $P$ turns out to be different from zero, then problem () has solutions. Next by using the multivalued version of the classical Liapunov-Krasnoselskǐ guiding potential method we calculate the topological degree of the Poincaré 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 (), 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: \[ {\left\lbrace \begin{array}{ll} x^{\prime }(\xi , t) = f(\xi , t, x(\xi ,t)),\\ x(\xi ,0) = x(\xi , a), \end{array}\right.} \] where $f:\Omega \times [0,a]\times R^n\rightarrow R^n$ is a random operator satisfying suitable assumptions. This paper stands a simplification of earlier works of F. S. De Blasi, G. Pianigiani and L. Górniewicz (see: [gor7], [gor8]), where the case of differential inclusions is considered.
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