The existence of periodic solutions to nonautonomous differential inclusions

Abstract:

For an $m$-dimensional differential inclusion of the form \[ \dot x \in A(t)x(t) + F[t,x(t)],\] with $A$ and $F$ $T$-periodic in $t$, we prove the existence of a nonconstant periodic solution. Our hypotheses require $m$ to be odd, and require $F$ to have different growth behavior for $\left | x \right |$ small and $\left | x \right |$ large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin.