The existence of periodic solutions to nonautonomous differential inclusions

Abstract

For an m m -dimensional differential inclusion of the form \[ x ˙ ∈ A ( t ) x ( t ) + F [ t , x ( t ) ] , \dot x \in A(t)x(t) + F[t,x(t)], \] with A A and F F T T -periodic in t t , we prove the existence of a nonconstant periodic solution. Our hypotheses require m m to be odd, and require F F to have different growth behavior for | x | \left | x \right | small and | x | \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.