Yamoto, J: Differential Equations

This thesis is mainly concerned about properties of the so-called Filippov operator that is associated with a di¿erential inclusion x(t) ¿ F(t) .e. t ¿ [0, T, where F : Rn is given set-valued map. The operator F produces a new set-valued map F [F ], which in e¿ect regularizes F so that F [F ] has nicer properties. After presenting its de¿nition, we show that F [F ] is always upper-semicontinuous as a map from Rn to the metric space of compact subsets of Rn endowed with the Hausdor¿metric. Our main approach is to study the operator via its support function, which we show is an upper semicontinuous function. We show that the support function can be used to characterize the operator, and prove a new result that characterizes those set-valued maps that are ¿xed by F ; this result was previously known to hold only in dimension one. We also generalize to higher dimensions a known result that characterizes those set-valued maps that are almost everywhere singleton-valued (that is, F (x) = {f (x)} where f : Rn ¿ Rn is an ordinary function).