Extremal solutions of differential inclusions via Baire category: A dual approach

摘要

Let F be a continuous multifunction on R~n with compact convex values. For any vector w, let F~w(x)?F(x) be the subset of points which maximize the inner product with w. Call W the set of all continuous functions w:[0,T]{mapping}Rn with the following property: all solutions to the Cauchy problem ?(t)∈F~w(t)(x(t)), x(0) = 0, are also solutions to ?(t)∈extF(x(t)). We prove that W is residual in C([0,T]; R~n).