Extremal solutions of differential inclusions via Baire category: a dual approach (Q2434681)

The paper studies the following Cauchy problems for differential inclusions \[ x'(t)\in \mathrm{ext}(F(x(t))),\quad x(0)=0,\tag{1} \] \[ x'(t)\in F^{w(t)}(x(t)),\quad x(0)=0,\tag{2} \] where \(F:{\mathbb R}^n\to {\mathcal P}({\mathbb R}^n)\) is a bounded Hausdorff continuous set-valued map with compact convex values, \(\mathrm{ext}(F(x))\) denotes the set of extreme points of \(F(x)\), \(F^{w}(x)=\{y\in F(x); \, <y,w>=\max_{z\in F(x)}<z,w>\}\) and \(w(.):[0,T]\to {\mathbb R}^n\) is a given continuous function. Using Baire category method it is proved that all solutions of problem (2) are also solutions of problem (1).