Extremal solutions and relaxation for second-order vector differential inclusions (Q2716328)

Let \(F\) be a multifunction mapping the interval \([0,b]\) into the collection of all nonempty, compact and convex subsets of \(\mathbb{R}^N\). In the first part of the paper, the authors give sufficient conditions for the solvability of periodic differential inclusions NEWLINE\[NEWLINEx''(t)-x(t)\in \operatorname {ext}F(t,x(t),x'(t)) \quad \text{a.e. on }[0,b],\qquad x(0)=x(b),\quad x'(0)=x'(b),\tag{1}NEWLINE\]NEWLINE where \(\operatorname {ext}F(t,x,y)\) denotes the extreme points of \(F(t,x,y)\). In the second part of the paper, it is shown that under some assumptions on the multifunction \(F\) every solution to the Dirichlet problem NEWLINE\[NEWLINEx''(t)-x(t)\in \operatorname {ext}F(t,x(t),x'(t)) \quad \text{a.e. on }[0,b],\qquad x(0)=x(b)=0,\tag{2}NEWLINE\]NEWLINE can be obtained as the \(C^1([0,b];\mathbb{R}^N)\)-limit of a sequence of solutions to the ``extremal'' Dirichlet problem (1), (2).