First and second order convex sweeping processes in reflexive smooth Banach spaces (Q977077)

In the paper are studied sweeping processes of the form \[ \begin{cases} -x'(t)\in N(C(t),J^*(x(t)))+F(t,J^*(x(t)))\quad a.e.\; ([0,T]),\\ x(0)=J(x_0)\in J(C(0)),\end{cases}\tag{1} \] \[ \begin{cases} -x(t)\in N(C(t),J^*(x'(t)))+F(t,J^*(x(t)))\quad a.e.\; ([0,T]),\\ x(0)=J(x_0)\in J(C(0)),\quad x'(0)\in J(C(0)),\end{cases}\tag{2} \] where \(X\) is a Banach space with topological dual space \(X^*\), \(C(.):I\to {\mathcal P}(X^*)\) and \(F(.,.):[0,T]\times X\to {\mathcal P}(X^*)\) are set-valued maps, \(N(S,u)\) is the convex normal cone of \(S\subset X\) at \(u\in S\), \(J:X\to {\mathcal P}(X^*)\) is the normalized duality mapping and \(J^*:X^*\to {\mathcal P}(X)\) is a normalized duality mapping in \(X^*\). When \(X\) is a separable \(p\)-uniformly convex and \(q\)-uniformly smooth Banach space, \(F(.,.)\) is upper semicontinuous with convex compact values, \(C(.)\) has nonempty closed values and satisfies a certain compactness assumption the authors prove the existence of Lipschitz continuous solutions for problems (1) and (2). The case of second-order convex sweeping process is also studied.