Unbounded state-dependent sweeping processes with perturbations in uniformly convex and \(q\)-uniformly smooth Banach spaces
Let \(X\) be a uniformly convex and \(q\)-uniformly smooth Banach space \((q\geq 2)\), \(X^*\) its dual space, \(J, J^*\) their duality mappings. \par The authors study the existence of solutions of the following problem \[ (y^*)'(t) \in - N_{C(t,J^*y^*(t))}(J^*y^*(t))-F(t,J^*y^*(t)) \; \text{a.e. } \; t\in [0, 1], \] \[ y^*(0) = J y_0, y_0\in C(0,y_0), \] where \(C:[0,T]\times X\multimap X\) with convex closed values satisfies Lipschitz type conditions, and the perturbation \(F:[0,T]\times X\multimap X^*\) is upper semicontinuous with convex weak\(^*\) compact values with linear growth conditions, \(N_C\) denotes outward normal cone. This may be seen as a sweeping process with a perturbation. Similar second-order problems are considered. \par Proofs use suitable modifications of Moreau' catching up algorithm.
unbounded state-dependent sweeping processes
uniformly convex and \(q\)-uniformly smooth Banach space
scientific article; zbMATH DE number 7006498
scientific article