Continuous selection of the solution map for one-sided Lipschitz differential inclusions
The author studies the following multivalued Cauchy problem \[ \dot{x}(t) \in F(x(t)),\quad a.e. \quad t\in I,\quad x(0)=x_0, \eqno(1) \] where \(I=[0,+\infty)\) and \(F:E\to 2^E\) is a multifunction with nonempty, convex and compact values in a Banach space \(E\) with a uniformly convex dual. By assuming that \(F\) is one-sided Lipschitz continuous and it has a sublinear growth, the author proves the existence of a continuous selection of the solution map, which is the multifunction that to every \(x_0 \in E\) assigns the set of all solutions to problem (1). In the last part of the paper, as an application, the problem of the existence of a viable trajectories, in a prescribed closed subset of \(E\), is studied.
scientific article; zbMATH DE number 5128005
scientific article