Well posedness for differential inclusions on closed sets

Abstract

For an initial value problem having uniqueness the well posedness, i.e., the continuous dependence on the initial condition, is expressed by saying that the map, assigning to the initial point the solution through it, is continuous. For problems lacking uniqueness, saying that any solution through a point can be embedded in a continuous, single valued, family of solutions depending on the initial point, can be considered as the natural extension of the well posedness. For a differential inclusion with Lipschitzean right-hand side, defined on an open set, several papers [3-6, 91 yield results on the existence of continuous selections from the map assigning to an initial point, or a parameter, the set of solutions to the corresponding Cauchy problem. The present note considers the same problem for a multifunction F defined on Rx K, where K is a closed subset of R, and satisfying a tangentiality condition, and proves an analogous result. Remark that the (generalized) successive approximations process that is the base of the construction, in the case under consideration requires at each step a projection over the (in general, non-convex) set K, since F is not defined outside K, and that this projection is not continuous. Moreover, the lack of an argument allowing the extension of a multivalued Lipschitzean map from a closed set to an open set containing it, prevents the possibility of exploiting the available techniques for the present case. As a side result we obtain the convergence of the sequence of generalized successive approximations for any initial function x0. As a corollary, we prove a result on the arcwise connectedness of the set of solutions and of