A continuous version of the relaxation theorem for nonlinear evolution inclusions
Νικόλαος Παπαγεωργίου · Kodai Mathematical Journal · 1995
ABSTRACT. We consider parametric nonlinear evolution inclusions defined on an evolution triple of spaces. First, we prove some continuous dependence results for the solution sets of both the convex and nonconvex problem and for the set of solution-selector pairs of the convex problem. Subsequently, we derive a parametrized version of the Filippov-Gronwall estimate in which the parameter varies in a continuous fashion. Using that estimate, we prove a continuous version of the nonlinear relaxation theorem. An example of a nonlinear parabolic control system is worked out in detail. 1. Introduction. One of the fundamental results in the theory of differential inclusions (set-valued differential equations), is the "relaxation theorem. " It says that if the orientor field (set-valued vector field) is h-Lipschitz in the state variable, then the solution set of the differential inclusion is dense in that of the