Some properties of viability problems depending on a parameter (Q1346275)

Let \(X\) be a Banach space, \(\Xi\) a separable metric space and \(K : \Xi - \circ X\) a compact-valued and lower semicontinuous multimap admitting a continuous selection \(x^ 0 (\cdot)\). Let \(\Gamma : [0,a] \times grK - \circ X\) be a closed, bounded-valued multimap such that \(\Gamma (\cdot, x, \xi)\) is measurable, \(\Gamma (t, \cdot, \cdot)\) is l.s.c for a.e. \(t \in [0,a]\), \(\Gamma (t, \cdot, \xi)\) satisfies some Lipschitz condition and the tangential condition \(\Gamma (t,x, \xi) \subset T_{K(\xi)} (x)\) where \(T\) denotes the Bouligand contingent cone is fulfilled. The author proves the existence and the continuous dependence on the parameter \(\xi\) of solutions of the problem \(x'(t, \xi) \in \Gamma (t,x, \xi)\), \(x(t, \xi) \in K (\xi)\), \(x(0, \xi) = x^ 0 (\xi)\). The relaxation property for this problem is also considered.