In the setting of 2-uniformly smooth and q -uniformly convex Banach spaces, we prove the existence of solutions of the following multivalued differential equation: -\frac{d}{dt} J(u(t)) \in N^C(C(t,u(t));u(t)) \text{ a.e. in } [0,T]. \:\:\: \mathrm{(SDNSP)} This inclusion is called State Dependent Nonconvex Sweeping Process (SDNSP). Here N^C(C(t,u(t)); u(t)) stands for the Clarke normal cone. The perturbed (SDNSPP) is also considered. Our results extend recent existing results from the setting of Hilbert spaces to the setting of Banach spaces. In our proofs we use some new results on V -uniformly generalized prox-regular sets in Banach spaces.

We prove an existence result, in the separable Banach spaces setting, for second order differential inclusions of type sweeping process. This type of differential inclusion is defined in terms of normal cones and it covers many dynamic quasi-variational inequalities. In the present paper, we prove in the nonconvex case an existence result of this type of differential inclusions when the separable Banach space is assumed to be q-uniformly convex and 2-uniformly smooth. In our proofs we use recent results on uniformly generalized prox-regular sets. Part of the novelty of the paper is the use of the usual Lipschitz continuity of the set-valued mapping which is very easy to verify contrarily to the ones used in the previous works. An example is stated at the end of the paper, showing the application of our existence result.