General existence results for third-order nonconvex state-dependent sweeping process with unbounded perturbations

Abstract

We prove the existence of solutions for third-order nonconvex state-dependent sweeping process with unbounded perturbations of the form: - A (x (3) (t)) N (K (t, x (t)); A (x (t))) + F (t, x (t), x (t), x (t)) + G (x (t), x (t), x (t)) a. e. [ 0, T ], A (x (t)) K (t, x (t)), a.e. t [ 0, T ], x (0) = x 0, x (0) = u 0, x (0) = 0, where T > 0, K is a nonconvex Lipschitz set-valued mapping, F is an unbounded scalarly upper semicontinuous convex set-valued mapping, and G is an unbounded uniformly continuous nonconvex set-valued mapping in a separable Hilbert space ℍ. © 2012 M. Bounkhel and B. Al-Senan.