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: $-A\left(x^{\left(3\right)}\right(t\left)\right)\in N\left(K\right(t,\dot{x}\left(t\right))$;$A\left(\ddot{x}\right(t\left)\right))+F(t,x\left(t\right),\dot{x}\left(t\right),\ddot{x}\left(t\right))+G(x\left(t\right),\dot{x}\left(t\right),\ddot{x}\left(t\right)\left)\text{a}\text{.}\text{e}\text{.}\right[0,T]$, $A\left(\ddot{x}\right(t\left)\right)\in K(t,\dot{x}(t\left)\right)$, $\text{a.e.}t\in [0,T]$, $x\left(0\right)=x_0,\dot{x}\left(0\right)=u_0$, $\ddot{x}\left(0\right)={\upsilon }_0$, 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 $\mathbb{H}$.