On a time and state dependent maximal monotone operator coupled with a sweeping process with perturbations
In this paper, the authors consider the following couple of evolution inclusions governed by time and state dependent maximal operator and closed convex sweeping process, with perturbations of the form \[ \left\{\begin{aligned} -&\dot u(t)\in A(t,v(t))u(t)+f(t,u(t), v(t)) \quad && \text{a.e. } t\in [0,T],\\ &u(t)\in D(A(t,v(t)))&&\forall t\in [0,T],\\ -&\dot v(t)\in N_{C(t,u(t))}v(t)+g(t, u(t), v(t)) &&\text{a.e. } t\in [0,T],\\ &u(0)=u_0\in D(A(0,v_0)), &&v(0)=v_0\in C(0,u_0), \end{aligned}\right. \] where \(A(t,x)\) is a time and state dependent maximal monotone operator, \(D(A(t, x))\) stands for the domain of the operator \(A(t, x),\) \( C(t, x)\) is a time and state dependent closed convex set, \(N_{C(t,x)}\) is a sweeping process, and \(f , g\) are Carathéodory mappings. The existence of absolutely continuous solutions in separable Hilbert spaces is studied. Applications to optimization and Skorohod problems are also presented.
scientific article; zbMATH DE number 7345316
scientific article