Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms (Q1706368)

In this paper, the authors consider the following problem: \[ \begin{aligned} a(u'(t))'\in A(u(t)) &+ \operatorname{ext} F(t,u(t),u'(t)) \quad\text{for a. a. } t\in [0,b],\\ u(0)&=u(b),\quad u'(0)=u'(b). \end{aligned} \] Here: \(a:\mathbb{R}^N\to \mathbb{R}^N\) and \(A:D(A)\subseteq\mathbb{R}^N\to 2^{\mathbb{R}^N}\) are two maximal monotone maps; \(F:[0,b]\times \mathbb{R}^N\times \mathbb{R}^N\to 2^{\mathbb{R}^N}\setminus\{\emptyset\}\) is a multifunction; \(\operatorname{ext} F(t,u,u')\) is the set of extreme points of the set \(F(t,u,u')\). When \(F(t,\cdot,\cdot)\) is \(h\)-continuous, the existence of a (extremal periodic) solution \(u\in C^1([0,b],\mathbb{R}^N)\) for the problem is proved. Then, strengthening the assumptions on \(a(\cdot)\) and \(F(t,\cdot,\cdot)\), a strong relaxation theorem is provided showing that each solution of the ``convex'' parametric Dirichlet problem \[ \begin{aligned} a(u'(t))'\in A(u(t)) &+ \mu F(t,u(t),u'(t)) \quad\text{for a. a. } t\in [0,b],\\ u(0)&=u(b)=0 \end{aligned} \] (\(\mu>0\) is a parameter) can be approximated in the \(C^1([0,b],\mathbb{R}^N)\)-norm by extremal trajectories, namely solutions of the Dirichlet problem \[ \begin{aligned} a(u'(t))'\in A(u(t)) &+ \mu\, \operatorname{ext} F(t,u(t),u'(t)) \quad\text{for a. a. } t\in [0,b],\\ u(0)&=u(b)=0. \end{aligned} \]