Density of the extremal solutions for a class of second order boundary value problem (Q1790569)

In the paper it is obtained the existence of extremal solutions for the following problem \[ -x''(t)\in \partial \varphi (x'(t))+ext(F(t,x(t),x'(t)))\quad \text{a.e. on } [0,1],\quad x(1)=x'(0)=0, \] where \(\varphi (.):{\mathbb R}^n\to {\mathbb R}\) is a convex function, \(\partial \varphi \) denotes the subdifferential in the sense of convex analysis, \(F(.):[0,1]\times {\mathbb R}^n\times {\mathbb R}^n\to {\mathcal P}({\mathbb R}^n)\) is a continuous set-valued map with convex compact values and \(ext(A)\) denotes the set of extreme points of a set \(A\subset {\mathbb R}^n\). At the same time, it is proved that if, in addition, \(F\) is Lipschitz in the second and third variable, then the solution set of the above problem is dense in the solution set of the problem \[ -x''(t)\in \partial \varphi (x'(t))+F(t,x(t),x'(t))\quad \text{a.e. on } [0,1],\quad x(1)=x'(0)=0. \]