Relaxation theorem for the evolution differential inclusions
Let \(X\) be a separable Banach space, \(A\) the infinitesimal generator of a linear \(C_0\) semigroup on \(X\), \(F:[0,T] \times X\to X\) a set-valued map with nonempty compact convex values, and \(x_0\in X\). The author compares the solution sets to the differential inclusions \[ \dot x(t)\in Ax(t)+ F\bigl(t,x(t)\bigr), \quad x(0)=x_0, \] and \[ \dot x(t)\in Ax(t)+ \text{ext} F \bigl(t,x(t)\bigr), \quad x(0)=x_0, \] where `\(\text{ext} F\)' denotes the set of extreme points of \(F\), and proves a density (relaxation) theorem. He seems to be unaware of earlier comparable results, that include the cases when \(A\) depends on time or is nonlinear. See, for instance, \textit{N. S. Papageorgiou} [Czech. Math. J. 44, No. 3, 481-500 (1994; Zbl 0868.34010); ibid., No. 4, 631-648 (1994; Zbl 0822.45008)].
scientific article; zbMATH DE number 1348144
scientific article