A continuous version of the relaxation theorem for nonlinear evolution inclusions (Q1902058)

The author proves some continuous dependence results for the parametrized Cauchy problem (P) \(x(t) + A(t,x(t)) \in F(t,x(t), \lambda)\), a.e. on \(T = [0,b]\), \(x(0) = v (\lambda)\), in a separable Hilbert space \(H\). Here \(\lambda \in \Lambda\) (a complete metric space), \(A(t,.)\) denotes (for each \(t \in T)\) a monotone hemicontinuous operator from \(X\) to \(X^*\) (where \(X\) is a reflexive Banach space of dual \(X^*\), such that \(X \subset H \subset X^*\), with dense and compact embeddings), \(F : T \times H \times \Lambda \to H\) is a closed valued multifunction, and \(v : \Lambda \to H\) is continuous. A convexified version of (P) is also considered, and a relaxation theorem relating the solution set of (P) to that of its convex counterpart is obtained. The theory is applied to the study of a parametrically controlled diffusion equation with nonlinear friction.