Differential inclusions with nonstationary maximal monotone operators (Q755957)

The paper deals with the differential equation (1) \(\dot x(t)+A^ tx(t)=0\), \(x(0)=x_ 0\in \overline{D(A^ 0)}\) in a Hilbert space H, where \(A^ t: D(A^ t)\to 2^ X\) is a time dependent maximal monotone multivalued operator, \(D(A^ t)\subset H\), \(t\in [0,T]\). For maximal monotone operators \(A_ 1\), \(A_ 2\) the author defines a distance \(d(A_ 1,A_ 2)\) (which is not a metric) and studies its properties. The main result is the proof of existence and uniqueness of problem (1) for an arbitrary regular operator \(A^ t\) (0\(\leq t\leq T)\). The operator \(A^ t\) is said to be regular if a) \(A^ t\) is maximal and monotone for \(0\leq t\leq T\), b) int D(A\({}^ t)\neq \emptyset\) for \(0\leq t\leq T\), c) the map \(t\to A^ t\) is continuous with respect to the distance d.