Compact evolution operators (Q1174379)

This article deals with the property of compactness of evolution operators \(U(t,s)\) \((0\leq s<t\leq T)\) for the linear differential equation \(x'=A(t)x\) \((0\leq t\leq T)\) where \(A(t)\) is the family of nonlinear (possibly multivalued) operators of dissipative type. Under some additional but natural conditions on \(A(t)\) the author proves that the compactness of operators \(U(t,s)\) \((0\leq s<t\leq T)\) is equivalent to the following properties: (i) for each \(s\in[0,T]\) and \(\lambda>0\) the operator \(J_ \lambda(s)=(I-\lambda A(s))^{-1}:X\to D(A(s))\) is compact and (ii) for each \(t_ 0\in(s,T]\) the function \(U(t,s)x\) \((x\in Y)\) are equicontinuous at \(t_ 0\) on a bounded subset \(Y\subseteq\overline{D(A(s))}\). This theorem is an extension of the Brezis result for semigroups.