Compact evolution operators
Nicolae H. Pavel · Differential and Integral Equations · 1989
In this paper, we give a characterization of compactness of the Evolution Operator U(t, s) generated by a family of nonlinear (possibly multivalued) operators {A(t),O:::; t:::; T} of dissipative type.This is an extension of a result of Brezis [2] on the compactness of the semigroup SA(t) generated by an m-dissipative operator A via the exponential formula of Crandall-Liggett [3].l.Preliminaries.Statements of the results.Let X be a real Banach space of norm 11.Recall some notations and definitions (for details we refer to [6], [7]).(where x andy are elements of X.Clearly,where J is the duality mapping of X.Now, let {A(t); 0 :::; t :::; T} be a family of nonlinear (possibly multivalued) operators A(t): D(A(t)) C X--> 2x satisfying the hypotheses (Cl) R(I-hA(t)) =X, for all h > 0 and t E [0, T] (C2) There are two continuous functions f: [0, T] -->X and L: R+ --> R+ such that (Yl-Yz, x1-xz); :S llf(t)-f(s)llllxl-xziiL(max{llxlll, llxzll}) for all 0:::; s:::; t:::