Compact Perturbations of m-Accretive Operators in General Banach Spaces

The evolution problem $\tfrac{{du}} {{dt}} + Au(t) \ni Bu(t),u(0) = x$, where A is a nonlinear m-accretive operator and B is a nonlinear completely continuous operator $B:C([0,T];X) \to L^p (0,T;X)$, is studied and various results on a local and global existence of solutions are established. A generalization of the Weyl–Riesz criterion for compactness of sets in the spaces $L^p (\mathbb{R},dt,X),1 \leq p < \infty $, X is a Banach space is proved.