The Constrained Krasnosel'skii Formula for Parabolic Differential Inclusions

We consider a constrained evolution inclusions of parabolic type \eqref{inkluzja-rozn} involving an $m$-dissipative linear operator and the source term of multivalued type in a Banach space and topological properties of the solution map. We show a relation between the constrained fixed point index of the Krasnosel'skii--Poincar\'{e} operator of translation along trajectories associated with \eqref{inkluzja-rozn} and the appropriately defined constrained degree of $A + F\le 0 , \cdot \pr$ of the right-hand side in \eqref{inkluzja-rozn}. Our results extend those of \cite{cw} and \cite{gab-krysz}.