The Constrained Krasnosel'skii Formula for Parabolic Differential Inclusions

TL;DR

This paper extends the Krasnosel'skii formula to constrained parabolic differential inclusions, linking fixed point index and degree theory for multivalued operators in Banach spaces.

Contribution

It introduces a new relation between the constrained fixed point index and the constrained degree for parabolic differential inclusions, extending previous results.

Findings

Established a relation between fixed point index and degree for constrained inclusions.

Extended Krasnosel'skii formula to a broader class of parabolic differential inclusions.

Provided topological properties of the solution map in Banach spaces.

Abstract

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}.