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The Krasnosel’skii formula for parabolic differential inclusions with state constraints. (English) Zbl 1374.34040

Summary: We consider a constrained semilinear evolution inclusion of parabolic type involving an \(m\)-dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of the solution map. We establish the \(R_\delta\)-description of the set of solutions surviving in the constraining area and show a relation between the fixed point index of the Krasnosel’skii-Poincaré operator of translation along trajectories associated with the problem and the appropriately defined constrained degree of the right-hand side in the equation. This provides topological tools appropriate to obtain results on the existence of periodic solutions to studied differential problems.

MSC:

34A60 Ordinary differential inclusions
35K58 Semilinear parabolic equations
47H04 Set-valued operators
47H11 Degree theory for nonlinear operators
47J35 Nonlinear evolution equations
54C60 Set-valued maps in general topology
34K09 Functional-differential inclusions
55M20 Fixed points and coincidences in algebraic topology
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