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The Krasnosel'skii formula for parabolic differential inclusions with state constraints

* The first author was partially supported by the Polish National Science Center under grant 2013/09/B/ST1/01963

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  • 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 $R_δ$-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.

    Mathematics Subject Classification: Primary: 34A60, 35K58, 47H04, 47H11, 47J35, 54C60; Secondary: 34K09, 55M20.


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