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

Wojciech Kryszewski; Dorota Gabor; Jakub Siemianowski

doi:10.3934/dcdsb.2018021

Article Contents

2018, Volume 23, Issue 1: 295-329. Doi: 10.3934/dcdsb.2018021

This issue Previous Article Arzelà-Ascoli's theorem in uniform spaces Next Article Optimal control applied to a generalized Michaelis-Menten model of CML therapy

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

1.
Institute of Mathematics, Technical University of Łódź, Poland
2.
Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, Poland

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

Received: December 2016

Published: November 2017

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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.

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

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