Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces

Piotr  Kokocki

doi:10.3934/cpaa.2015.14.2315

Article Contents

2015, Volume 14, Issue 6: 2315-2334. Doi: 10.3934/cpaa.2015.14.2315

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Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces

Piotr Kokocki^1,

1.
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń

Received: December 31, 2014

Revised: April 30, 2015

Published: October 2015

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Abstract

We provide a global continuation principle of periodic solutions for the equation $\dot u = - Au + F(t,u)$, where $ A:D(A) \to X$ is a sectorial operator on a Banach space $X$ and $F:[0, +\infty) \times X^\alpha \to X$ is a nonlinear map defined on a fractional space $X^\alpha$. The approach that we use in this paper is based upon the theory of topological invariants that applies in the situation when the Poincaré operator associated with the equation is endowed with some form of compactness.

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Mathematics Subject Classification: Primary: 37B30, 47J35; Secondary: 35B10, 47J15.

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