# American Institute of Mathematical Sciences

September  2008, 22(3): 605-628. doi: 10.3934/dcds.2008.22.605

## Krasnosel'skii type formula and translation along trajectories method for evolution equations

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

Received  October 2007 Revised  April 2008 Published  August 2008

The Krasnosel'skii type degree formula for the equation

$\dot u = - Au + F(u)$

where $A:D(A)\to E$ is a linear operator on a separable Banach space $E$ such that $-A$ is a generator of a $C_0$ semigroup of bounded linear operators of $E$ and $F:E\to E$ is a locally Lipschitz $k$-set contraction, is provided. Precisely, it is shown that if $V$ is an open bounded subset of $E$ such that $0$∉$(-A+F)(\partial V \cap D(A))$, then the topological degree of $-A+F$ with respect to $V$ is equal to the fixed point index of the operator of translation along trajectories for sufficiently small positive time. The obtained degree formula is crucial for the method of translation along trajectories. It is applied to the nonautonomous periodic problem and an average principle is derived. As an application a first order system of partial differential equations is considered.

Citation: Aleksander Ćwiszewski, Piotr Kokocki. Krasnosel'skii type formula and translation along trajectories method for evolution equations. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 605-628. doi: 10.3934/dcds.2008.22.605
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