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Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$
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, 87100, Toruń, Poland, Poland 
$\dot u =  Au + F(u)$
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.
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