Krasnosel'skii type formula and translation along trajectories method for evolution equations
Aleksander Ćwiszewski, Piotr Kokocki · Discrete and Continuous Dynamical Systems · 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 boundedlinear 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 $0$∉$(-A+F)(\partialV \cap D(A))$, then the topological degree of $-A+F$ with respect to$V$ is equal to the fixed point index of the operator of translationalong trajectories for sufficiently small positive time. Theobtained degree formula is crucial for the method of translationalong trajectories. It is applied to the nonautonomous periodicproblem and an average principle is derived. As an application afirst order system of partial differential equations is considered.