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| The classical Krasnosel'skii formula relates the Brouwer degree of the
right-hand side of a differential equation in $\mathbb{R}^n$ with the Brouwer
fixed point index of the corresponding Poincar\'{e} operator (of translations
along trajectories). We apply the similar idea to the parameterized semilinear
differential inclusion \[(*)\hspace{20mm} \dot{u}\in Au+F_\lambda (t,u),
\hspace{5mm} t\in J, u\in E\] where $J=[0,\infty)$, $\lambda \in
\Lambda\subset \mathbb{R}^k$, $E$ is a Banach space, $A:D(A)\to E$ is the
infinitesimal generator of the $C_0$-semigroup of bounded linear operators on
$E$ and $F:\Lambda \times J \times E\to 2^E$ is a set-valued weakly upper
semicontinuous map with convex weakly compact values. Namely we compare the
respective homotopy invariants of the right-hand side and of the operator
$\Phi _t$ which assigns to the initial value $x\in E$ the set $\{u(t) ;$
$u$ is a (mild) solution of $(*)$ starting at $x\}$. It allows to
detect bifurcations of closed orbits to $(*)$. We also present how one can
apply this method to a concrete reaction-diffusion inclusion. |
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