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| In the talk we will study the relations between the fixed point index of the Poincar\'e translation
operator generated by the the following semilinear differential equation
$$\dot u\in Au+F(t,u),\;\; t\in J,\;, u\in C,$$
where $J:=[0,T]$, $T>0$, $C\subset E$, $E$ is a Banach space, is a closed convex set, $A:D(A)\to E$, $D(A)\subset E$, is the generator of a compact $C_0$-semigroup
$S=\{S(t)\}_{t\geq 0}$ of bounded linear operators on $E$ and $F:J\times E\to E$ is a continuous map or a set-valued weakly upper semicontinuous map with convex weakly compact values. It is to be noted that many partial differential equations (or systems of such equations), e.g. of parabolic (in particular nonlinear reaction-diffusion equations) or hyperbolic type (or inclusions), can be transformed so as to have this form. The presence of the constraining set $C$ is justified by applications. With this problem one associates the Poincare translation operator $\Phi_t$, where $t>0$, which assigns to each initial value $x\in E$ the set of values $u(t)$, where $u$ the (mild) solution starting at $x$. We show that under some natural assumptions $C$ is viable with respect to $\Phi$, i.e., solutions stay in $C$ and then we show that, in the spirit of the Krasnosel'skii formula which relates the Brouwer degree of the right-hand side of an ordinary differential equation in $R^n$ with the Brouwer fixed point index
of the Poincar\'e operator, also in the considered situation the right-hand side of the equation is in an appropriate sense homotopic to $\Phi_t$ with sufficiently small $t>0$, which gives the formula relating the Granas fixed point index of $\Phi_t$ on $C$ with the appropriately defined constrained topological degree of the right hand side. |
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