Abstract
We define the concept of a Conley index and a homology index braid class for ordinary differential equations of the form \begin{equation} \dot x= F_1(x), \tag{$E$} \end{equation} where $\mathcal{M}$ is a $C^2$-manifold and $F_1$ is the principal part of a continuous vector field on $\mathcal{M}$. This allows us to extend our previously obtained results from [M.C. Carbinatto and K.P. Rybakowski, On the suspension isomorphism for index braids in a singular perturbation problem, Topological Methods in Nonl. Analysis 32 (2008), 199-225] on singularly perturbed systems of ordinary differential equations \begin{equation} \begin{split} \varepsilon\dot y&=f(y,x,\varepsilon),\\ \dot x&=h(y,x,\varepsilon) \end{split} \tag($E_\varepsilon$) \end{equation} on $Y\times \mathcal{M}$, where $Y$ is a finite dimensional Banach space and $\mathcal{M}$ is a $C^2$-manifold, to the case where the vector field in $(E_\varepsilon)$ is continuous, but not necessarily locally Lipschitzian.
Citation
Maria C. Carbinatto. Krzysztof P. Rybakowski. "Conley index and homology index braids in singular perturbation problems without uniqueness of solutions." Topol. Methods Nonlinear Anal. 35 (1) 1 - 32, 2010.
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