### A class of strongly cooperative systems without compactness

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It is shown that the uniform exponential stability and the uniform stability at permanently acting disturbances of a sufficiently smooth but not necessarily steady-state solution of a general variational inequality is a consequence of the uniform exponential stability of a zero solution of another (so called linearized) variational inequality.

The following problem of Markus and Yamabe is answered affirmatively: Let f be a local diffeomorphism of the euclidean plane whose jacobian matrix has negative trace everywhere. If f(0) = 0, is it true that 0 is a global attractor of the ODE dx/dt = f(x)? An old result of Olech states that this is equivalent to the question if such an f is injective. Here the problem is treated in the latter form by means of an investigation of the behaviour of f near infinity.

The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient ${\lambda}^{2}q\left(s\right),\phantom{\rule{4pt}{0ex}}s\in [{s}_{0},\infty )$ is investigated, where $\lambda \in \mathbb{R}$ and $q\left(s\right)$ is a nondecreasing step function tending to $\infty $ as $s\to \infty $. Let $S$ denote the set of those $\lambda $’s for which the corresponding differential equation has a solution not tending to 0. It is proved that $S$ is an additive group. Four examples are given with $S=\left\{0\right\}$, $S=\mathbb{Z}$, $S=\mathbb{D}$ (i.e. the set of dyadic numbers), and $\mathbb{Q}\subset S\u2acb\mathbb{R}$.