### A class of singularly perturbed evolution systems.

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A differential equation is a Hilbert space with all solutions bounded but with so finite nontrivial invariant measure is constructed. In fact, it is shown that all solutions to this equation converge weakly to the origin, nonetheless, there is no stationary point. Moreover, so solution has a non-empty $\Omega $-set.

In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.

In LaSalle's book "The Stability of Dynamical Systems", the author gives four conditions which imply that the origin of a discrete dynamical system defined on ℝ is a global attractor, and proposes to study the natural extensions of these conditions in ℝⁿ. Although some partial results are obtained in previous papers, as far as we know, the problem is not completely settled. In this work we first study the four conditions and prove that just one of them implies that the origin is a global attractor...

Under suitable hypotheses on $\gamma \left(t\right)$, $\lambda \left(t\right)$, $q\left(t\right)$ we prove some stability results which relate the asymptotic behavior of the solutions of ${u}^{\text{'}\text{'}}+\gamma \left(t\right){u}^{\text{'}}+(q\left(t\right)+\lambda \left(t\right))u=0$ to the asymptotic behavior of the solutions of ${u}^{\text{'}\text{'}}+q\left(t\right)u=0$.