### 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.

Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This...

In the paper some existence results for periodic boundary value problems for the ordinary differential equation of the second order in a Hilbert space are given. Under some auxiliary assumptions the set of solutions is compact and connected or it is convex.

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.

It is shown that the approximating equations whose existence is required in the author's previous work on partially regular weak solutions can be constructed without any additional assumption about the equation itself. This leads to a variation of a Galerkin method.