We study non-compact surfaces obtained by gluing strips R × (−1, 1) with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the group of homeomorphisms of that foliation is contractible.2010 Mathematics Subject Classification. 30F15, 57R30.

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We propose a new criterion of stability in the sense of Lyapunov of the equilibrium point of autonomic differential equations using discrete approach. This approach includes a consideration of a family of hypersurfaces instead of the Lyapunov functions, and conditions on the right-hand part of the differential equation on them instead of conditions on a Lyapunov function along trajectories of the equation.

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