The Poincaré-Bendixson theorem and compact families of spaces of solutions of ordinary differential equations (Q1326042)

Continuing his previous work [e.g., The theory of the Cauchy problem for an ordinary differential equation from the point of view of general topology, in ``General Topology. Mappings of Topological Spaces'' (Russian), Izdat. MGU, Moscow (1966); Mat. Zametki 38, 832-851 (1985; Zbl 0626.34014), etc.], the author uses a concept of convergence of a sequence of spaces of functions to prove an abstract theorem that may be interpreted as a generalization of the Poincaré-Bendixson theorem to some classes of nonautonomous differential equations. As examples, the spaces of solutions of some classes of differential equations are shown to satisfy certain hypotheses of the theorem. It should be noted that an interested reader of this note must be familiar with the concepts and notations introduced by the author in some 10 of his previously published papers.