Homology properties of the sets of solutions to ordinary

The recent studies concerned with existence theorems for solutions to the Dirichlet problem for a second-order ordinary differential equation maintain the tradition of using either the Leray–Schauder theory or the properties of the shift operator along trajectories. The same holds for a large part of the research devoted to existence theorems for periodic solutions. However, the first method involves heavy constraints on the right-hand side of the equation under consideration, while the results delivered by the second have been weaker so far. In the present paper, results on the persistence of the solutions to the problem in question under a homotopy of the right-hand side (usually derived from the Leray–Schauder theory) are proved using ideas relating to the second method and the topological structures suggested earlier by the author, which are adequate for the construction of the sets of solutions to ordinary differential equations. The method put forward is insensitive to complexities in the structure of the right-hand side.