Partial mappings, Čech homology and ordinary differential equations

Consideration of partial mappings allows us to expose a noticeable part of the theory of ordinary differential equations at the axiomatic level. This gives us a possibility to cover equations with discontinuous right-hand sides and differential inclusions by standard well-known (but a little revised) theory. Homological properties of the corresponding solution sets allow us to create a new version of the shift method in the theory of boundary value problems as powerful as the application here of the Leray-Schauder theory.