We prove that the inclusion of the space of proper gradient local maps into the space of proper local maps induces a bijection between the sets of the respective otopy classes of these maps.

We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classi cations are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from R^n to R^n and proper gradient maps from R^2 to R^2 with the Brouwer degree greater or equal to zero.

We prove that the inclusion of the space of proper gradient local maps into the space of proper local maps induces a bijection between the sets of the respective otopy classes of these maps.

We show that there exist two proper gradient vector fields on Rn which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.

Consider the Euclidean space Rn with the orthogonal action of a compact Lie group G. We prove that a locally Lipschitz G-invariant mapping f from Rn to R can be uniformly approximated by G-invariant smooth mappings g in such a way that the gradient of g is a graph approximation of Clarke’s generalized gradient of f . This result enables a proper development of equivariant gradient degree theory for a class of set-valued gradient...