Degree of locally condensing perturbations of Fredholm maps with positive index and applications

We give a construction of the topological degree for locally condensing perturbations of C1-Fredholm maps of positive index with values in the GLc-framed cobordism group. The construction of the degree extends constructions of the degree for Fredholm maps of positive index and also its finite-dimensional, compact, and compactly restricted perturbations. It based on the finite-dimensional reduction method. We apply the constructed degree to prove the existence theorem for a system of ordinary differential equations with Hopf's boundary conditions.