On the oriented degree of a certain class of perturbations of Fredholm mappings, and on bifurcation of solutions of a nonlinear boundary value problem with noncompact perturbations

The concept of oriented degree is extended to the class of mappings of the form $f-g$, where $f$ is a proper Fredholm mapping of nonnegative index and $g$ a continuous $f$-compactly restrictable mapping. In the case when $f$ is a Fredholm mapping of zero index and $f$ and $g$ are equivariant with respect to the action of the circle and the torus, formulas are obtained which express the degree of these mappings in terms of invariants of representations of the corresponding groups. An application to the investigation of the global behavior of a bifurcation branch of a certain nonlinear boundary value problem is given.