We develop an additive, integer-valued degree theory for the class of
quasilinear Fredholm mappings. This class is sufficiently large so that
within its framework one can study general fully nonlinear elliptic boundary
value problems. In contrast to the Leray-Schauder degree, which is homotopy
invariant, a degree for the whole class of quasilinear Fredholm mappings must
necessarily accommodate sign-switching of the degree along admissible
homotopies. We introduce a homotopy invariant of paths of linear Fredholm
operators having invertible end-points, which we call the parity. The parity
provides a complete description of the possible changes in sign of the degree
and thereby enables us to use the degree to prove multiplicity and bifurcation
theorems for quasilinear Fredholm mappings. Applications are given to the
study of fully nonlinear elliptic boundary value problems.
Key Words and Phrases: Nonlinear elliptic boundary value problem, topological
degree, linear Fredholm operators, the parity, global bifurcation.