A generalization of the implicit function theorem for mappings from Rn + 1 into Rn and its applications

Some regularity results about the local structure of the zero set of a mapping of n + 1 real variables with values in $\text{R}$ⁿ are presented. They all provide improved and generalized versions of the Morse lemma for plane curves as well as generalizations of the implicit function theorem. We describe their applications to one-parameter nonlinear problems, including an analysis of bifurcation phenomena at simple or multiple eigenvalues.