Abstract:

Generalized transversality theorem for $ C^r $ mapping $ F(u, s):M\times S\rightarrow N $ is established in infinite dimensional Banach manifolds $ M, S, N $. If the mapping $ F(u, s) $ is generalized transversal to a single point set $ \{\hat{\theta}\} $, and $ f_s(u)=F(u, s) $ is a Fredholm operator in the sense of parameter s, then there exists a residual set $ \Sigma\subset S, $ such that $ f_s(u) $ are generalized transversal to $ \{\hat{\theta}\} $, for all $ s\in \Sigma. $

Key words: Transversality, Generalized inverse, Banach manifold, Singularities