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