Global Implicit Function Theorems and Critical Point Theory in Fr\'{e}chet Spaces

We prove two versions of a global implicit function theorem, which involve no loss of derivative, for Keller's $C_c^1$-mappings between arbitrary Fr\'{e}chet spaces. Subsequently, within this framework, we apply these theorems to establish the global existence and uniqueness of solutions to initial value problems that involve the loss of one derivative. Moreover, we prove a Lagrange multiplier theorem by employing indirect applications of the global implicit function theorems through submersions and transversality.