  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.
