The first part of this paper considers the problem of solving an equation of the form
F(x, y) = 0, for
y = 
(x) as a function of
x, where
F : X x Y 
Z is a smooth nonlinear mapping between Banach spaces. The focus is on the case in which the mapping
F is degenerate at some point
(x*, y*) with respect to
y, i.e., when
F
y(x*, y*), the derivative of
F with respect to
y, is not invertible and, hence, the classical Implicit Function Theorem is not applicable. We present
pth-order generalizations of the Implicit Function Theorem for this case.
The second part of the paper uses these
pth-order implicit function theorems to derive sufficient conditions for the existence of a solution of degenerate nonlinear boundary-value problems for second-order ordinary differential equations in cases close to resonance.
The last part of the paper presents a modified perturbation method for solving degenerate second-order boundary value problems with a small parameter.
The results of this paper are based on the constructions in
p-regularity theory, whose basic concepts and main results are given in the paper
Factor-analysis of nonlinear mappings: p-regularity theory by Tret'yakov and Marsden(Communications on Pure and Applied Analysis,
2 (2003), 425-445).
