Higher--order implicit function theorems and degenerate nonlinear boundary-value problems

The first part of this paper considers the problem of solving an equation of the form $F(x, y)=0$, for $y = \varphi (x)$ as a function of $x$, where $F: X \times Y \rightarrow Z$ 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 $p$th-order generalizations of the Implicit Function Theorem for this case. The second part of the paper uses these $p$th-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 of $p$-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).