Towards Nonlinearity: The p-Regularity Theory

We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of <i>p</i>-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of <i>p</i>-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré's method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems.