New numerical methods and some applied aspects of the $p$-regularity theory

The theory of $p$-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu. G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the $p$-factor method for solving ODEs with a small parameter.