A Remark on the Existence of Solutions to Nonlinear Equations with Degenerate Mappings

Abstract

The paper studies nonlinear mappings using methods of p-regularity theory and some concepts and technics of set-valued analysis. The main result addresses to the problem of existence of solutions to nonlinear equations in the degenerate case where a linear part may be singular at the considered initial point.

On the Existence of Solutions to Nonlinear Equations Involving Singular Mappings with Non-zero p-Kernel

“…We quote one modification of the theorem on existence of solutions to the equations with non-degenerate mappings (see in [15]). The essential idea of the proof in the singular case is based on the similar construction to the one in the regular case.…”

“…In the previous paper [15] we have considered the case of the trivial p-kernel of porder derivatives of the mapping F(x) at the initial point x 0 (for completely degenerate case). For these purposes we required much stronger assumptions on the mapping F(x) such as invertibility of nonlinear operator F ( p) (x 0 )[·] p , uniform boundedness of F ( p) (x 0 )[·] p −1 and so on.…”

“…In this paper we continue consideration of the solution existence problem to the nonlinear equation which was studied in [15].…”

“…In present paper in the main result (Theorem 2) we give sufficient conditions for existence of local solutions to some equations in non-regular (singular) case when p-kernel of pth order derivative of F at the initial point x 0 is nonempty. Earlier we have considered the trivial p-kernel case (see [15]). This result can be applied for instance to the following nonlinear boundary value problem:…”

Iterative method for solving nonlinear singular problems

“…We use some of the methods to prove the conditions of the existence of local solutions to nonlinear singular equations in [4] and then some other results in the case with non-zero p-kernel (see [5]) and also give a description of a tangent cone in singular case [6].…”