On the equivalence of singular and ill-posed problems: The $p$-factor regularization method

The local equivalence of singular and ill-posed problems in a class of sufficiently smooth mappings is shown, which justifies the use of the $p$-factor regularization method to solve them. The main constructions in $p$-regularity theory that are necessary for stable solution of approximate problems are described, and estimation theorems for regularizing algorithms are proved.