Fields Institute Communications Volume 25, 00 In the book ”Operator theory and Applications”, Fields Institute Communica- tions vol. 25, AMS, Providence, 2000, pp.111-138 (Ed.A.G.Ramm, P.N.Shivakumar, A.V.Strauss). Continuous Methods for Solving Nonlinear Ill-Posed Problems Ruben G. Airapetyan Department of Mathematics Kansas State University Manhattan, Kansas 66506-2602 airapet@math.ksu.edu Alexander G. Ramm Department of Mathematics Kansas State University Manhattan, Kansas 66506-2602 ramm@math.ksu.edu Alexandra B. Smirnova Department of Mathematics Kansas State University Manhattan, Kansas 66506-2602 smirn@math.ksu.edu Abstract. The goal of this paper is to develop a general approach to solution of ill-posed nonlinear problems in a Hilbert space based on continuous processes with a regularization procedure. To avoid the ill- posed inversion of the Fr´ echet derivative operator a regularizing one- parametric family of operators is introduced. Under certain assumptions on the regularizing family a general convergence theorem is proved. The proof is based on a lemma describing asymptotic behavior of solutions of a new nonlinear integral inequality. Then the applicability of the theorem to the continuous analogs of the Newton, Gauss-Newton and simple iteration methods is demonstrated. 1 Introduction Let us consider a nonlinear operator equation F (z)=0,F : H → H, (1.1) in a real Hilbert space H (equation (1.1) in a complex Hilbert space can be treated similarly). Assume that (1.1) is solvable (not necessarily uniquely). If the Fr´ echet deriv- ative of the operator F has nontrivial null-space at the solution to (1.1), then one can use the classical Newton method for solution to (1.1) only under some strong assumptions on the operator F (see [11, 6]). Otherwise in order to construct a numerical method for solution to (1.1) one needs some regularization procedure. In the theory of ill-posed problems many different discrete methods based on a regularization are known. Many different convergence theorems for such schemes 1991 Mathematics Subject Classification. Primary 47H17; Secondary 65J15, 58C15. c 00 American Mathematical Society 111