---
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 Fréchet 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.

    [**AMS subject classification:**]{} Primary: 47H17. Secondary: 65J15, 58C15.
---

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¶ ł Ł

[**Continuous methods for solving nonlinear ill-posed problems**]{}\

Ruben G. Airapetyan\
E-mail: airapet@math.ksu.edu\
Department of Mathematics\
Kansas State University\
Manhattan, Kansas 66506-2602, U.S.A.\

Alexander G. Ramm\
E-mail: ramm@math.ksu.edu\
Department of Mathematics\
Kansas State University\
Manhattan, Kansas 66506-2602, U.S.A.\

Alexandra B. Smirnova\
E-mail: smirn@math.ksu.edu\
Department of Mathematics\
Kansas State University\
Manhattan, Kansas 66506-2602, U.S.A.\

Introduction
============

Let us consider a nonlinear operator equation $$\label{noneq}
F(z)=0,\q F: H\to H,$$ in a real Hilbert space $H$ (equation (\[noneq\]) in a complex Hilbert space can be 