## A convergence analysis of the Landweber iteration for nonlinear ill-posed problems.(English)Zbl 0840.65049

Let $$X$$ and $$Y$$ be Hilbert spaces, $$F : {\mathcal D}(F) \subset X \to Y$$ a nonlinear differentiable operator. To solve the equation $$F(x) = y$$, the authors apply the Landweber method $x_{k + 1} = x_k + F'(x_k)(y - F(x_k)),\quad k = 0,1,2,\dots.$ The authors prove the convergence of the method (with the discrepancy stopping corresponding to inexact data $$y$$) under the conditions $|F'(x)|\leq 1,\quad |F(x) - F(\widetilde{x}) - F'(x) (x - \widetilde{x}) |\leq \eta |F(x) - F(\widetilde{x})|,\quad \eta < 1/2.$ They present examples of different problems ( Hammerstein integral equation, distributed parameter problem) where the last condition is fulfilled. Under further conditions, they also study the convergence rate of the method.
Reviewer: G.Vainikko (Espoo)

### MSC:

 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 65J15 Numerical solutions to equations with nonlinear operators 45G10 Other nonlinear integral equations 65R30 Numerical methods for ill-posed problems for integral equations 47J25 Iterative procedures involving nonlinear operators
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