### Abstract

George and Shobha (2012) considered the finite dimensional realization of an iterative method for non-linear ill-posed Hammerstein type operator equation KF(x) = f, when the Fréchet derivative F' of the non-linear operator F is not invertible. In this pa- per we consider the special case i.e., F'^{-1} exists and is bounded. We analyze the convergence using Lipschitz-type conditions used in [10], [13], [22] and also analyze the convergence using a center type Lipschitz condition. The center type Lipschitz con- dition provides a tighter error estimate and expands the applicability of the method. Using a logarithmic-type source condition on F(x0)-F(◯) (here ◯ is the actual solution of KF(x) = f) we obtain an optimal order convergence rate. Regularization param- eter is chosen according to the balancing principle of Pereverzev and Schock (2005). Numerical illustrations are given to prove the reliability of our approach.

Original language | English |
---|---|

Pages (from-to) | 34-55 |

Number of pages | 22 |

Journal | Communications on Applied Nonlinear Analysis |

Volume | 23 |

Issue number | 1 |

Publication status | Published - 01-01-2016 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Communications on Applied Nonlinear Analysis*,

*23*(1), 34-55.

}

*Communications on Applied Nonlinear Analysis*, vol. 23, no. 1, pp. 34-55.

**Discretized Newton-Tikhonov method for ill-posed hammerstein type equations.** / Argyros, Ioannis K.; George, Santhosh; Shobha, Monnanda Erappa.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Discretized Newton-Tikhonov method for ill-posed hammerstein type equations

AU - Argyros, Ioannis K.

AU - George, Santhosh

AU - Shobha, Monnanda Erappa

PY - 2016/1/1

Y1 - 2016/1/1

N2 - George and Shobha (2012) considered the finite dimensional realization of an iterative method for non-linear ill-posed Hammerstein type operator equation KF(x) = f, when the Fréchet derivative F' of the non-linear operator F is not invertible. In this pa- per we consider the special case i.e., F'-1 exists and is bounded. We analyze the convergence using Lipschitz-type conditions used in [10], [13], [22] and also analyze the convergence using a center type Lipschitz condition. The center type Lipschitz con- dition provides a tighter error estimate and expands the applicability of the method. Using a logarithmic-type source condition on F(x0)-F(◯) (here ◯ is the actual solution of KF(x) = f) we obtain an optimal order convergence rate. Regularization param- eter is chosen according to the balancing principle of Pereverzev and Schock (2005). Numerical illustrations are given to prove the reliability of our approach.

AB - George and Shobha (2012) considered the finite dimensional realization of an iterative method for non-linear ill-posed Hammerstein type operator equation KF(x) = f, when the Fréchet derivative F' of the non-linear operator F is not invertible. In this pa- per we consider the special case i.e., F'-1 exists and is bounded. We analyze the convergence using Lipschitz-type conditions used in [10], [13], [22] and also analyze the convergence using a center type Lipschitz condition. The center type Lipschitz con- dition provides a tighter error estimate and expands the applicability of the method. Using a logarithmic-type source condition on F(x0)-F(◯) (here ◯ is the actual solution of KF(x) = f) we obtain an optimal order convergence rate. Regularization param- eter is chosen according to the balancing principle of Pereverzev and Schock (2005). Numerical illustrations are given to prove the reliability of our approach.

UR - http://www.scopus.com/inward/record.url?scp=85030983035&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030983035&partnerID=8YFLogxK

M3 - Article

VL - 23

SP - 34

EP - 55

JO - Communications on Applied Nonlinear Analysis

JF - Communications on Applied Nonlinear Analysis

SN - 1074-133X

IS - 1

ER -