Open Access Research

Expanding the applicability of Lavrentiev regularization methods for ill-posed problems

Ioannis K Argyros1, Yeol Je Cho2* and Santhosh George3

Author Affiliations

1 Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

2 Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju, 660-701, Korea

3 Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Karnataka, 757 025, India

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Boundary Value Problems 2013, 2013:114  doi:10.1186/1687-2770-2013-114

Published: 7 May 2013

Abstract

In this paper, we are concerned with the problem of approximating a solution of an ill-posed problem in a Hilbert space setting using the Lavrentiev regularization method and, in particular, expanding the applicability of this method by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). Numerical examples are given to show that our convergence criteria are weaker and our error analysis tighter under less computational cost than the corresponding works given in (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009).

MSC: 65F22, 65J15, 65J22, 65M30, 47A52.

Keywords:
Lavrentiev regularization method; Hilbert space; ill-posed problems; stopping index; Fr├ęchet-derivative; source function; boundary value problem