Faculty Publications
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Item Expanding the applicability of Lavrentiev regularization methods for ill-posed problems(2013) Argyros, I.K.; Cho, Y.J.; George, S.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). © 2013 Argyros et al.; licensee Springer.Item Local convergence of a uniparametric halley-type method in banach space free of second derivative(International Publications internationalpubls@yahoo.com, 2015) Argyros, I.K.; George, S.; Mohapatra, R.N.We present a local convergence analysis of a uniparametric Halley-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative [26]. Numerical examples are also provided in this study.Item On a sixth-order Jarratt-type method in Banach spaces(World Scientific, 2015) Argyros, I.K.; George, S.We present a local convergence analysis of a sixth-order Jarratt-type method in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies such as [X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms 57 (2011) 441-456.] require hypotheses up to the third Fréchet-derivative. Numerical examples are also provided in this study. © 2015 World Scientific Publishing Company.Item Local convergence of deformed Jarratt-type methods in Banach space without inverses(World Scientific Publishing Co. Pte Ltd wspc@wspc.com.sg, 2016) Argyros, I.K.; George, S.We present a local convergence analysis for the Jarratt-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative. Hence, the applicability of these methods is expanded under weaker hypotheses and less computational cost for the constants involved in the convergence analysis. Numerical examples are also provided in this study. © World Scientific Publishing Company.Item On the local convergence of a secant like method in a banach space under weak conditions(International Publications internationalpubls@yahoo.com, 2016) Argyros, I.K.; Khattri, S.K.; George, S.We present a local convergence analysis of a Secant-like method in a Banach space setting. The method is used to approximate a solution of a nonlinear equation. The sufficient convergence conditions are weaker than in earlier studies. Numerical examples are also given in this work. © 2016 International Publications. All rights reserved.Item Extending the applicability of Newton’s method using Wang’s– Smale’s ?–theory(North University of Baia Mare Office_CJEES@yahoo.ro 76 Victoriei Baia Mare 430 122, 2017) Argyros, I.K.; George, S.We improve semilocal convergence results for Newton’s method by defining a more precise domain where the Newton iterate lies than in earlier studies using the Smale’s ?– theory. These improvements are obtained under the same computational cost. Numerical examples are also presented in this study to show that the earlier results cannot apply but the new results can apply to solve equations. © 2017, North University of Baia Mare. All rights reserved.Item On the local convergence of newton-like methods with fourth and fifth order of convergence under hypotheses only on the first fréchet derivative(Institute of Mathematics nsjom@dmi.uns.ac.rs, 2017) Argyros, I.K.; Padikkal, P.; George, S.We present a local convergence analysis of several Newton-like methods with fourth and fifth order of convergence in order to approximate a locally unique solution of an equation in Banach space setting. Earlier studies have used hypotheses up to the fifth derivative although only the first derivative appears in the definition of these methods. In this study we only use the hypothesis on the first derivative. This way we expand the applicability of these methods. Moreover, we provide a radius of convergence, a uniqueness ball and computable error bounds based on Lipschitz constants. Numerical examples computing the radii of the convergence balls as well as examples where earlier results cannot apply to solve equations but our results can apply are also given in this study. © 2017, Institute of Mathematics. All rights reserved.Item Local convergence of a fifth convergence order method in Banach space(Elsevier B.V., 2017) Argyros, I.K.; George, S.We provide a local convergence analysis for a fifth convergence order method to find a solution of a nonlinear equation in a Banach space. In our paper the sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative. Previous works use conditions reaching up to the fourth Fréchet-derivative. This way, the applicability of these methods is extended under weaker conditions and less computational cost for the Lipschitz constants appearing in the convergence analysis. Numerical examples are also given in this paper. © 2016 The AuthorsItem Local convergence for a family of sixth order Chebyshev-Halley -type methods in Banach space under weak conditions(Tusi Mathematical Research Group (TMRG) moslehian@memeber.ams.org, 2018) Argyros, I.K.; George, S.We present a local convergence analysis for a family of super- Halley methods of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first and second Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet derivative. Numerical examples are also provided in this study. © 2017 Khayyam Journal of Mathematics.Item An improved semilocal convergence analysis for the Halley's method(International Publications internationalpubls@yahoo.com, 2018) Argyros, I.K.; Khattri, S.K.; George, S.We expand the applicability of the Halley's method for approximating a locally unique solution of nonlinear equations in a Banach space setting. Our majorizing sequences are finer than in earlier studies such as [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 21, 23] and furthermore developed convergence criteria can be weaker. Finally numerical work is reported that compares favorably to the existing approaches in the literature [6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 24, 25, 26, 28]. © 2018 International Publications. All rights reserved.