3. Book Chapters
Permanent URI for this collectionhttps://idr.nitk.ac.in/handle/1/8
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Item Ball convergence theorem for a fifth-order method in banach spaces(2019) Argyros I.K.; George S.We present a local convergence analysis for a fifth-order 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 use hypotheses up to the fourth Fréchet-derivative [1]. 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. © 2020 by Nova Science Publishers, Inc. All rights reserved.Item Weaker convergence conditions of an iterative method for nonlinear ill-posed equations(2019) Argyros I.K.; George S.In this chapter we expand the applicability of an iterative method which converges to the unique solution xα of the method of Lavrentiev regularization, i.e., F(x) + α(x - x0) = y, approximating the solution x of the ill-posed problem F(x) = y where F: D(F) - X - X is a nonlinear monotone operator defined on a real Hilbert space X. We use a center-Lipschitz instead of a Lipschitz condition used in [1-3]. The convergence analysis and the stopping rule are based on the majorizing sequence. The choice of the regularization parameter is the crucial issue. We show that the adaptive scheme considered by Perverzev and Schock [4] for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate. Numerical examples are presented to show that older convergence conditions [1-3] are not satisfied but the new ones are satisfied. © 2020 by Nova Science Publishers, Inc. All rights reserved.Item Local convergence of osada’s method for finding zeros with multiplicity(2019) Argyros I.K.; George S.We provide an extended local convergence of Osada’s method for approximating a zero of a nonlinear equation with multiplicitym, where m is a natural number. The new technique provides a tighter convergence analysis under the same computational cost as in earlier works. This technique can be used on other iterative methods too. Numerical examples further validate the theoretical results. © 2020 by Nova Science Publishers, Inc. All rights reserved.Item Improved qualitative analysis for newton-like methods with r-order of convergence at least three in banach spaces(2019) Argyros I.K.; George S.The aim of this study is to extend the applicability of a certain family of Newton- like methods with R-order of convergence at least three. By using our new idea of restricted convergence, we find a more precise location where the iterates lie leading to smaller constants and functions than in earlier studies which in turn lead to a tighter semi-local convergence for these methods. This idea can be used on other iterative methods as well as in the local convergence analysis of these methods. Numerical examples further show the advantages of the new results over the ones in earlier studies. © 2020 by Nova Science Publishers, Inc. All rights reserved.Item Modified newton-type compositions for solving equations in banach spaces(2019) Argyros I.K.; George S.We compare the radii of convergence as well as the error bounds of two efficient sixth convergence order methods for solving Banach space valued operators. The convergence criteria invlove conditions on the first derivative. Earlier convergence criteria require the existence of derivatives up to order six. Therefore, our results extended the usage of these methods. Numerical examples complement the theoretical results. © 2020 by Nova Science Publishers, Inc. All rights reserved.Item On an eighth order steffensen-type solver free of derivatives(2019) Argyros I.K.; George S.We expand the applicability of an eighth convergence order Steffensen-type solver for equations involvingBanach space valued operators using only the first order derivative in contrast to earlier works using derivatives of order five which do not appear in the method, and in the special case of the i-dimensional Euclidean space. © 2020 by Nova Science Publishers, Inc. All rights reserved.Item Extended convergence of king-werner-like methods without derivatives(2019) Argyros I.K.; George S.We provide a semilocal as well as a local convergence analysis of some efficient King-Werner-likemethods of order 1+2 free of derivatives for Banach space valued operators. We use our new idea of the restricted convergence region to find a smaller subset than before containing the iterates. Consequently the resulting Lipschitz parameters are smaller than in earlier works. Hence, to a finer convergence analysis is obtained. The extensions involve no new constants, since the new ones specialize to the ones in previous works. Examples are used to test the convergence criteria. © 2020 by Nova Science Publishers, Inc. All rights reserved.Item Developments on the convergence region of newton-like methods with generalized inverses in banach spaces(2019) Argyros I.K.; George S.The convergence region of Newton-like methods involving Banach space valued mappings and generalized inverses is extended. To achieve this task, a region is found inside the domain of the mapping containing the iterates. Then, the semi-local as well as local convergence analysis is finer, since the new Lipschitz parameters are at least as small and in earlier work using the same information. We compare convergence criteria using numerical examples. © 2020 by Nova Science Publishers, Inc. All rights reserved.