Faculty Publications
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Item Ball convergence theorem for a fifth-order method in banach spaces(Nova Science Publishers, Inc., 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 Improved qualitative analysis for newton-like methods with r-order of convergence at least three in banach spaces(Nova Science Publishers, Inc., 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 Developments on the convergence region of newton-like methods with generalized inverses in banach spaces(Nova Science Publishers, Inc., 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.Item Extended convergence of king-werner-like methods without derivatives(Nova Science Publishers, Inc., 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 On an eighth order steffensen-type solver free of derivatives(Nova Science Publishers, Inc., 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 Modified newton-type compositions for solving equations in banach spaces(Nova Science Publishers, Inc., 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 Modification of the kantorovich-type conditions for newton's method involving twice frechet differentiable operators(2013) Argyros, I.K.; George, S.We expand the applicability of Newton's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The nonlinear operator involved is twice Fréchet differentiable. We introduce more precise majorizing sequences than in earlier studied (see [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal. 11 (2004) 103-119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math. 79 (1997) 131-145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217]). This way, our convergence criteria can be weaker; the error estimates tighter and the information on the location of the solution more precise. Numerical examples are presented to show that our results apply in cases not covered before such as [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal. 11 (2004) 103-119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math. 79 (1997) 131-145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217]. © 2013 World Scientific Publishing Company.Item On extended convergence domains for the newton-kantorovich method(Publishing House of the Romanian Academy Calea 13 Septembrie nr. 13, Sector 5, 050711. P.O. Box 5-42, Bucuresti, 2014) Argyros, I.K.; George, S.We present results on extended convergence domains and their applications for the Newton-Kantorovich method (NKM), using the same information as in previous papers. Numerical examples are provided to emphasize that our results can be applied to solve nonlinear equations using (NKM), in contrast with earlier results which are not applicable in these cases. © 2014, Publishing House of the Romanian Academy. All rights reserved.Item On the "terra incognita" for the newton-kantrovich method with applications(2014) Argyros, I.K.; Cho, Y.J.; George, S.In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fréchet-derivative of the operator involved is p-Hölder continuous (p ?(0, 1]). Numerical examples involving two boundary value problems are also provided. © 2014 Korean Mathematical Society.Item Expanding the convergence domain for chun-stanica-neta family of third order methods in banach spaces(Korean Mathematical Society kms@kms.or.kr, 2015) Argyros, I.K.; George, S.; Magreñán Ruiz, Á.A.We present a semilocal convergence analysis of a third order method for approximating a locally unique solution of an equation in a Banach space setting. Recently, this method was studied by Chun, Stanica and Neta. These authors extended earlier results by Kou, Li and others. Our convergence analysis extends the applicability of these methods under less computational cost and weaker convergence criteria. Numerical examples are also presented to show that the earlier results cannot apply to solve these equations. © 2015 Korean Mathematical Society.