Faculty Publications
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Item Ball convergence comparison between three iterative methods in Banach space under hypothese only on the first derivative(Elsevier Inc. usjcs@elsevier.com, 2015) Argyros, I.K.; George, S.Abstract We present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for these methods under hypotheses only on the first Fréchet derivative in contrast to earlier studies such as Adomian (1994) [1], Babajee et al. (2008) [13], Cordero and Torregrosa (2007) [17], Cordero et al. [18], Darvishi and Barati (2007) [19], using hypotheses reaching up to the fourth Fréchet derivative although only the first derivative appears in these methods. This way we expand the applicability of these methods. Numerical examples are also presented in this study. © 2015 Elsevier Inc.Item Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions(Springer-Verlag Italia s.r.l., 2016) Argyros, I.K.; George, S.We present a local convergence analysis of a sixth order iterative method for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Sharma et al. (Appl Math Comput 190:111–115, 2007) have shown convergence of these methods under hypotheses up to the fifth derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study. © 2015, Springer-Verlag Italia.Item Local convergence of a fast Steffensen-type method on Banach space under weak conditions(Inderscience Publishers, 2017) Argyros, I.K.; George, S.This paper is devoted to the study of the seventh-order Steffensen-type methods for solving nonlinear equations in Banach spaces. Using the idea of a restricted convergence domain, we extended the applicability of the seventh-order Steffensen-type methods. Our convergence conditions are weaker than the conditions used in the earlier studies. Numerical examples are also given in this study. © © 2017 Inderscience Enterprises Ltd.Item Local Convergence of Jarratt-Type Methods with Less Computation of Inversion Under Weak Conditions(Taylor and Francis Ltd., 2017) Argyros, I.K.; George, S.We present a local convergence analysis for Jarratt-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Earlier studies cannot be used to solve equations using such methods. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study. © 2017, © Vilnius Gediminas Technical University, 2017.Item Ball convergence of some iterative methods for nonlinear equations in Banach space under weak conditions(Springer-Verlag Italia s.r.l., 2018) Argyros, I.K.; George, S.The aim of this paper is to expand the applicability of a fast iterative method in a Banach space setting. Moreover, we provide computable radius of convergence, error bounds on the distances involved and a uniqueness of the solution result based on Lipschitz-type functions not given before. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. The convegence order is determined using the computational order of convergence or the approximate order of convergence. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study. © 2017, Springer-Verlag Italia S.r.l.Item Kantorovich-Like Convergence Theorems for Newton’s Method Using Restricted Convergence Domains(Taylor and Francis Inc. 325 Chestnut St, Suite 800 Philadelphia PA 19106, 2019) Argyros, I.K.; George, S.The convergence set for Newton’s method is small in general using Lipschitz-type conditions. A center-Lipschitz-type condition is used to determine a subset of the convergence set containing the Newton iterates. The rest of the Lipschitz parameters and functions are then defined based on this subset instead of the usual convergence set. This way the resulting parameters and functions are more accurate than in earlier works leading to weaker sufficient semi-local convergence criteria. The novelty of the paper lies in the observation that the new Lipschitz-type functions are special cases of the ones given in earlier works. Therefore, no additional computational effort is required to obtain the new results. The results are applied to solve Hammerstein nonlinear integral equations of Chandrasekhar type in cases not covered by earlier works. © 2018, © 2019 Taylor & Francis Group, LLC.Item On the complexity of choosing majorizing sequences for iterative procedures(Springer-Verlag Italia s.r.l., 2019) Argyros, I.K.; George, S.The aim of this paper is to introduce general majorizing sequences for iterative procedures which may contain a non-differentiable operator in order to solve nonlinear equations involving Banach valued operators. A general semi-local convergence analysis is presented based on majorizing sequences. The convergence criteria, if specialized can be used to study the convergence of numerous procedures such as Picard’s, Newton’s, Newton-type, Stirling’s, Secant, Secant-type, Steffensen’s, Aitken’s, Kurchatov’s and other procedures. The convergence criteria are flexible enough, so if specialized are weaker than the criteria given by the aforementioned procedures. Moreover, the convergence analysis is at least as tight. Furthermore, these advantages are obtained using Lipschitz constants that are least as tight as the ones already used in the literature. Consequently, no additional hypotheses are needed, since the new constants are special cases of the old constants. These ideas can be used to study, the local convergence, multi-step multi-point procedures along the same lines. Some applications are also provided in this study. © 2018, Springer-Verlag Italia S.r.l., part of Springer Nature.Item Extended Newton-type iteration for nonlinear ill-posed equations in Banach space(Springer Verlag service@springer.de, 2019) Sreedeep, C.D.; George, S.; Argyros, I.K.In this paper, we study nonlinear ill-posed equations involving m-accretive mappings in Banach spaces. We produce an extended Newton-type iterative scheme that converges cubically to the solution which uses assumptions only on the first Fréchet derivative of the operator. Using general Hölder type source condition we obtain an error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005) for choosing the regularization parameter. © 2018, Korean Society for Computational and Applied Mathematics.Item Improved semi-local convergence of the Newton-HSS method for solving large systems of equations(Elsevier Ltd, 2019) Argyros, I.K.; George, S.; Magreñán Ruiz, A.The aim of this article is to present the correct version of the main theorem 3.2 given in Guo and Duff (2011), concerning the semi-local convergence analysis of the Newton-HSS (NHSS)method for solving systems of nonlinear equations. Our analysis also includes the corrected upper bound on the initial point. © 2019Item Expanding the applicability of an iterative regularization method for ill-posed problems(Biemdas Academic Publishers, 2019) Argyros, I.K.; George, S.An iteratively regularized projection method, which converges quadratically, is considered for stable approximate solutions to a nonlinear ill-posed operator equation F(x) = y, where F : D(F) ? X ? X is a nonlinear monotone operator defined on the real Hilbert space X. We assume that only a noisy data y? with ky? y? k ? ? are available. Under the assumption that the Fréchet derivative F0 of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that, under a general source condition on x0 ? x, the error kxn h ? ? ? xk between the regularized approximation xn h ? ? , (x0 h ? ? := Phx0, where Ph is an orthogonal projection on to a finite dimensional subspace Xh of X) and the solution x is of optimal order. © 2019 Journal of Nonlinear and Variational Analysis