Faculty Publications
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Item Weaker convergence conditions of an iterative method for nonlinear ill-posed equations(Nova Science Publishers, Inc., 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 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 Local convergence for some high convergence order Newton-like methods with frozen derivatives(Springer Nature, 2015) Argyros, I.K.; George, S.We present a local convergence analysis of some families of Newton-like methods with frozen derivatives in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Amat et al. (Appl Math Lett. 25:2209–2217, 2012), Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Xiao and Yin (Appl Math Comput, 2015) the local convergence was proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these methods. In this paper we expand the applicability of these methods using only hypotheses on the first derivative and Lipschitz constants. Numerical examples are also presented in this study. © 2015, Sociedad Española de Matemática Aplicada.Item Ball convergence of some fourth and sixth-order iterative methods(World Scientific Publishing Co. Pte Ltd wspc@wspc.com.sg, 2016) Argyros, I.K.; George, S.We present a local convergence analysis for some families of fourth and sixth-order methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Earlier studies [V. Candela and A. Marquina, Recurrence relations for rational cubic methods II: The Chebyshev method, Computing 45 (1990) 355-367; C. Chun, P. Stanica and B. Neta, Third order family of methods in Banach spaces, Comput. Math. Appl. 61 (2011) 1665-1675; J. M. Gutiérrez and M. A. Hernández, Recurrence relations for the super-Halley method, Comput. Math. Appl. 36 (1998) 1-8; M. A. Hernández and M. A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math. 126 (2000) 131-143; M. A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433-455; M. A. Hernández, Second-derivative-free variant of the Chebyshev method for nonlinear equations, J. Optim. Theory Appl. 104(3) (2000) 501-515; J. L. Hueso, E. Martinez and C. Teruel, Convergence, efficiency and dynamics of new fourth and sixth-order families of iterative methods for nonlinear systems, J. Comput. Appl. Math. 275 (2015) 412-420; Á. A. Magre nán, Estudio de la dinámica del método de Newton amortiguado, Ph.D. Thesis, Servicio de Publicaciones, Universidad de La Rioja (2013), http://dialnet.unirioja.es/servlet/tesis?codigo=38821; J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970); M. S. Petkovic, B. Neta, L. Petkovic and J. Džunic, Multi-Point Methods for Solving Nonlinear Equations (Elsevier, 2013); J. F. Traub, Iterative Methods for the Solution of Equations, Automatic Computation (Prentice-Hall, Englewood Cliffs, NJ, 1964); X. Wang and J. Kou, Semilocal convergence and R-order for modified Chebyshev-Halley methods, Numer. Algorithms 64(1) (2013) 105-126] have used hypotheses on the fourth Fréchet derivative of the operator involved. We use hypotheses only on the first Fréchet derivative in our local convergence analysis. This way, the applicability of these methods is extended. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples illustrating the theoretical results are also presented in this study. © 2016 World Scientific Publishing Company.Item Ball Convergence for two-parameter chebyshev-halley-like methods in banach space using hypotheses only on the first derivative(International Publications internationalpubls@yahoo.com, 2017) Argyros, I.K.; George, S.; Verma, R.U.We present a local convergence analysis of a sixth-order method for approximate a locally unique solution of an equation in the Banach space setting. The convergence of this methods is shown in Narang et al. (2016) under hypotheses up to the fourth Fréchet-derivative and the Lipschitz continuity of the third derivative, although only the first derivative appears in the method. In this study we expand the applicability of this method using only hypotheses on the first derivative of the function. Numerical examples are also presented in this study.Item Ball Convergence for an Inverse Free Jarratt-Type Method Under Hölder Conditions(Springer, 2017) Argyros, I.K.; George, S.We present a local convergence analysis of an inverse free Jarratt-type method in order to approximate a locally unique solution of an equation in a Banach space setting. Earlier studies have used hypotheses up to the third Fréchet-derivative of the operator involved to show convergence although only the first derivative is used in the method. We show convergence using only the first Fréchet derivative under Hölder conditions. This way we expand the applicability of the method. Numerical examples are also provided in this study. © 2015, Springer India Pvt. Ltd.Item Ball Convergence for Second Derivative Free Methods in Banach Space(Springer, 2017) Argyros, I.K.; Padikkal, P.; George, S.Hueso et al. (Appl Math Comput 211:190–197, 2009) considered a third and fourth order iterative methods for nonlinear systems. The methods were shown to of order third and fourth if the operator equation is defined on the j-dimensional Euclidean space (Hueso et al. in Appl Math Comput 211:190–197, 2009). The order of convergence was shown using hypotheses up to the third Fréchet derivative of the operator involved although only the first derivative appears in these methods. In the present study we only use hypotheses on the first Fréchet-derivative. This way the applicability of these methods is expanded. Moreover we present a radius of convergence a uniqueness result and computable error bounds based on Lipschitz constants. Numerical examples are also presented in this study. © 2015, Springer India Pvt. Ltd.Item Expanding the applicability of generalized high convergence order methods for solving equations(Tusi Mathematical Research Group (TMRG) moslehian@memeber.ams.org, 2018) Argyros, I.K.; George, S.The local convergence analysis of iterative methods is important since it indicates the degree of difficulty for choosing initial points. In the present study we introduce generalized three step high order methods for solving nonlinear equations. The local convergence analysis is given using hypotheses only on the first derivative, which actually appears in the methods in contrast to earlier works using hypotheses on higher derivatives. This way we extend the applicability of these methods. The analysis includes computable radius of convergence as well as error bounds based on Lipschitz-type conditions, which is not given in earlier studies. Numerical examples conclude this study. © 2018, Khayyam Journal of Mathematice.Item Ball comparison for three optimal eight order methods under weak conditions(Babes-Bolyai University oeconomica@econ.ubbcluj.ro, 2019) Argyros, I.K.; George, S.We considered three optimal eighth order method for solving nonlinear equations. In earlier studies Taylors expansions and hypotheses reaching up to the eighth derivative are used to prove the convergence of these methods. These hypotheses restrict the applicability of the methods. In our study we use hypotheses on the first derivative. Numerical examples illustrating the theoretical results are also presented in this study. © 2019, Babes-Bolyai University.Item Local comparison of two sixth order solvers in banach space under weak conditions(Erdal Karapinar, 2019) Argyros, I.K.; George, S.Two efficient sixth order solvers are compared involving Banach space valued operators. Earlier papers use hypotheses up to the seventh derivative that do not appear in the solver in the local convergence analysis. But we use hypotheses only on the first derivative. Hence, we expand the applicability of these solvers. We use examples to test the older as well as our results. © 2019, Erdal Karapinar. All rights reserved.
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