Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces

[EN] Thesemilocal and local convergence analyses of a two-step iterative method for nonlinear nondifferentiable operators are described in Banach spaces. The recurrence relations are derived under weaker conditions on the operator. For semilocal convergence, the domain of the parameters is obtained to ensure guaranteed convergence under suitable initial approximations. The applicability of local convergence is extended as the differentiability condition on the involved operator is avoided. The region of accessibility and a way to enlarge the convergence domain are provided. Theorems are given for the existence-uniqueness balls enclosing the unique solution. Finally, some numerical examples including nonlinear Hammerstein type integral equations are worked out to validate the theoretical results.This research was supported in part by the project of Generalitat Valenciana Prometeo/2016/089 and MTM2014-52016-C2-2-P of the Spanish Ministry of Science and Innovation.Kumar, A.; Gupta, D.; Martínez Molada, E.; Singh, S. (2018). Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces. Complexity. 1-11. https://doi.org/10.1155/2018/7352780S11

A Unified Convergence Analysis for Some Two-Point Type Methods for Nonsmooth Operators

The aim of this paper is the approximation of nonlinear equations using iterative methods. We present a unified convergence analysis for some two-point type methods. This way we compare specializations of our method using not necessarily the same convergence criteria. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and nonsmooth operators.Research of the first and third authors supported in part by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by MTM2015-64382-P. Research of the fourth and fifth authors supported by Ministerio de Economía y Competitividad under grant MTM2014-52016-C2-1P. This research received no external funding

Increasing the applicability of Steffensen's method

We present an original alternative to the majorant principle of Kantorovich to study the semilocal convergence of Steffensen's method when it is applied to solve nonlinear systems which are differentiable. This alternative allows choosing starting points from which the convergence of Steffensen's method is guaranteed, but it is not from the majorant principle. Moreover, this study extends the applicability of Steffensen's method to the solution of nonlinear systems which are nondifferentiable and improves a previous result given by the authors. © 2014 Elsevier Inc. All rights reserved

Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems

[EN] In this paper, we analyze the semilocal convergence of k-steps Newton's method with frozen first derivative in Banach spaces. The method reaches order of convergence k + 1. By imposing only the assumption that the Fr,chet derivative satisfies the Lipschitz continuity, we define appropriate recurrence relations for obtaining the domains of convergence and uniqueness. We also define the accessibility regions for this iterative process in order to guarantee the semilocal convergence and perform a complete study of their efficiency. Our final aim is to apply these theoretical results to solve a special kind of conservative systems.Hernández-Verón, MA.; Martínez Molada, E.; Teruel-Ferragud, C. (2017). Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems. Numerical Algorithms. 76(2):309-331. https://doi.org/10.1007/s11075-016-0255-zS309331762Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Comput. 25, 2209–2217 (2012)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical Modelling with Applications in Biosciences and Engineering. Nova Publishers, New York (2011)Argyros, I.K., George, S.: A unified local convergence for Jarratt-type methods in Banach space under weak conditions. Thai. J. Math. 13, 165–176 (2015)Argyros, I.K., Hilout, S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Argyros, I.K., Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Hilout, S.: On the semilocal convergence of efficient Chebyshev–Secant-type methods. J. Comput. Appl. Math. 235, 3195–2206 (2011)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation. Math. Comput. Mod. 57, 1950–1956 (2013)Ezquerro, J.A., Grau-Sánchez, M., Hernández, M. A., Noguera, M.: Semilocal convergence of secant-like methods for differentiable and nondifferentiable operators equations. J. Math. Anal. Appl. 398(1), 100–112 (2013)Honorato, G., Plaza, S., Romero, N.: Dynamics of a higher-order family of iterative methods. J. Complexity 27(2), 221–229 (2011)Jerome, J.W., Varga, R.S.: Generalizations of Spline Functions and Applications to Nonlinear Boundary Value and Eigenvalue Problems, Theory and Applications of Spline Functions. Academic Press, New York (1969)Kantorovich, L.V., Akilov, G.P.: Functional analysis Pergamon Press. Oxford (1982)Keller, H.B.: Numerical Methods for Two-Point Boundary-Value Problems. Dover Publications, New York (1992)Na, T.Y.: Computational Methods in Engineering Boundary Value Problems. Academic Press, New York (1979)Ortega, J.M.: The Newton-Kantorovich theorem. Amer. Math. Monthly 75, 658–660 (1968)Ostrowski, A.M.: Solutions of Equations in Euclidean and Banach Spaces. Academic Press, New York (1973)Plaza, S., Romero, N.: Attracting cycles for the relaxed Newton’s method. J. Comput. Appl. Math. 235(10), 3238–3244 (2011)Porter, D., Stirling, D.: Integral Equations: A Practical Treatment, From Spectral Theory to Applications. Cambridge University Press, Cambridge (1990)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall. Englewood Cliffs, New Jersey (1964)Argyros, I.K., George, S.: Extending the applicability of Gauss-Newton method for convex composite optimization on Riemannian manifolds using restricted convergence domains. Journal of Nonlinear Functional Analysis 2016 (2016). Article ID 27Xiao, J.Z., Sun, J., Huang, X.: Approximating common fixed points of asymptotically quasi-nonexpansive mappings by a k+1-step iterative scheme with error terms. J. Comput. Appl. Math 233, 2062–2070 (2010)Qin, X., Dehaish, B.A.B., Cho, S.Y.: Viscosity splitting methods for variational inclusion and fixed point problems in Hilbert spaces. J. Nonlinear Sci. Appl. 9, 2789–2797 (2016

Semilocal Convergence of the Extension of Chun's Method

[EN] In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun's iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Frechet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE) and FONDOCYT 027-2018 Republica Dominicana.Cordero Barbero, A.; Maimó, JG.; Martínez Molada, E.; Torregrosa Sánchez, JR.; Vassileva, MP. (2021). Semilocal Convergence of the Extension of Chun's Method. Axioms. 10(3):1-11. https://doi.org/10.3390/axioms10030161S11110

Extended convergence analysis of Newton-Potra solver for equations

In the paper a local and a semi-local convergence of combined iterative process for solving nonlinear operator equations is investigated. This solver is built based on Newton solver and has R-convergence order 1.839.... The radius of the convergence ball and convergence order of the investigated solver are determined in an earlier paper. Modifications of previous conditions leads to extended convergence domain. These advantages are obtained under the same computational effort. Numerical experiments are carried out on the test examples with nondifferentiable operator

Nonexpansive mappings and monotone vector fields in Hadamard manifolds

This paper briefly surveys some recent advances in the investigation of nonexpansive mappings and monotone vector fields, focusing in the extension of basic results of the classical nonlinear functional analysis from Banach spaces to the class of nonpositive sectional curvature Riemannian manifolds called Hadamard manifolds. Within this setting, we first analyze the problem of finding fixed points of nonexpansive mappings. Later on, different classes of monotonicity for set-valued vector fields and the relationship between some of them will be presented, followed by the study of the existence and approximation of singularities for such vector fields. We will discuss about variational inequality and minimization problems in Hadamard manifolds, stressing the fact that these problems can be solved by means of the iterative approaches for monotone vector fields