Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators

[EN] In this paper, the convergence and dynamics of improved Chebyshev-Secant-type iterative methods are studied for solving nonlinear equations in Banach space settings. Their semilocal convergence is established using recurrence relations under weaker continuity conditions on first-order divided differences. Convergence theorems are established for the existence-uniqueness of the solutions. Next, center-Lipschitz condition is defined on the first-order divided differences and its influence on the domain of starting iterates is compared with those corresponding to the domain of Lipschitz conditions. Several numerical examples including Automotive Steering problems and nonlinear mixed Hammerstein-type integral equations are analyzed, and the output results are compared with those obtained by some of similar existing iterative methods. It is found that improved results are obtained for all the numerical examples. Further, the dynamical analysis of the iterative method is carried out. It confirms that the proposed iterative method has better stability properties than its competitors.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22.Kumar, A.; Gupta, DK.; Martínez Molada, E.; Hueso, JL. (2021). Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators. Numerical Algorithms. 86(3):1051-1070. https://doi.org/10.1007/s11075-020-00922-9S10511070863Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3-4), 433–445 (2001)Ezquerro, J.A., Grau-Sánchez, Miquel, Hernández, M.A.: Solving non-differentiable equations by a new one-point iterative method with memory. J. Complex. 28(1), 48–58 (2012)Ioannis , K.A., Ezquerro, J.A., Gutiérrez, J.M., hernández, M.A., saïd Hilout: On the semilocal convergence of efficient Chebyshev-Secant-type methods. J. Comput. Appl. Math. 235(10), 3195–3206 (2011)Hongmin, R., Ioannis, K.A.: Local convergence of efficient Secant-type methods for solving nonlinear equations. Appl. Math. comput. 218(14), 7655–7664 (2012)Ioannis, Ioannis K.A., Hongmin, R.: On the semilocal convergence of derivative free methods for solving nonlinear equations. J. Numer. Anal. Approx. Theory 41 (1), 3–17 (2012)Hongmin, R., Ioannis, K.A.: On the convergence of King-Werner-type methods of order $1+\sqrt {2}$ free of derivatives. Appl. Math. Comput. 256, 148–159 (2015)Kumar, A., Gupta, D.K., Martínez, E., Sukhjit, S.: Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces. J. Comput. Appl. Math. 330, 732–741 (2018)Grau-Sánchez, M., Noguera, M., Gutiérrez, J.M.: Frozen iterative methods using divided differences “à la Schmidt–Schwetlick”. J. Optim. Theory Appl. 160 (3), 931–948 (2014)Louis, B.R.: Computational Solution of Nonlinear Operator Equations. Wiley, New York (1969)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Parisa, B., Cordero, A., Taher, L., Kathayoun, M., Torregrosa, J.R.: Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics 87 (2), 913–938 (2017)Chun, C., Neta, B.: The basins of attraction of Murakami’s fifth order family of methods. Appl. Numer. Math. 110, 14–25 (2016)Magreñán, Á. A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)Ramandeep, B., Cordero, A., Motsa, S.S., Torregrosa, J.R.: Stable high-order iterative methods for solving nonlinear models. Appl. Math. Comput. 303, 70–88 (2017)Pramanik, S.: Kinematic synthesis of a six-member mechanism for automotive steering. Trans Ame Soc. Mech. Eng. J. Mech. Des. 124(4), 642–645 (2002

Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations

[EN] In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Frechet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Holder condition; however, these operators verify the omega condition established. Then, the semilocal convergence balls are obtained and the R-order of convergence and error bounds can be obtained by following thee main theorem. Finally, we perform a numerical experience by solving a nonlinear Hammerstein integral equations in order to show the applicability of the theoretical results by obtaining the existence and uniqueness balls.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22.Singh, S.; Martínez Molada, E.; Kumar, A.; Gupta, DK. (2020). Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations. Mathematics. 8(3):1-11. https://doi.org/10.3390/math8030384S11183Hernández, M. A. (2001). Chebyshev’s approximation algorithms and applications. Computers & Mathematics with Applications, 41(3-4), 433-445. doi:10.1016/s0898-1221(00)00286-8Amat, S., Hernández, M. A., & Romero, N. (2008). A modified Chebyshev’s iterative method with at least sixth order of convergence. Applied Mathematics and Computation, 206(1), 164-174. doi:10.1016/j.amc.2008.08.050Argyros, I. K., Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., & Hilout, S. (2011). On the semilocal convergence of efficient Chebyshev–Secant-type methods. Journal of Computational and Applied Mathematics, 235(10), 3195-3206. doi:10.1016/j.cam.2011.01.005Hueso, J. L., & Martínez, E. (2013). Semilocal convergence of a family of iterative methods in Banach spaces. Numerical Algorithms, 67(2), 365-384. doi:10.1007/s11075-013-9795-7Zhao, Y., & Wu, Q. (2008). Newton–Kantorovich theorem for a family of modified Halley’s method under Hölder continuity conditions in Banach space. Applied Mathematics and Computation, 202(1), 243-251. doi:10.1016/j.amc.2008.02.004Parida, P. K., & Gupta, D. K. (2007). Recurrence relations for a Newton-like method in Banach spaces. Journal of Computational and Applied Mathematics, 206(2), 873-887. doi:10.1016/j.cam.2006.08.027Parida, P. K., & Gupta, D. K. (2008). Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces. Journal of Mathematical Analysis and Applications, 345(1), 350-361. doi:10.1016/j.jmaa.2008.03.064Cordero, A., Ezquerro, J. A., Hernández-Verón, M. A., & Torregrosa, J. R. (2015). On the local convergence of a fifth-order iterative method in Banach spaces. Applied Mathematics and Computation, 251, 396-403. doi:10.1016/j.amc.2014.11.084Argyros, I. K., & Hilout, S. (2013). On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 245, 1-9. doi:10.1016/j.cam.2012.12.002Argyros, I. K., George, S., & Magreñán, Á. A. (2015). Local convergence for multi-point-parametric Chebyshev–Halley-type methods of high convergence order. Journal of Computational and Applied Mathematics, 282, 215-224. doi:10.1016/j.cam.2014.12.023Wang, X., Kou, J., & Gu, C. (2012). Semilocal Convergence of a Class of Modified Super-Halley Methods in Banach Spaces. Journal of Optimization Theory and Applications, 153(3), 779-793. doi:10.1007/s10957-012-9985-9Argyros, I. K., & Magreñán, Á. A. (2015). A study on the local convergence and the dynamics of Chebyshev–Halley–type methods free from second derivative. Numerical Algorithms, 71(1), 1-23. doi:10.1007/s11075-015-9981-xWu, Q., & Zhao, Y. (2007). Newton–Kantorovich type convergence theorem for a family of new deformed Chebyshev method. Applied Mathematics and Computation, 192(2), 405-412. doi:10.1016/j.amc.2007.03.018Martínez, E., Singh, S., Hueso, J. L., & Gupta, D. K. (2016). Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces. Applied Mathematics and Computation, 281, 252-265. doi:10.1016/j.amc.2016.01.036Kumar, A., Gupta, D. K., Martínez, E., & Singh, S. (2018). Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces. Journal of Computational and Applied Mathematics, 330, 732-741. doi:10.1016/j.cam.2017.02.042Singh, S., Gupta, D. K., Martínez, E., & Hueso, J. L. (2016). Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces. Mediterranean Journal of Mathematics, 13(6), 4219-4235. doi:10.1007/s00009-016-0741-

Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces

[EN] Semilocal convergence for an iteration of order five for solving nonlinear equations in Banach spaces is established under second-order Fr,chet derivative satisfying the Lipschitz condition. It is done by deriving a number of recurrence relations. A theorem for the existence-uniqueness along with the estimation of error bounds of the solution is established. Its R-order is shown to be equal to five. Both efficiency and computational efficiency indices are given. A variety of examples are worked out to show its applicability. In comparison to existing methods having similar R-orders, improved results in terms of computational efficiency index and error bounds are found using our methodology.The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.Singh, S.; Gupta, D.; Martínez Molada, E.; Hueso Pagoaga, JL. (2016). Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces. Mediterranean Journal of Mathematics. 13(6):4219-4235. doi:10.1007/s00009-016-0741-5S42194235136Cordero A., Hueso J.L., Martinez E., Torregrosa J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)Chen, L., Gu, C., Ma Y.: Semilocal convergence for a fifth order Newton’s method using Recurrence relations in Banach spaces. J. Appl. Math. 2011, 1–15 (2011)Wang X., Kou J., Gu C.: Semilocal convergence of a sixth order Jarrat method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)Zheng L., Gu C.: Semilocal convergence of a sixth order method in Banach spaces. Numer. Algorithms 61, 413–427 (2012)Zheng L., Gu C.: Recurrence relations for semilocal convergence of a fifth order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)Proinov P.D., Ivanov S.I.: On the convergence of Halley’s method for multiple polynomial zeros. Mediterr. J. Math. 12, 555–572 (2015)Ezquerro, J.A., Hernández-Verón M.A.: On the domain of starting points of Newton’s method under center lipschitz conditions. Mediterr. J. Math. (2015). doi: 10.1007/s00009-015-0596-1Cordero A., Hernández-Verón M.A., Romero N., Torregrosa J.R.: Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces. J. Comput. Appl. Math. 273, 205–213 (2015)Parida P.K., Gupta D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)Hueso J.L., Martínez E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)Argyros, I.K., Hilout S.: Numerical methods in nonlinear analysis. World Scientific Publ. Comp., New Jersey (2013)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical modelling with applications in biosciences and engineering. Nova Publishers, New York (2011)Argyros I.K., Khattri S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2004)Argyros I.K., Hilout A.S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Kantorovich, L.V., Akilov G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros I.K., George S., Magreñán A.A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of higher convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)Argyros I.K., Magreñán A.A.: A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2015)Amat S., Hernández M.A., Romero N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Chun, C., St $${\breve{a}}$$ a ˘ nic $${\breve{a}}$$ a ˘ , P., Neta, B.: Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)Ostrowski, A.M.: Solution of equations in Euclidean and Banach spaces, 3rd edn. Academic Press, New-York (1977)Jaiswal J.P.: Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency. Numer. Algorithms 71, 933–951 (2015)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964

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

Extending the Applicability of an Efficient Fifth Order Method Under Weak Conditions in Banach Space

We extend the applicability of an efficient fifth order method for solving Banach space valued equations. To achieve this we use weaker Lipschitz-type conditions in combination with our idea of the restricted convergence region. Numerical examples are used to compare our results favorably to the ones in earlier works

Semilocal convergence of a family of iterative methods in Banach spaces

[EN] In this work, we prove a third and fourth convergence order result for a family of iterative methods for solving nonlinear systems in Banach spaces. We analyze the semilocal convergence by using recurrence relations, giving the existence and uniqueness theorem that establishes the R-order of the method and the priori error bounds. Finally, we apply the methods to two examples in order to illustrate the presented theory.This work has been supported by Ministerio de Ciencia e Innovaci´on MTM2011-28636-C02-02 and by Vicerrectorado de Investigaci´on. Universitat Polit`ecnica de Val`encia PAID-SP-2012-0498Hueso Pagoaga, JL.; Martínez Molada, E. (2014). Semilocal convergence of a family of iterative methods in Banach spaces. Numerical Algorithms. 67(2):365-384. https://doi.org/10.1007/s11075-013-9795-7S365384672Traub, J.F.: Iterative Methods for the Solution of Nonlinear Equations. Prentice Hall, New York (1964)Kantorovich, L.V.: On the newton method for functional equations. Doklady Akademii Nauk SSSR 59, 1237–1240 (1948)Candela, V., Marquina, A.: Recurrence relations for rational cubic methods, I: The Halley method. Computing 44, 169–184 (1990)Candela, V., Marquina, A.: Recurrence relations for rational cubic methods, II: The Chebyshev method. Computing 45, 355–367 (1990)Hernández, M.A.: Reduced recurrence relations for the Chebyshev method. J. Optim. Theory Appl. 98, 385–397 (1998)Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for super-Halley method. J. Comput. Math. Appl. 7, 1–8 (1998)Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-like methods. Appl. Math. Optim. 41, 227–236 (2000)Ezquerro, J.A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49, 325–342 (2009)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. 235–10, 3195–3206 (2011)Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newtons method. J. Complex. 28(3), 364–387 (2012)Wang, X., Gu, C., Kou, J.: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algoritm. 54, 497–516 (2011)Kou, J., Li, Y., Wang, X.: A variant of super Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)Zheng, L., Gu, C.: Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces. Numer. Algoritm. 59, 623–638 (2012)Amat, S., Hernández, M.A., Romero, N.: A modified Chebyshevs iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Wang, X., Kou, J., Gu, C.: Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algoritm. 57, 441–456 (2011)Hernández, M.A.: The newton method for operators with hlder continuous first derivative. J. Optim. Appl. 109, 631–648 (2001)Ye, X., Li, C.: Convergence of the family of the deformed Euler-Halley iterations under the Hlder condition of the second derivative. J. Comput. Appl. Math. 194, 294–308 (2006)Zhao, Y., Wu, Q.: Newton-Kantorovich theorem for a family of modified Halleys method under Hlder continuity conditions in Banach spaces. Appl. Math. Comput. 202, 243–251 (2008)Argyros, I.K.: Improved generalized differentiability conditions for Newton-like methods. J. Complex. 26, 316–333 (2010)Hueso, J.L., Martínez. E., Torregrosa, J.R.: Third and fourth order iterative methods free from second derivative for nonlinear systems. Appl. Math. Comput. 211, 190–197 (2009)Taylor, A.Y., Lay, D.: Introduction to Functional Analysis, 2nd edn.New York, Wiley (1980)Jarrat, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)Cordero, A., Torregrosa, J.R.: Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

On the semilocal convergence of derivative free methods for solving nonlinear equations

We introduce a Derivative Free Method (DFM) for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for DFM using recurrence relations. Numerical examples validating our theoretical results are also provided in this study to show that DFM is faster than other derivative free methods [9] using similar information

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

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

Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces

In this work we introduce a new form of setting the general assumptions for the local convergence studies of iterative methods in Banach spaces that allows us to improve the convergence domains. Specifically a local convergence result for a family of higher order iterative methods for solving nonlinear equations in Banach spaces is established under the assumption that the Frechet derivative satisfies the Lipschitz continuity condition. For some values of the parameter, these iterative methods are of fifth order. The importance of our work is that it avoids the usual practice of boundedness conditions of higher order derivatives which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We have considered a number of numerical examples including a nonlinear Hammerstein equation and computed the radii of the convergence balls. It is found that the radius of convergence ball obtained by our approach is much larger when compared with some other existing methods. The global convergence properties of the family are explored by analyzing the dynamics of the corresponding operator on complex quadratic polynomials.Martínez Molada, E.; Singh, S.; Hueso Pagoaga, JL.; Gupta, D. (2016). Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces. Applied Mathematics and Computation. 281:252-265. doi:10.1016/j.amc.2016.01.036S25226528