34 research outputs found

    Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations

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    [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 of a family of iterative methods in Banach spaces

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    [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

    Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces

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    [EN] The convergence analysis both local under weaker Argyros-type conditions and semilocal under. omega-condition is established using first order Frechet derivative for an iteration of fifth order in Banach spaces. This avoids derivatives of higher orders which are either difficult to compute or do not exist at times. The Lipchitz and the Holder conditions are particular cases of the omega-condition. Examples can be constructed for which the Lipchitz and Holder conditions fail but the omega-condition holds. Recurrence relations are used for the semilocal convergence analysis. Existence and uniqueness theorems and the error bounds for the solution are provided. Different examples are solved and convergence balls for each of them are obtained. These examples include Hammerstein-type integrals to demonstrate the applicability of our approach.Singh, S.; Gupta, D.; Singh, R.; Singh, M.; Martínez Molada, E. (2018). Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces. International Journal of Computational Methods. 15(6):1-18. https://doi.org/10.1142/S0219876218500482S11815

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

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    [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 a˘{\breve{a}} a ˘ nic a˘{\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

    Semilocal convergence of a continuation method with Hölder continuous second derivative in Banach spaces

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    AbstractIn this paper, the semilocal convergence of a continuation method combining the Chebyshev method and the convex acceleration of Newton’s method used for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that the second Frëchet derivative satisfies the Hölder continuity condition. This condition is mild and works for problems in which the second Frëchet derivative fails to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α∈[0,1] for the solution x∗ is given. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences under Hölder continuity condition on F″, it is found that our analysis gives improved results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Chebyshev method (α=0) and the convex acceleration of Newton’s method (α=1) respectively with improved results

    Convergence Theorem for a Family of New Modified Halley’s Method in Banach Space

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    We establish convergence theorems of Newton-Kantorovich type for a family of new modified Halley’s method in Banach space to solve nonlinear operator equations. We present the corresponding error estimate. To show the application of our theorems, two numerical examples are given

    Semilocal and local convergence of a fifth order iteration with Frechet derivative satisfying Holder condition

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    The semilocal and local convergence in Banach spaces is described for a fifth order iteration for the solutions of nonlinear equations when the Frechet derivative satisfies the Holder condition. The Holder condition generalizes the Lipschtiz condition. The importance of our work lies in the fact that many examples are available which fail to satisfy the Lipschtiz condition but satisfy the Holder condition. The existence and uniqueness theorem is established with error bounds for the solution. The convergence analysis is finally worked out on different examples and convergence balls for each of them are obtained. These examples include nonlinear Hammerstein and Fredholm integral equations and a boundary value problem. It is found that the larger radius of convergence balls is obtained for all the examples in comparison to existing methods using stronger conditions. (C) 2015 Elsevier Inc. All rights reserved.Singh, S.; Gupta, D.; Martínez Molada, E.; Hueso Pagoaga, JL. (2016). Semilocal and local convergence of a fifth order iteration with Frechet derivative satisfying Holder condition. Applied Mathematics and Computation. 276:266-277. doi:10.1016/j.amc.2015.11.062S26627727

    Local convergence of a parameter based iteration with Holder continuous derivative in Banach spaces

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    [EN] The local convergence analysis of a parameter based iteration with Hölder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives. An existence-uniqueness theorem along with the derivation of error bounds for the solution is established. Different numerical examples including nonlinear Hammerstein equation are solved. The radii of balls of convergence for them are obtained. Substantial improvements of these radii are found in comparison to some other existing methods under similar conditions for all examples considered.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, DK.; Badoni, RP.; Martínez Molada, E.; Hueso Pagoaga, JL. (2017). Local convergence of a parameter based iteration with Holder continuous derivative in Banach spaces. CALCOLO. 54(2):527-539. doi:10.1007/s10092-016-0197-9S527539542Argyros, 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)Singh, S., Gupta, D.K., Martínez, E., Hueso, J.L.: Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition. Appl. Math. Comput. 276, 266–277 (2016)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)Rall, L.B.: Computational solution of nonlinear operator equations, reprint edn. R. E. Krieger, New York (2007)Cordero, A., Ezquerro, J.A., Hernández-Verón, M.A., Torregrosa, J.R.: On the local convergence of a fifth-order iterative method in Banach spaces. Appl. Math. Comput. 251, 396–403 (2015)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)Argyros, I.K., Behl, R., Motsa, S.S.: Local convergence of an efficient high convergence order method using hypothesis only on the first derivative. Algorithms 8, 1076–1087 (2015)Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros, I.K., Magreñán, A.A.: A study on the local convergence and dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2016)Li, D., Liu, P., Kou, J.: An improvement ofthe Chebyshev-Halley methods free from second derivative. Appl. Math. Comput. 235, 221–225 (2014)Argyros, I.K., George, S.: Local convergence of deformed Halley method in Banach space under Holder continuity conditions. J. Nonlinear Sci. Appl. 8, 246–254 (2015)Argyros, I.K., Khattri, S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2014)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., George, S.: Local convergence of modified Halley-like methods with less computation of inversion. Novi. Sad. J. Math. 45, 47–58 (2015)Xiao, X.Y., Yin, H.W.: Increasing the order of convergence for iterative methods to solve nonlinear systems. Calcolo (2015). doi: 10.1007/s10092-015-0149-9Martínez, E., Singh, S., Hueso, J.L., Gupta, D.K.: Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces. Appl. Math. Comput. 281, 252–265 (2016

    Third-Order Newton-Type Methods Combined with Vector Extrapolation for Solving Nonlinear Systems

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    We present a third-order method for solving the systems of nonlinear equations. This method is a Newton-type scheme with the vector extrapolation. We establish the local and semilocal convergence of this method. Numerical results show that the composite method is more robust and efficient than a number of Newton-type methods with the other vector extrapolations

    Semilocal Convergence for a Fifth-Order Newton's Method Using Recurrence Relations in Banach Spaces

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    We study a modified Newton's method with fifth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existence-uniqueness theorem is given to establish the R-order of the method to be five and a priori error bounds. Finally, a numerical application is presented to demonstrate our approach
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