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

  • Santhosh George National Institute of Technology Karnataka
  • Ioannis K. Argyros Cameron University
Keywords: Banach space, Three step metho, Local-semi-local convergence, Weak conditions

Abstract

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.

Downloads

Download data is not yet available.

References

Argyros, I. K., Convergence and applications of Newton-type methods, Springer Verlag, New York, 2008.
I. K. Argyros, A. A. Magre\~na\~n, Iterative Methods and their dynamics with applications: A Contemporary Study, CRC Press, 2017.
{Argyros, I.K.}, \'{A}. A. Magre\~{n}\'{a}n, A contemporary study of iterative methods convergence, dynamics and applications, N.Y., 2018.
Argyros, I. K., George, S., Thapa, N., Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications, Volume-I, Nova Publishes, NY, 2018.
Argyros, I. K., George, S., Thapa, N.,, Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications, Volume-II, Nova Publishes, NY, 2018.
Argyros, I. K., Ezquerro, J. A., Guti\'{e}rrez, J. M.,Hern\'{a}ndez, M. A., Hilout, S.: On the semilocal convergence of efficient Chebyshev-secant-type methods. J. Comput. Appl. Math. 235, 3195-3206 (2011)
Argyros, I. K., Hilout, S., Tabatabai, M. A.: Mathematical Modelling with Applications in Biosciences and Engineering. Nova Publishers, New York, (2011)
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., Hilout, S.: Numerical methods in Nonlinear Analysis. World Scientific Publ. Comp. New Jersey (2013)
Argyros, I. K., Khattri, S. K.: Local convergence for a family of third order methods in Banach spaces. Punjab Univ. J. Math. (Lahore). 46, 53-62 (2014)
Argyros, I. K., George, S.,Magre\~{n}\'{a}n, \'{A}. A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of high convergence order. J. Comput. Appl. Math. 282, 215-224 (2015)
Argyros, I. K., Magre\~{n}\'{a}n, \'{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 (2016)
Amat, S., Hern\'{a}ndez, M. A., Romero, N.: A modified Chebyshev's iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164-174 (2008)
Cordero, A., Ezquerro, J. A., Hern\'{a}ndez-Ver\'{o}n, 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)
Hern\'{a}ndez, M. A.: Chebyshev's approximation algorithms and applications. Comput. Math. Appl. 41, 433-445 (2001)
Hueso, J. L., Mart\'{i}nez, E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms. 67, 365-384 (2014)
Kumar, A., Gupta, D. K., Mart\'{i}nez, E., Singh, S.: Semilocal convergence of a secant-type method under weak Lipschitz conditions in Banach spaces. J. Comput. Appl. Math. 330, 732-741 (2018)
Mart\'{i}nez, 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)
Parida, P. K., Gupta, D. K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873-887 (2007)
Parida, P. K., Gupta, D. K.: Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces. J. Math. Anal. Appl. 345, 350-361 (2008)
Singh, S., Gupta, D. K., Mart\'{i}nez, E., Hueso, J. L.: Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces. Mediterr. J. Math. 13, 4219-4235 (2016)
Mart\'{i}nez, E.,Gupta, D. K., Domain of existence and uniqueness for nonlinear Hammerstein integral equations, (communicated).
Traub, J. F.: Iterative Methods for the Solutions of Equations. Prentice-Hall, Englewood Cliffs, New Jersey (1964)
Published
2019-04-18
How to Cite
George, S., & Argyros, I. K. (2019). Extending the Applicability of an Efficient Fifth Order Method Under Weak Conditions in Banach Space. MathLAB Journal, 2(1), 63-72. Retrieved from https://purkh.com/index.php/mathlab/article/view/289
Section
Research Articles