Open Access Subscription Access
Open Access Subscription Access
Semilocal Convergence of a Seventh-Order Method in Banach Spaces Under HÖlder Continuity Condition
The motive of this article is to analyze the semilocal convergence of a well existing iterative method in the Banach spaces to get the solution of nonlinear equations. The condition, we assume that the nonlinear operator fulfills the Hölder continuity condition which is softer than the Lipschitz continuity and works on the problems in which either second order Frèchet derivative of the nonlinear operator is challenging to calculate or does not hold the Lipschitz condition. In the convergence theorem, the existence of the solution x* and its uniqueness along with prior error bound are established. Also, the R-order of convergence for this method is proved to be at least 4+3q. Two numerical examples are discussed to justify the included theoretical development followed by an error bound expression.
Banach Space, Nonlinear Operator, Semilocal Convergence, Hölder Condition, Frèchet Derivative, Recurrence Relation, Error Bound.
- S. Amat and S. Busquier, Third-order iterative methods under Kantorovich conditions, J. Math. Anal. Appl. 336, 243-261 (2007).
- L. Chen, C. Gu and Y. Ma, Semilocal convergence for a fifth-order Newton’s method using recurrence relations in Banach spaces, J. Appl. Math., Volume 2011, Article ID 786306, 15 pages (2011).
- A. Cordero, M. A. Hern´andez, N. Romero and J. R. Torregrosa, Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces, J. Comput. Appl. Math. 273, 205-213 (2015).
- M. A. Hern´andez and M. A. Salanova, Sufficient conditions for semilocal convergence of a fourthorder multipoint iterative method for solving equations in Banach spaces, Southwest J. Pure Appl. Math. 1, 29-40 (1999).
- J. P. Jaiswal, Semilocal convergence and its computational efficiency of a seventh-order method in Banach spaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., DOI: 10.1007/s40010-018-0590-7 (2019).
- L. V. Kantorovich and G. P. Akilov, Functional Analysis. Pergamon Press, Oxford (1982).
- P. K. Parida and D. K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. Appl. Math. 206, 873-887 (2007).
- M. Prashanth and D. K. Gupta, Recurrence relations for super-Halley’s method with H¨ older continuous second derivative in Banach spaces, Kodai Math. J. 36(1), 119-136 (2013).
- L. B. Rall, Computational solution of nonlinear operator equations. Robert E Krieger, New York (1979).
- X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algor. 57, 441-456 (2011).
- X.Wang and J. Kou, Analysis of convergence for improved Chebyshev-Halley methods under different conditions, J. Optimiz. Theory App., 162(3), 920-930 (2013).
- X. Xiao and H. Yin, A new class of methods with higher order of convergence for solving systems of nonlinear equations, Appl. Math. Comput. 26, 300-309 (2015).
- L. Zheng and C. Gu, Semilocal convergence of a sixth-order method in Banach spaces, Numer. Algor. 61, 413-427 (2012).
- L. Zheng and C. Gu, Fourth-order convergence theorem by using majorizing functions for superHalley method in Banach spaces, Int. J. Comp. Math. 90, 423-434 (2013).
Abstract Views: 3
PDF Views: 1