BROADENING THE CONVERGENCE DOMAIN OF SEVENTH-ORDER METHOD SATISFYING LIPSCHITZ AND HOLDER CONDITIONS

Abstract

The local convergence analysis of a seventh order algorithm for solving nonlinear equations is presented in
the current discussion by assuming that the ?rst-order Fréchet derivative belongs to the Lipschitz class. This
approach yields radii of convergence ball, error bound and uniqueness of the solution. Further, generalization
of the study extended by considering Hölder continuity condition. At last, we estimated the radii of the
convergence balls using a variety of numerical examples, including a nonlinear Hammerstein equation.

Keywords

• Nonlinear equation
• Banach space
• local convergence
• Lipschitz continuity condition
• Holder continuity condition.

References

1. [1] J.F. Traub, Iterative Methods for the solution of equations, Chelsea Publishing Company, New York (1977).
2. [2] L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford (1982).
3. [3] I.K. Argyros, S. George, Local convergence of two competing third order methods in Banach spaces, Appl. Math., 41 (2016) 341-350.
4. [4] I.K. Argyros, S.K. Khattri, Local convergence for a family of third order methods in Banach spaces, Punjab Univ. J. Math., 46 (2016) 52-63.
5. [5] I.K. Argyros, D. Gonzalez, S.K. Khattri, Local convergence of a one parameter fourth-order Jarratt-type method in Banach spaces, Comment. Math. Univ. Carolin, 57 (2016) 289-300.
6. [6] A. Cordero, J.A. Ezquerro, M.A. Hernández, J. Torregrosa, On the local convergence of a ?fth-order iterative method in Banach spaces, Appl. Math. Comput, 251 (2015) 396-403.
7. [7] E. Martinez, S. Singh, J.L. Hueso, D.K. Gupta, Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces. Applied Mathematics and Computation, 281 (2016) 252-265.
8. [8] R. Behl, S.S. Motsa, Geometric construction of eighth-order optimal families of Ostrowski's method. The Scienti?c World Journal, (2015).

9. [9] J.R. Sharma, H. Arora, A new family of optimal eighth order methods with dynamics for nonlinear equations, Appl. Math. Comput., 273 (2016) 924-933.
10. [10] L.B. Rall, Computational solution of nonlinear operator equations, Robert E Krieger, New York (1979).
11. [11] J.R. Sharma, P. Gupta, An eficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl., 67(3) (2014) 591-601.
12. [12] B. Panday, J.P. Jaiswal, On the local convergence of modified Homeier-like method in Banach spaces. Numerical Analysis and Applications, 11(4) (2018) 332-345.
13. [13] X. Xiao, H. Yin, A new class of methods with higher order of convergence for solving systems of nonlinear equations. Applied Mathematics and Computation, 264 (2015) 300-309.
14. [14] J.P. Jaiswal, Semilocal convergence and its computational eficiency of a seventh-order method in Banach spaces, Proceed- ings of the National Academy of Sciences, India Section A: Physical Sciences, 90(2) (2020) 271-279.
15. [15] N. Gupta, J.P. Jaiswal, Semilocal convergence of a seventh-order method in Banach spaces under Holder continuity con- dition, The Journal of the Indian Mathematical Society, 87 (2020) 56-69.
16. [16] N. Gupta, J.P. Jaiswal, Semilocal convergence of a seventh-order method in Banach spaces under W-continuity condition, Surveys in Mathematics and its Applications, 15 (2020) 325-339.
17. [17] M.A. Noor, M. Waseem, Some iterative methods for solving a system of nonlinear equations. Computers Mathematics with Applications, 57(1) (2009) 101-106.
18. [18] D. Sharma, S.K. Parhi, Extending the applicability of a third-order scheme with Lipschitz and Hölder continuous derivative in Banach spaces. Journal of the Egyptian Mathematical Society, 28(1) (2020) 1-13.
19. [19] I.K. Argyros, S. George, Local convergence of a fifth convergence order method in Banach space. Arab Journal of Mathe- matical Sciences, 23 (2017) 205-214.
20. [20] I.K. Argyros, S. Hilout, Computational methods in nonlinear analysis, World Scientific Publishing Company, New Jersey (2013).
21. [21] S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third- order Newton-type method, J. Math. Anal. Appl., 366 (2010) 24-32.
22. [22] I.K. Argyros, S. George, Local convergence for some high convergence order Newton-like methods with frozen derivatives. SeMA Journal, 70(1) (2015) 47-59.
23. [23] I.K. Argyros, S. George, Local convergence of deformed Halley method in Banach space under Holder continuity conditions. J. Nonlinear Sci. Appl, 8 (2015) 246-254.
24. [24] A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A family of iterative methods with sixth and seventh order conver- gence for nonlinear equations. Mathematical and Computer Modelling, 52(9-10) (2010) 1490-1496.
25. [25] T. Liu, X. Qin, P. Wang, Local convergence of a family of iterative methods with sixth and seventh order convergence under weak conditions. International Journal of Computational Methods, 16(08) (2019) 1850120.
26. [26] I.K. Argyros, M.J. Legaz, A.A. Magreñán, D. Moreno, J.A. Sicilia, Extended local convergence for some inexact methods with applications, Journal of Mathematical Chemistry, 57(5) (2019) 1508-1523.
27. [27] D. Jain, Families of Newton-like methods with fourth-order convergence. International Journal of Computer Mathematics, 90(5) (2013) 1072-1082.