
Twopoint method for solving nonlinear equation with nondifferentiable
operator (in Ukrainian) 
Author 
S. M. Shakhno, H. P. Yarmola
Lviv Ivan Franko National University

Abstract 
In the paper we study a combined differentialdifference method for solving nonlinear equations
with nondifferentiable operator. The semilocal convergence of the method is investigated
and the order of convergence is established. 
Keywords 
nonlinear equation; differentialdifference method; semilocal convergence; nondifferentiable
operator 
Reference 
1. Kurchatov V.A. On one method of linear interpolation for solving functional equations // Dokl. AN
SSSR. Ser. Mathematics. Physics. 1971. V.198. Ή3. P. 524526. (in Russian)
2. Ortega J.M., Rheinboldt W.C. Iterative solution of nonlinear equations in several variables. Academic
Press, New York, 1970.
3. Traub J.F. Iterative methods for the solution of equations. Prentice Hall, Englewood Cliffs, 1964.
4. Shakhno S.M. On the difference method with quadratic convergence for solving nonlinear operator equations//
Mat. Stud. 2006. V.26. P. 105110. (in Ukrainian)
5. Amat S., Busquier S. A modified secant method for semismooth equations // Appl. Math. Lett. 2003.
V.16. P. 877881.
6. Argyros I.K. A unifying localsemilocal convergenceanalysis and applications for twopoint Newtonlike
methods in Banach space// J. Math. Anal. Appl. 2004. V.298. P. 374397.
7. Argyros I.K. Improving the rate of convergence of Newton methods on Banach spaces wth a convergence
structure and applications// Appl. Math. Lett. 1997. V.6. P. 2128.
8. Ρhen X. On the convergence of Broydenlike methods for nonlinear equations with nondifferentiable
terms// Ann. Inst. Statist. Math. 1990. V.42, Ή2. P. 387401.
9. Hernandez M.A., Rubio M.J. The Secant method for nondifferentiable operators// Appl. Math. Lett.
2002. V.15. P. 395399.
10. Ren H., Argyros I.K. A new semilocal convergence theorem for a fast iterative method with nondifferentiable
operators// J. Appl. Math. Comp. 2010. V.34. Ή12. P. 3946.
11. Shakhno S.M. On an iterative algorithm with superquadratic convergence for solving nonlinear operator
equations// J. Comp. App. Math. 2009. V.231. P. 222235.
12. Wang X. Convergence of Newtons method and uniquiness of the solution of equations in Banach space//
IMA Journal of Numerical Analysis. 2000. V.20. P. 123134.

Pages 
213220 
Volume 
36 
Issue 
2 
Year 
2011 
Journal 
Matematychni Studii 
Full text of paper 
PDF 
Table of content of issue 
HTML 