Abstract
In the present article, we establish a semilocal convergence theorem for the S-iteration process of Newton–Kantorovich like in Banach space setting for solving nonlinear operator equations and discuss its semilocal convergence analysis. We apply our result to solve the Fredholm-integral equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Smooke, M.D.: Error estimate for the modified Newton’s method with applications to the solution of nonlinear, two-point boundary-value problems. J. Optim. Theory Appl. 39(4), 489–511 (1983)
Angasarian, O.L.M.: A Newton’s method for linear programming. J. Optim. Theory Appl. 121(1), 1–18 (2004)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis in Normed spaces. Pregamon Press, Oxford (1964)
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Series: Topological Fixed Point Theory and its Applications, vol. 6. Springer, New York (2009)
RallL, B.: Computational Solution of Nonlinear Operator Equations. Wiley, New York (1969)
Zeidler, E.: Applied Functional Analysis Main Principles and Their application. Springer, New York (1995)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Cajori, F.: Historical note on the Newton–Raphson method of approximation. Am. Math. Monthly 18(2), 29–32 (1911)
Kantorovich, L.V.: On Newton’s method for functional equations. Dokl. Akad. Nauk. SSSR 59, 1237–1240 (1948). (in Russian)
Kantorovich, L.V.: The majorant principle and Newton’s method. Dokl. Akad. Nauk SSSR 76, 17–20 (1951). (in Russian)
Chen, M., Khan, Y., Yildirim, A.: Newton-Kantorovich convergence theorem of a modified Newton’s method under the \(\gamma \) -condition in a Banach space. J. Optim. Theory appl. 157, 651–662 (2013)
Proinov, P.D.: New general convergence theory for iterative process and its applications to Newton- Kantorovich type theorems. J. Complex. 26, 3–42 (2010)
Argyros, I.K.: On Newton’s method under mild differentiability conditions and applications. Appl. Math. Comput. 102, 177–183 (1999)
Argyros, I.K., Hilout, S.: Improved generaliged differentiability conditions for Newton-like methods. J. Complex. 26, 316–333 (2010)
Argyros, I.K., Hilout, S.: Majorizing sequences for iterative methods. J. Comput. Appl. Math. 236, 1947–1960 (2012)
Argyros, I.K.: An improved error analysis for Newton-like methods under generalized conditions. J. Comput. Appl. Math. 157, 169–185 (2003)
Argyros, I.K., Hilout, S.: On the convergence of Newton-type methods under mild differentiability conditions. Number Algorithms 52, 701–726 (2009)
Sahu, D.R., Singh, K.K., Singh Vipin, K.: Some Newton-like methods with sharper error estimates for solving operator equations in Banach spaces. Fixed Point Theory Appl. 78, 1–20 (2012)
Sahu, D.R., Singh, K.K., Singh Vipin, K.: A Newton-like method for generalized operator equations in Banach spaces. Number Algorithms 67, 289–303 (2014)
Ezquerro, J.A., González, D., Hernández, M.A.: Majorizing sequences for Newton’s method from initial value problems. J. Comput. Appl. Math. 236(9), 2246–2258 (2012)
Ezquerro, J.A., González, D., Hernández, M.A.: A modification of the classic conditions of Newton–Kantorovich for Newton’s method. Math. Comput. Modelling 57, 584–594 (2013)
Ezquerro, J.A., González, D., Hernández, M.A.: A semilocal convergence result for Newton’s method under generalized conditions of Kantorovich. J. Complex. 30, 309–324 (2014)
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61–79 (2007)
Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–610 (1953)
Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)
Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12, 187–204 (2011)
Sahu, D.R.: Strong convergence of a fixed point iteration process with applications. International Conference on Recent Advances in Mathematical Sciences and Applications, pp. 100–116 (2009)
Sahu, D.R., Singh, K.K.: On generalized Newton’s method for solving operator equations. Filomat 26(5), 1055–1063 (2012)
Wang, X.H.: Convergence of Newton’s method and inverse function theorem in Banach space. Math. Comput. 68, 169–186 (1999)
Wang, X.H.: Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal. 20, 123–134 (2000)
Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C. (eds.) The Merging of Disciplines: New Directions, pp. 185–196. Springer, New York (1986). (Pure Appl. Comput. Math.)
Shen, W., Li, C.: Smale’s \(\alpha \)-theory for inexact Newton methods under the \(\gamma \)-condition. J. Math. Anal. Appl. 369, 29–42 (2010)
Argyros, I.K., Khattri, S.K., Hilout, S.: Expanding the applicability of Inexact Newton methods under Smale’s \((\alpha,\gamma )\) -theory. Appl. Math. Comput. 224, 224–237 (2013)
Zhao, Y., Wu, Q.: Convergence analysis for a deformed Newton’s method with third-order in Banach space under \(\gamma \)-condition. Int. J. Comput. Math. 86, 441–450 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sahu, D.R., Yao, J.C., Singh, V.K. et al. Semilocal Convergence Analysis of S-iteration Process of Newton–Kantorovich Like in Banach Spaces. J Optim Theory Appl 172, 102–127 (2017). https://doi.org/10.1007/s10957-016-1031-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-1031-x