|1Chuiko, SM |
1Donbass State Pedagogical University, Slovyansk
|Dopov. Nac. akad. nauk Ukr. 2018, 6:22-31|
We present a modification of the Newton—Kantorovich method for nonlinear operator equations in a Banach space. We prove, under certain conditions, that this modified Newton—Kantorovich method has quadratic convergence. The modified Newton—Kantorovich method is used to solve some nonlinear integral and integral- differential equations.
|Keywords: Banach space, modification of the Newton—Kantorovich method, nonlinear operator equations, quadratic convergence|
- Boichuk, A. A. & Samoilenko, A. M. (2004). Generalized inverse operators and Fredholm boundary-value problems. Berlin; Boston: Walter de Gruyter.
- Azbelev, N. V., Maksimov, V. P. & Rakhmatullina, L. F. (1991). An introduction to the theory of functional differential equations. Moscow: Nauka (in Russian).
- Kantorovich, L. V. & Akilov, G. P. (1977). Functional analysis. Moscow: Nauka (in Russian).
- Bogolyubov, N. N., Mitropol'skii, Yu. A. & Samoilenko, A. M. (1969). Method of accelerated convergence in nonlinear mechanics. Kiev: Naukova Dumka (in Russian).
- Dennis, J. E. & Schnabel, R. B. (1996). Numerical methods for unconstrained optimization and nonlinear equations. Colorado: Society for Industrial and Applied Mathematics.
- Polyak, B. T. (2006). Newton's method and its role in optimization and computational mathematics. Trudy ISA RAN, 28, pp. 48-66 (in Russian).
- Chuiko, S.M., Boichuk, I.A. (2013). Autonomous Noetherian boundary value problem in the critical case. Nonlinear Oscillations. 2009, 12, Iss. 3, pp. 417-428.
- Chuiko, S. M., Boichuk, I. A., & Pirus, O. E. (2013). On the approximate solution of an autonomous boundary-value problem the Newton—Kantorovich method. J. Math. Sci., 189, No. 5, pp. 867-881.
- Boichuk, A. A. & Holovats'ka, I. A. (2014). Boundary-value problems for systems of integro-differential equations. J. Math. Sci., 203, No. 3, pp. 306-321.
- Chuiko, S. M. & Pirus, O. E. (2013). On the approximate solution of autonomous boundary-value problems by the Newton method. J. Math. Sci., 191, No. 3, pp. 449-464.
- Samoilenko, A. M., Boichuk, A. A., &Krivosheya, S. A. (1996). Boundary-value problems for systems of integro-differential equations with degenerate kernel. Ukr. Math. J., 48, No. 11, pp. 1785-1789.
- Chuiko, S. M. (2017). To the generalization of the Newton—Kantorovich theorem. Visnyk of V.N. Karazin Kharkiv National University. Ser. "Mathematics, Applied Mathematics and Mechanics", 85, pp. 62-68.
- Boichuk, A. A., & Pokutnyi, A. A. (2014). Solutions of the Schrödinger equation in a Hilbert space. Boundary Value Problems. doi: https://doi.org/10.1186/1687-2770-2014-4
- Gutlyanskii, V., Ryazanov, V., Srebro, U. & Yakubov, E. (2012). The Beltrami equation: a geometric approach, developments in mathematics, Vol. 26. New York etc: Springer.
- Gutlyanskii, V., Ryazanov, V. & Yakubov, E. (2015) Toward the theory of the Dirichlet problem for the Bel t ra mi equations. Dopov. Nac. akad. nauk Ukr., No. 11, pp. 23-29. doi: https://doi.org/10.15407/dopovidi2015.11.023