Abstract
The main focus of this paper is the analysis of the semilocal convergence (S.C.) of a three-step Newton-type scheme (TSNTS) used for finding the solution of nonlinear operators in Banach spaces (B.S.). A novel S.C. analysis of the TSNTS is introduced, which is based on the assumption that a generalized Lipschitz condition (G.L.C.) is satisfied by the first derivative of the operator. The findings contribute to the theoretical understanding of TSNTS in B.S. and have practical implications in various applications, such as integral equation further validating our results.
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REFERENCES
Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, SIAM, 2000.
Rall, L.B., Computational Solution of Nonlinear Operator Equations, New York: Robert E. Krieger, 1979.
Kantorovich, L.V. and Akilov, G.P., Functional Analysis, Oxford: Pergamon Press, 1982.
Hernandez, M.A. and Salanova, M.A., Sufficient Conditions for Semilocal Convergence of a Fourth-Order Multipoint Iterative Method for Solving Equations in Banach Spaces, Southwest J. Pure Appl. Math., 1999, vol. 1, pp. 29–40.
Chen, L., Gu, C., and Ma, Y., Semilocal Convergence for a Fifth-Order Newton’s Method Using Recurrence Relations in Banach Spaces, J. Appl. Math., 2011, vol. 2011, Article no. 786306.
Wang, X., Gu, C., and Kou, J., Semilocal Convergence of a Multipoint Fourth-Order Super Halley Method in Banach Spaces, Numer. Algor., 2011, vol. 56, pp. 497–516.
Jaiswal, J.P., Semilocal Convergence of an Eighth-Order Method in Banach Spaces and Its Computational Efficiency, Numer. Algor., 2016, vol. 71, pp. 933–951.
Jaiswal, J.P., Semilocal Convergence Analysis and Comparison of Alternative Computational Efficiency of the Sixth-Order Method in Banach Spaces, Novi Sad J. Math., 2019, vol. 49, no. 2, pp. 1–16.
Jaiswal, J.P., Semilocal Convergence and Its Computational Efficiency of a Seventh-Order Method in Banach Spaces, Proc. Nat. Acad. Sci., India, Sect. A: Phys. Sci., 2020, vol. 90, no. 2, pp. 271–279.
Prashanth, M. and Gupta, D.K., Recurrence Relations for Super-Halley’s Method with Holder Continuous Second Derivative in Banach Spaces, Kodai Math. J., 2013, vol. 36, no. 1, pp. 119–136.
Singh, S., Gupta, D.K., Martı́nez, E., and Hueso, J.L., Semilocal and Local Convergence of a Fifth Order Iteration with Fréchet Derivative Satisfying Hölder Condition, Appl. Math. Comput., 2016, vol. 276, no. 5, pp. 266–277.
Jaiswal, J.P., Semilocal Convergence of a Computationally Efficient Eighth-Order Scheme in Banach Spaces under Hölder Condition on Third Derivative, J. Analysis, 2020, vol. 28, pp. 141–154.
Gupta Neha and Jaiswal, J.P., Semilocal Convergence of a Seventh-Order Method in Banach Spaces under Hölder Continuity Condition, J. Indian Math. Soc., 2020, vol. 87, pp. 56–69.
Prashanth, M. and Gupta, D.K., Semilocal Convergence for Super-Halley’s Method under w-Differentiability Condition, Japan J. Indust. Appl. Math., 2015, vol. 32, no. 1, pp. 77–94.
Jaiswal, J.P., Analysis of Semilocal Convergence in Banach Spaces under Relaxed Continuity Condition and Computational Efficiency, Num. An. Appl., 2017, vol. 10, no. 2, pp. 129–139.
Jaiswal, J.P., Semilocal Convergence of a Computationally Efficient Eighth-Order Method in Banach Spaces under w-Continuity Condition, Iranian J. Sci. Technol., Transact. A: Sci., 2018, vol. 42, no. 2, pp. 819–826.
Jaiswal, J.P., Panday Bhavna, and Choubey Neha, Analysis of Semilocal Convergence under w-Continuity Condition on Second Order Frechet Derivative in Banach Space, Acta Mathematica Universitatis Comenianae, 2019, vol. 88, no. 2, pp. 173–185.
Gupta Neha and Jaiswal, J.P., Semilocal Convergence of a Seventh-Order Method in Banach Spaces under w-Continuity Condition, Surveys Math. Its Appl., 2020, vol. 15, pp. 325–339.
Wang, X., Convergence of Newton’s Method and Uniqueness of the Solution of Equations in Banach sPace, IMA J. Num. An., 2000, vol. 20, pp. 123–134.
Saxena, A., Argyros, I.K., Jaiswal, J.P., Argyros, C., and Pardasani, K.R., On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions, Mathematics, 2021, vol. 9, no. 6, Article no. 669, https://doi.org/10.3390/math9060669
Xinghua, W., Convergence of Newton’s Method and Inverse Function Theorem in Banach Space, Math. Comput., 1999, vol. 68, no. 225, pp. 169–186.
Xu, X. and Li, C., Convergence Criterion of Newton Method for Singular Systems with Constant Rank Derivatives, J. Math. An. Appl., 2008, vol. 345, no. 2, pp. 689–701.
Ferreira, O.P., A Robust Semi-Local Convergence Analysis of Newton’s Method for Cone Inclusion Problems in Banach Spaces under Affine Invariant Majorant Condition, J. Comput. Appl. Math., 2015, vol. 279, pp. 318–335.
Bittencourt, T. and Ferreira, O.P., Kantorovich’s Theorem on Newton’s Method under Majorant Condition in Riemannian Manifolds, J. Global Optim., 2017, vol. 68, no. 2, pp. 387–411.
Ferreira, O.P. and Svaiter, B.F., A Robust Kantorovich’s Theorem on the Inexact Newton Method with Relative Residual Error Tolerance, J. Complex., 2012, vol. 28, no. 3, pp. 346–363.
Argyros, I.K., Jidesh, P., and George, S., Ball Convergence for Second Derivative Free Methods in Banach Space, Int. J. Appl. Comput. Math., 2017, vol. 3, pp. 713–720.
Jaiswal, J.P., Existence and Uniqueness Theorems for a Three-Step Newton-Type Method under L-Average Conditions, J. Nonlin. Model. An., 2022, vol. 4, no. 4, pp. 650–657.
Ferreira, O.P. and Svaiter, B.F., Kantorovich’s Majorants Principle for Newton’s Method, J. Comput. Appl. Math., 2009, vol. 42, no. 2, pp. 213–229.
Li, C. and Ng, K.F., Majorizing Functions and Convergence of the Gauss–Newton Method for Convex Composite Optimization, SIAM J. Optim., 2007, vol. 18, no. 2, pp. 613–642.
Wang, J., Hu, Y., Wai Yu, C.K., Li, C., and Yang, X., Extended Newton Methods for Multi-Objective Optimization: Majorizing Function Technique and Convergence Analysis, SIAM J. Optim., 2019, vol. 29, no. 3, pp. 2388–2421.
Ling, Y., Liang, J., and Lin, W., On Semilocal Convergence Analysis for Two-Step Newton Method under Generalized Lipschitz Conditions in Banach Spaces, Numer. Algor., 2022, vol. 90, pp. 577–606.
Li, C., Hu, N., and Wang, J., Convergence Behavior of Gauss–Newton’s Method and Extensions of the Smale Point Estimate Theory, J. Complex., 2010, vol. 26, no. 3, pp. 268–295.
Hiriart-Urruty, J.B. and Lemaréchal, C., Convex Analysis and Minimization Algorithms I: Fundamentals, Springer Science & Business Media, 1993.
Ling, Y. and Xu, X., On the Semilocal Convergence Behavior for Halley’s Method, Comput. Optim. Appl., 2014, vol. 58, no. 3, pp. 597–618.
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Translated from Sibirskii Zhurnal Vychislitel’noi Matematiki, 2023, Vol. 27, No. 1, pp. 11-32. https://doi.org/10.15372/SJNM20240102.
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Jaiswal, J.P. Analyzing the Semilocal Convergence of a Fourth-Order Newton-Type Scheme with Novel Majorant and Average Lipschitz Conditions. Numer. Analys. Appl. 17, 8–27 (2024). https://doi.org/10.1134/S1995423924010026
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DOI: https://doi.org/10.1134/S1995423924010026