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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.55 no.2 Bogotá July/Dec. 2021 Epub May 31, 2022
https://doi.org/10.15446/recolma.v55n2.102622
ORIGINAL ARTICLES
New linearization method for nonlinear problems in Hilbert space
Nuevo método de linealización para problemas no lineales en espacios de Hilbert
1Université 8 Mai 1945, Guelma, Algeria
2Derzhavin Tambov State University, Tambov, Russia
In this paper, we build a Newton-like sequence to approach the zero of a nonlinear Fréchet differentiable function defined in Hilbert space. This new iterative sequence uses the concept of the adjoint operator, which makes it more manageable in practice compared to the one developed by Kantorovich which requires the calculation of the inverse operator in each iteration. Because the calculation of the adjoint operator is easier compared to the calculation of the inverse operator which requires in practice solving a system of linear equations, our new method makes the calculation of the term of our new sequence easier and more convenient for numerical approximations. We provide an a priori convergence theorem of this sequence, where we use hypotheses equivalent to those constructed by Kantorovich, and we show that our new iterative sequence converges towards the solution.
Keywords: Nonlinear problems; Newton-like method; Fréchet differentiability; Adjoint Operator
En este artículo, construimos una sucesión similar a la de Newton para acercarnos al cero de una función diferenciable en el sentido Fréchet no lineal definida en un espacio de Hilbert. Esta nueva sucesión utiliza el concepto del adjunto del operador, que hace que el proceso iterativo sea más manejable en la práctica en comparación al desarrollado por Kantorovich que requiere el cálculo del operador inverso en cada iteración. Dado que el cálculo del operador adjunto es más fácil en comparación con el cálculo del operador inverso que en la práctica equivale a resolver un sistema de ecuaciones, nuestra nuevo método hace que el cálculo del término de nuestra nueva sucesión sea más fácil y conveniente para la aproximación numérica. Proporcionamos un teorema de convergencia a priori de esta sucesión, donde usamos unas hipótesis equivalentes a las construidas por Kantorovich, y mostramos que nuestra nueva sucesión iterativa converge hacia la solución.
Palabras clave: Problemas no lineales; método tipo Newton; diferenciabilidad de Fréchet; operador adjunto
REFERENCES
1. M. Ahues, A note on perturbed fixed slope iterations, Appl. Math. Letters 18 (2005), no. 4, 375-380. [ Links ]
2. I. K. Argyros and Á. A. Magreńán, Extended convergence results for the newton-kantorovich iteration, J. of Comp. and Appl. Math 286 (2015), no. 3, 54-67. [ Links ]
3. F. Cianciaruso and E. De Pascale, Estimates of majorizing sequences in the newton kantorovich method: A further improvement, J. of Math. Anal. And Appl. 322 (2006), no. 1, 329-335. [ Links ]
4. Z. K. Eshkuvatov, H. H. Hameed, and N. M. A. Nik Long, One dimensional nonlinear integral operator with newton kantorovich method, J. of King Saud University - Science 28 (2016), no. 2, 172-177. [ Links ]
5. A. J. Ezquerro, V. Hernández, and M. Angel, Newton's Method: an Updated Approach of Kantorovich's Theory, Birkhäuser, 2017. [ Links ]
6. L. Grammont, M. Ahues, and F. D. D'Almeida, For nonlinear infinite dimensional equations, which to begin with: Linearization or discretization?, J. Integral Equations Applications 26 (2014), no. 3, 413-436. [ Links ]
7. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, 1982. [ Links ]
8. L. V. Kantorovich and V. I. Krilov, Approximate Methods of Higher Analysis, Interscience Publishers Inc, 1958. [ Links ]
9. A. Khellaf, W. Merchela, and S. Benarab, New numerical process solving nonlinear infinite-dimensional equations, Comp. Appl. Math. 39 (2020), no. 93. [ Links ]
Received: November 01, 2020; Accepted: November 27, 2021
This is an open-access article distributed under the terms of the Creative Commons Attribution License
