Abstract
In a Hilbert space we construct a regularized continuous analog of the Newton method for nonlinear equation with a Fréchet differentiable and monotone operator. We obtain sufficient conditions of its strong convergence to the normal solution of the given equation under approximate assignment of the operator and the right-hand of the equation.
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References
Vainberg, M. M. Variational Method and the Monotone Operator Method in the Theory of Nonlinear Equations (Nauka, Moscow, 1972) [in Russian].
Kantorovich, L. V. and Akilov, G. P. Functional Analysis (Nauka, Moscow, 1984) [in Russian].
Gavurin, M. K. Nonlinear Functional Equations and Continuous Analogues of Iteration Methods, Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 18–31 (1958) [in Russian].
Zhidkov, E. P., Makarenko, G. I., and Puzynin, I. V. A Continuous Analogue of Newton’s Method in Nonlinear Problems of physics, Physics of Elementary Particles and Atomic Nuclei, 4 (1), 127–166 (1973) [in Russian].
Zhidkov, E. P. and Kozlova, O. V. The Continuous Analog of the Newton Method in Theory of Scattering Inverse Problem Under Presence of Eigenfunctions and Eigenvalues, Mat. Model. 18, No. 2, 120–128 (2006) [in Russian].
Alber, Ya. and Ryazantseva, I. Nonlinear Ill-Posed Problems of Monotone Type (Springer, Dordrecht, 2006).
Ryazantzeva, I. P. Selected Chapters of Theory of Operators ofMomotone Type (Nizhnii Novgorod Univ., Nizhnii Novgorod, 2008) [in Russian].
Kachurovskii, R. I. Non-Linear Monotone Operators in Banach Spaces, Russian Mathematical Surveys 23, No. 2, 121–168 (1968).
Bakushinsky, A. and Goncharsky, A. Ill-Posed Problems: Theory and Applications (MoscowUniv. Press, Moscow, 1989; Kluwer Acad Publ., Dordrecht–Boston–London, 1994).
Antipin, A. S. Continuous and Iterative Processes with Operators of Projection and of Projection Type, in Problems of Cybernetics. Numerical Problems of Analysis of Large Systems (Sci. Council on the Complex Problem ‘Cybernetics’, AN SSSR, Moscow, 5–43 (1989)) [in Russian].
Vasil’ev, F. P. Methods for Solving Extremal Problems (Moscow, Nauka, 1981) [in Russian].
Ryazantseva, I. P. On Continuous First-Order Methods and Their Regularized Versions for Mixed Variational Inequalities, Differential Equations 45, No. 7, 1005–1012 (2012).
Ryazantseva, I. P. Continuous First-Order Methods for Monotone Inclusions in a Hilbert Space, Computational Mathematics and Mathematical Physics 53, No. 8, 1070–1077 (2013).
Tikhonov, A. N., Goncharsky, A. V., Stepanov, V. V., and Yagola, A. G. Regularizing Algorithms and Apriori Information (Mir, Moscow, 1983) [in Russian].
Tikhonov, A. N., Goncharsky, A. V., Stepanov, V. V., and Yagola, A. G. NumericalMethods for the Solution of Ill-Posed Problems (Nauka, Moscow, 1990; Kluwer Academic Publishers, Dordrecht, 1995).
Trenogin, B. A. Functional Analysis (Mir, Moscow, 1988) [in Russian].
Ryazantseva, I. P. Some Continuous Regularization Methods for Monotone Equations, Computational Mathematics and Mathematical Physics 34, No. 1, 1–7 (1994).
Ramm, A. G. Dynamical Systems Method for Solving Operator Equations (Elsevier, Amsterdam, 2007).
Kokurin, M. Yu. Continuous Methods of Stable Approximation of Solutions to Non-Linear Equations in Hilbert Space Based on a Regularized Gauss–Newton Scheme, Comp. Math. and Math. Physics 44, No. 1, 6–15 (2004).
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Original Russian Text © I.P. Ryazantseva, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 11, pp. 53–67.
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Ryazantseva, I.P. Regularized continuous analog of the newton method for monotone equations in a Hilbert space. Russ Math. 60, 45–57 (2016). https://doi.org/10.3103/S1066369X16110050
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DOI: https://doi.org/10.3103/S1066369X16110050