Abstract
We study equations with multiple-valued operators in a Hilbert space. We understand their solutions in the sense of inclusion. We reduce such equations to mixed variational inequalities or to equations with single-valued operators. For constructed problems we propose implicit iterative processes of the second order and establish sufficient conditions for their strong convergence.
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References
Ya. Alber and I. Ryazantseva, Nonlinear Ill-Posed Problems of Monotone Type (Springer, Dodrecht, 2006).
I. P. Ryazantseva, Selected Chapters in the Theory of Monotone Operators (NGTU, Nizhny Novgorod, 2008) [in Russian].
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities. Applications to Free Boundary Problems (John Wiley & Sons, New York, NY, USA, 1984; Nauka, Moscow, 1988).
J.-P. Lions, Some Methods for Solving Nonlinear Boundary-Value Problems (Dunod, Gauthier-Villars, Paris, 1969; Mir, Moscow, 1972).
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics (Springer, Berlin, 1976; Mir, Moscow, 1980).
M.A. Noor and T.M. Rassias, Resolvent Equations for Set-Valued Mixed Variational Inequalities, Nonlinear Anal. 42(1), 71–83 (2000).
I. B. Badriev and O. A. Zadvornov, “Iteration Methods of Solving Variational Inequalities of the Second Kind with Inverse Strongly Monotone Operators,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 1, 20–28 (2003) [Russian Mathematics (Iz. VUZ) 47 (1), 18–26 (2003)].
I. V. Konnov, “Combined Relaxation Method for Generalized Variational Inequalities,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 12, 46–54 (2001) [Russian Mathematics (Iz. VUZ) 45 (12), 43–51 (2001)].
I. V. Konnov and O. V. Pinyagina, “Descent Method with Respect to the Gap Function for Nonsmooth Equilibrium Problems,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 12, 71–77 (2003) [Russian Mathematics (Iz. VUZ) 47 (12), 67–73 (2003)].
A. S. Antipin and F. P. Vasil’ev, “Regularization Methods for Solving Equilibrium Programming Problems with Coupled Constraints,” Zhurn. Vychisl. Matem. i Matem. Fiziki 45(1), 27–40 (2005).
I. V. Konnov and O. V. Pinyagina, “ADescent Method with Inexact Linear Search for Nonsmooth Equilibrium Problems,” Zhurn. Vychisl. Matem. iMatem. Fiziki 48(10), 1812–1818 (2008).
I. P. Ryazantseva, “Certain First-Order Iterative Methods for Mixed Variational Inequalities in a Hilbert Space,” Zhurn. Vychisl. Matem. i Matem. Fiziki 51(5), 762–770 (2011).
I. P. Ryazantseva, “Second-Order Methods for Some Quasivariational Inequalities,” Differents. Uravneniya 44(7), 976–987 (2008).
F. P. Vasil’ev, Numerical Methods for Solving Extremal Problems (Nauka, Moscow, 1988) [in Russian].
A. D. Lyashko and M. M. Karchevskii, “On the Solution of Some Nonlinear Problems of Seepage Theory,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 6, 73–81 (1975) [Soviet Mathematics (Iz. VUZ) 19 (6), 73–81 (1975)].
I. B. Badriev, O. A. Zadvornov, and A. M. Saddeek, “Convergence Analysis of Iterative Methods for Some Variational Inequalities with Pseudomonotone Operators,” Differents. Uravneniya 37(7), 1009–1016 (2001).
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Original Russian Text © I.P. Ryazantseva, 2013, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2013, No. 7, pp. 52–61.
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Ryazantseva, I.P. Iterative processes of the second order for monotone inclusions in a Hilbert space. Russ Math. 57, 45–52 (2013). https://doi.org/10.3103/S1066369X13070050
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DOI: https://doi.org/10.3103/S1066369X13070050