Abstract
The convergence of iterative methods for solving variational inequalities with monotone-type operators in Banach spaces is studied. Such inequalities arise in the description of deformation processes of soft rotational network shells. Certain properties of these operators, such as coercivity, potentiality, bounded Lipschitz continuity, pseudomonotonicity, and inverse strong monotonicity, are determined. An iterative method for solving these variational inequalities is proposed, its convergence is investigated, and the boundedness of the iterative sequence is proved. Moreover, it is proved that any weakly convergent subsequence of the iterative sequence converges to a solution of the original variational inequality.
Similar content being viewed by others
References
G.R. Abdyusheva, I. B. Badriev, V. V. Banderov, O. A. Zadvornov, and R. R. Tagirov, “Mathematical modeling of the problem on the equilibrium of a soft biological shell. I. Generalized setting,” Uchen. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauk 154(4), 57–73 (2012).
I. B. Badriev, O. A. Zadvornov, and A. M. Saddek, “Convergence analysis of iterative methods for some variational inequalities with pseudomonotone operators,” Differ. Equations 37(7), 934–942 (2001).
J.-L. Lions, Quelques methodes de résolution des problémes aux limites nonlinéaires (Dunod, Paris, 1969; Mir, Moscow, 1972).
H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974; Mir, Moscow, 1978).
P. Tseng, “Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming,” Math. Program. 48(1–3), 249–263 (1990).
D. Zhu and P. Marcotte, “New classes of generalized monotonicity,” J. Optimiz. Theory Appl. 87(2), 457–471 (1995).
E. G. Gol’shtein and N. V. Tret’yakov, Modified Lagrangian functions (Nauka, Moscow, 1989) [in Russian].
M. M. Vainberg and I. M. Lavrent’ev, “Nonlinear quasi-potential operators,” Dokl. Akad. Nauk 205(5), 1022–1024 (1972).
I. B. Badriev, O. A. Zadvornov, A. D. Lyashko, “A Study of Variable Step Iterative Methods for Variational Inequalities of the Second Kind,” Differ. Equations 40(7), 971–983 (2004).
A. D. Lyashko and M. M. Karchevskii, “Solution of certain nonlinear problems of filtration theory,” Soviet Math. (Izv. VUZ. Mat.), 19(6), 60–66 (1975).
I. B. Badriev and M. M. Karchevskii, “On the convergence of an iterative process in a Banach,” Journal of Mathematical Sciences 71(6), 2727–2735 (1994).
R. Glowinski, G.-L. Lions, and R. Trémolières, Analyse numérique des inéquations variationneles (Dunod, Paris, 1976; Mir, Moscow, 1979).
M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems (North-Holland, Amsterdam, 1983).
D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2(1), 17–40 (1976).
P. L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal. 16(6), 964–979 (1979).
M. Fortin and R. Glowinski, Méthodes de lagrangien augmenté: applications à la résolution numérique de problèmes aux limites (Dunod, Paris, 1982).
I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976; Mir, Moscow, 1979).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I.B. Badriev, V.V. Banderov, 2013, published in Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2013, Vol. 155, No. 2, pp. 18–32.
Rights and permissions
About this article
Cite this article
Badriev, I.B., Banderov, V.V. Iterative methods for solving variational inequalities of the theory of soft shells. Lobachevskii J Math 35, 371–383 (2014). https://doi.org/10.1134/S1995080214040015
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080214040015