Lobachevskii Journal of Mathematics

, Volume 35, Issue 4, pp 371–383 | Cite as

Iterative methods for solving variational inequalities of the theory of soft shells

  • I. B. BadrievEmail author
  • V. V. Banderov


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.

Keywords and phrases

variational inequality pseudomonotone operator potential operator iterative method soft network shell 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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).Google Scholar
  2. 2.
    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).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    J.-L. Lions, Quelques methodes de résolution des problémes aux limites nonlinéaires (Dunod, Paris, 1969; Mir, Moscow, 1972).Google Scholar
  4. 4.
    H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974; Mir, Moscow, 1978).zbMATHGoogle Scholar
  5. 5.
    P. Tseng, “Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming,” Math. Program. 48(1–3), 249–263 (1990).CrossRefzbMATHGoogle Scholar
  6. 6.
    D. Zhu and P. Marcotte, “New classes of generalized monotonicity,” J. Optimiz. Theory Appl. 87(2), 457–471 (1995).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    E. G. Gol’shtein and N. V. Tret’yakov, Modified Lagrangian functions (Nauka, Moscow, 1989) [in Russian].Google Scholar
  8. 8.
    M. M. Vainberg and I. M. Lavrent’ev, “Nonlinear quasi-potential operators,” Dokl. Akad. Nauk 205(5), 1022–1024 (1972).MathSciNetGoogle Scholar
  9. 9.
    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).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    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).Google Scholar
  11. 11.
    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).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    R. Glowinski, G.-L. Lions, and R. Trémolières, Analyse numérique des inéquations variationneles (Dunod, Paris, 1976; Mir, Moscow, 1979).Google Scholar
  13. 13.
    M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems (North-Holland, Amsterdam, 1983).zbMATHGoogle Scholar
  14. 14.
    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).CrossRefzbMATHGoogle Scholar
  15. 15.
    P. L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal. 16(6), 964–979 (1979).CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    M. Fortin and R. Glowinski, Méthodes de lagrangien augmenté: applications à la résolution numérique de problèmes aux limites (Dunod, Paris, 1982).Google Scholar
  17. 17.
    I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976; Mir, Moscow, 1979).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Kazan (Volga Region) Federal UniversityKazanTatarstan, Russia

Personalised recommendations