Regularization of monotone variational inequalities with Mosco approximations of the constraint sets

Set-Valued Analysis

Volumn 13, Issue 3, 2005, Pages 265-290

  Alber, Yakov ^a   Butnariu, Dan ^b   Ryazantseva, Irina ^c  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b UNIVERSITY OF HAIFA   (Israel)

^c NIZHNY NOVGOROD STATE TECHNICAL UNIVERSITY N A R E ALEKSEEV   (Russian Federation)

Author keywords

Demiclosed operator; Fast Mosco convergence relative to a sequence of positive real numbers; Monotone operator; Mosco convergence; Regularization method; Variational inequality

Indexed keywords

EID: 22044447326     PISSN: 09276947     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11228-004-3030-6     Document Type: Article

References (33)

1. Alber, Y.: On a solution of nonlinear equations with monotone operators in Banach spaces, Siberian Math. J. 16 (1975), 1-8.
2. Alber, Y.: The regularization method for variational inequalities with nonsmooth unbounded operators in Banach space, Appl. Math. Lett. 6 (1993), 63-68.
3. Alber, Y., Kartsatos, A. G. and Litsyn, E.: Iterative solutions of unstable variational inequalities on approximately given sets, Abstr. Appl. Anal. 1 (1996), 45-64.
4. Alber, Y., Butnariu, D. and Kassay, G.: Convergence and stability of a regularization method for maximal monotone inclusions and its applications to convex optimization, In: F. Giannessi and A. Maugeri (eds), Variational Analysis and Applications, Kluwer Academic Publishers, Dordrecht, 2004.
5. Alber, Y., Butnariu, D. and Ryazantseva, I.: Regularization methods for ill-posed inclusions and variational inequalities with domain perturbations, J. Nonlinear Conv. Anal. 2 (2001), 53-79.
6. Alber, Y. and Notik, A. I.: Perturbed unstable variational inequalities with unbounded operators on approximately given sets, Set-Valued Anal. 1 (1993), 393-402.
7. Bakushinskii, A. B. and Gontscharskii, A. V.: Ill-posed Problems - Numerical Methods and Applications (in Russian), Nauka, Moscow, 1989.
8. Bonnans, J. F. and Shapiro, A.: Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
9. Browder, F. E.: Multivalued monotone nonlinear mappings and duality mappings in Banach spaces, Trans. Amer. Math. Soc. 118 (1965), 338-351.
10. Browder, F. E.: Existence and approximation of solutions of nonlinear variational inequalities, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1080-1086.
11. Djafari Rouhani, B. and Khan, A. A.: On the embedding of variational inequalities, Proc. Amer. Math. Soc. 131 (2003), 3861-3871.
12. Engl, H. W., Hanke, M. and Neubauer, A.: Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996.
13. Holmes, P. R.: Geometric Functional Analysis, Springer, New York, 1975.
14. Kaplan, A. and Tichatschke, R.: Stable Methods for Ill-posed Variational Problems: Prox-Regularization of Elliptic Variational Inequalities and Semi-Infinite Problems, Akademie Verlag, Berlin, 1994.
15. Kinderlehrer, D. and Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
16. Lavrent'ev, M. M.: Some Ill-posed Problems of Mathematical Physics (in Russian), Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1962.
17. Lavrent'ev, M. M., Zerkal, S. M. and Trofimov, O. E.: Computer Modelling in Tomography, VSP BV, Utrecht, 2001.
18. Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites nonlineaires, Dunod, Paris, 1976.
19. Liskovets, O. A.: Variational Methods for Solving Unstable Problems (in Russian), Nauka i Tekhnika, Minsk, 1981.
20. Liskovets, O. A.: Regularization of problems with discontinuous monotone arbitrarily perturbed operators, Soviet Math. Dokl. 28 (1983), 324-327.
21. Liskovets, O. A.: Discrete convergence of elements and operators for ill-posed problems with a monotone operator, Soviet Math. Dokl. 31 (1985), 202-206.
22. Liskovets, O. A.: External approximations for the regularization of monotone variational inequalities, Soviet Math. Dokl. 36 (1988), 220-224.
23. Liskovets, O. A.: Regularized variational inequalities with pseudo-monotone operators on approximately given sets (in Russian), Differential Equations 11 (1989), 1970-1977.
24. Liskovets, O. A.: Regularization of ill-posed variational inequalities (in Russian), private communication, 1990.
25. Mosco, U.: Convergence of convex sets and solutions of variational inequalities, Adv. in Math. 3 (1969), 510-585.
26. Mosco, U.: Perturbations of variational inequalities, In: Nonlinear Functional Analysis, Proc. Sympos. Pure Math. 18, Part 1, Amer. Math. Soc., Providence, RI, 1970
27. Nirenberg, L.: Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York, 1974.
28. Pascali, D. and Sburlan, S.: Nonlinear Mappings of Monotone Type, Stijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1978.
29. Rockafellar, R. T.: On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88.
30. Rockafellar, R. T.: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216.
31. Ryazantseva, L.: Variational inequalities with monotone operators on sets which are specified approximately, USSR Comp. Math. Math. Phys. 24 (1984), 194-197.
32. Tikhonov, A. N.: Regularization of incorrectly posed problems, Soviet Math. Dokl. 4 (1963), 1035-1038.
33. Tikhonov, A. N. and Arsenin, V. Y.: Solutions of Ill-posed Problems, V. H. Winston & Sons, New York, 1977.