Regularization methods for a class of variational inequalities in Banach spaces
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 11 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we introduce two regularization methods, based on the Browder–Tikhonov and iterative regularizations, for finding a solution of variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gateaux differentiate norm.
@article{ZVMMF_2012_52_11_a2,
     author = {Ngyuen Buong and Ngyuen Thi Hong Phuong},
     title = {Regularization methods for a class of variational inequalities in {Banach} spaces},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1951},
     year = {2012},
     volume = {52},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_11_a2/}
}
TY  - JOUR
AU  - Ngyuen Buong
AU  - Ngyuen Thi Hong Phuong
TI  - Regularization methods for a class of variational inequalities in Banach spaces
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2012
SP  - 1951
VL  - 52
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_11_a2/
LA  - en
ID  - ZVMMF_2012_52_11_a2
ER  - 
%0 Journal Article
%A Ngyuen Buong
%A Ngyuen Thi Hong Phuong
%T Regularization methods for a class of variational inequalities in Banach spaces
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2012
%P 1951
%V 52
%N 11
%U http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_11_a2/
%G en
%F ZVMMF_2012_52_11_a2
Ngyuen Buong; Ngyuen Thi Hong Phuong. Regularization methods for a class of variational inequalities in Banach spaces. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 11. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_11_a2/

[1] S. Antman, “The Influence of Elasticity in Analysis: Modern Developments”, Bull. Am. Math. Soc., 9 (1983), 267–291 | DOI | MR | Zbl

[2] G. Fichera, “La nascita della teoria delle diseguaglianze variazionali ricordata dopo trent'anni”, Accademia Nazionale dei Lincei, 114 (1995), 47–53

[3] G. Stampacchia, “Formes bilineares coercitives sur les ensembles convexes”, Computes Rendus Hebdom. des Seances de l'Academie des Sci., 258 (1964), 4413–4416 | MR | Zbl

[4] J. L. Lions, G. Stampacchia, “Variational Inequalities”, Commun. Pure Appl. Math., 20 (1967), 493–519 | DOI | MR | Zbl

[5] D. Duvaut, J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976 | MR

[6] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic, New York, 1980 | MR | Zbl

[7] I. Hlavacek, J. Haslinger, J. Necas, J. Lovicek, Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York, 1982

[8] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-verlag, New York, 1984 | MR | Zbl

[9] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, 1985 | MR | Zbl

[10] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1985 | MR | Zbl

[11] K. Aoyama, H. Iiduka, W. Takahashi, “Weak Convergence of an Iterative Sequence for Accretive Operators in Banach Spaces”, Fixed Point Theory Appl., 2006 | MR

[12] Y. Yamada, “The Hybrid Steepest-Descent Method for Variational Inequalities Problems over the Intersection of the Fixed Point Sets of Nonexpansive Mappings”, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, eds. D. Butnariu, Y. Censor, S. Reich, North-Holland, Amsterdam, 2001 | Zbl

[13] H. K. Xu, T. H. Kim, “Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities”, J. Optim. Theory Appl., 119 (2003), 185–201 | DOI | MR | Zbl

[14] Y. Yao, M. A. Noor, Y. C. Liou, “Convergence of Hybrid Steepest-Descent Method for Variational Inequalities”, Appl. Math. Comput., 216 (2010), 822–829 | DOI | MR | Zbl

[15] Sh. Wang, “Convergence and Weaker Control Conditions for Hybrid Iterative Algorithms”, FPTA, 2011 | MR

[16] Ya. I. Alber, “On the Stability of Iterative Approximations to Fixed Points of Nonexpansive Mappings”, J. Math. Anal. Appl., 328 (2007), 958–971 | DOI | MR | Zbl

[17] Ya. I. Alber, Ir. Ryazantseva, Nonlinear Ill-posed Problems of Monotone Type, Springer-Verlag, Berlin, 2006 | MR | Zbl

[18] Bakushinsky A., Goncharsky A., Ill-Posed Problems: Theory and Applications, Kluwer Academic, Dordrecht, 1994 | MR

[19] F. E. Browder, “Existence and Approximation of Solutions of Nonlinear Variational Inequalities”, Proc. Natl. Acad. Sci. USA, 56 (1966), 1080–1086 | DOI | MR | Zbl

[20] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings, and Nonlinear Problems, Kluwer Academic, Dordrecht, 1990 | MR | Zbl

[21] W. Takahashi, Y. Ueda, “On Reich's Strong Convergence Theorem for Resolvents of Accretive Operators”, J. Math. Anal. Appl., 104 (1984), 546–553 | DOI | MR

[22] V. V. Vasin, A. L. Ageev, Ill-Posed Problems with a priori Information, Nauka, Yekaterinburg, 1993 ; VSP, Utrecht, 1995 | MR | Zbl

[23] H. K. Xu, “Inequalities in Banach Spaces with Applications”, Nonlinear Anal., 16 (1991), 1127–1138 | DOI | MR | Zbl

[24] T. Suzuki, “Strong Convergence of Approximated Sequences for Nonexpansive Mappings in Banach Spaces”, Proc. Am. Math. Soc., 135 (2007), 99–106 | DOI | Zbl

[25] W. Takahashi, “Weak and Strong Convergence Theorems for Families of Nonexpansive Mappings and Their Applications”, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 51 (1997), 277–292 | MR | Zbl

[26] W. Takhashi, K. Shimoji, “Convergence Theorems for Nonexpansive Mappings and Feasibility Problems”, Math. Comput. Model., 32 (2000), 1463–1471 | DOI | MR

[27] K. Shimoji, W. Takhashi, “Strong Convergence to Common Fixed Points of Infinite Nonexpansive Mappings and Applications”, Taiwanese J. Math., 5 (2001), 387–404 | MR | Zbl

[28] E. F. Browder, “Nonlinear Mappings of Nonexpansive and Accretive Type in Banach Spaces”, Bull. Am. Math. Soc., 73 (1967), 875–882 | DOI | MR | Zbl

[29] R. P. Agarwal, D. O'Regan, D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Springer-Verlag, New York, 2009 | MR