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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 832548, 8 pages
Extragradient Method for Solutions of Variational Inequality Problems in Banach Spaces
1Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 27 March 2013; Accepted 11 May 2013
Academic Editor: Ru Dong Chen
Copyright © 2013 H. Zegeye and N. Shahzad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- R. P. Agarwal, D. ORegan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Springer, New York, NY, USA, 2000.
- G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964.
- H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 3, pp. 341–350, 2005.
- G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.
- P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.
- M. A. Noor, “A class of new iterative methods for general mixed variational inequalities,” Mathematical and Computer Modelling, vol. 31, no. 13, pp. 11–19, 2000.
- I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently parallel algorithms in feasibility and optimization and their applications, vol. 8, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
- Y. Yao, Y.-C. Liou, C.-L. Li, and H.-T. Lin, “Extended extragradient methods for generalized variational inequalities,” Journal of Applied Mathematics, vol. 2012, Article ID 237083, 14 pages, 2012.
- Y. Yao and H.-K. Xu, “Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications,” Optimization, vol. 60, no. 6, pp. 645–658, 2011.
- H. Zegeye, E. U. Ofoedu, and N. Shahzad, “Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3439–3449, 2010.
- H. Zegeye and N. Shahzad, “A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 1, pp. 263–272, 2011.
- H. Zegeye and N. Shahzad, “Strong convergence theorems for a common zero of a countably infinite family of -inverse strongly accretive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2, pp. 531–538, 2009.
- K. Aoyama, H. Iiduka, and W. Takahashi, “Weak convergence of an iterative sequence for accretive operators in Banach spaces,” Fixed Point Theory and Applications, vol. 2006, Article ID 35390, 2006.
- F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
- H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.
- Y. I. Alber, A. N. Iusem, and M. V. Solodov, “Minimization of nonsmooth convex functionals in Banach spaces,” Journal of Convex Analysis, vol. 4, no. 2, pp. 235–254, 1997.
- F. E. Browder, “Nonlinear Operators and nonlinear equations of evolution in Banach spaces,” in Nonlinear Functional Analysis, pp. 1–308, American Mathematical scociety, Rhode Island, New England, 1976.
- H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
- C. H. Morales and J. S. Jung, “Convergence of paths for pseudocontractive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 128, no. 11, pp. 3411–3419, 2000.
- C. E. Chidume, H. Zegeye, and N. Shahzad, “Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings,” Fixed Point Theory and Applications, no. 2, pp. 233–241, 2005.
- Y. Zhang and Y. Guo, “Weak convergence theorems of three iterative methods for strictly pseudocontractive mappings of Browder-Petryshyn type,” Fixed Point Theory and Applications, vol. 2008, Article ID 672301, 13 pages, 2008.
- H. Zhang and Y. Su, “Strong convergence theorems for strict pseudo-contractions in -uniformly smooth Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3236–3242, 2009.
- S. Saejung, K. Wongchan, and P. Yotkaew, “Another weak convergence theorems for accretive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, article 26, 2011, http://www.fixedpointtheoryandapplications.com/content/2011/1/26.