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An Extragradient Method for Fixed Point Problems and Variational Inequality Problems
Journal of Inequalities and Applications volume 2007, Article number: 038752 (2007)
We present an extragradient method for fixed point problems and variational inequality problems. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for monotone mapping.
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967,20(2):197–228. 10.1016/0022-247X(67)90085-6
Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Analysis 1998,6(4):313–344. 10.1023/A:1008643727926
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.
Yao J-C: Variational inequalities with generalized monotone operators. Mathematics of Operations Research 1994,19(3):691–705. 10.1287/moor.19.3.691
Yao J-C, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optim. Appl.. Volume 76. Edited by: Crouzeix JP, Haddjissas N, Schaible S. Springer, New York, NY, USA; 2005:501–558. 10.1007/0-387-23393-8_12
Zeng LC, Schaible S, Yao J-C: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Optimization Theory and Applications 2005,124(3):725–738. 10.1007/s10957-004-1182-z
Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003,118(2):417–428. 10.1023/A:1025407607560
Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2006,128(1):191–201. 10.1007/s10957-005-7564-z
Korpelevič GM: An extragradient method for finding saddle points and for other problems. Èkonomika i Matematicheskie Metody 1976,12(4):747–756.
Zeng L-C, Yao J-C: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2006,10(5):1293–1303.
Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970,149(1):75–88. 10.1090/S0002-9947-1970-0282272-5
Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Computers & Mathematics with Applications 2000,40(4–5):559–567. 10.1016/S0898-1221(00)00179-6
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017
Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
Yao Y, Yao J-C: On modified iterative method for nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2007.
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Yao, Y., Liou, YC. & Yao, JC. An Extragradient Method for Fixed Point Problems and Variational Inequality Problems. J Inequal Appl 2007, 038752 (2007). https://doi.org/10.1155/2007/38752
- Variational Inequality
- Monotone Mapping
- Nonexpansive Mapping
- Common Element
- Point Problem