A HYBRID EXTRAGRADIENT METHOD FOR FINDING VARIATIONAL INEQUALITY PROBLEMS, FIXED POINT PROBLEMS AND EQUILIBRIUM PROBLEMS

Chanan Sudsukh (Thailand)

Received April 4, 2009

References:

[1] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), 123-145. [2] A. Bnouhachem, M. Aslam Noor and Z. Hao, Some new extragradient iterative methods for variational inequalities, Nonlinear Anal. 70 (2009), 1321-1329. [3] P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117-136. [4] S. D. Flåm and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming 78 (1997), 29-41. [5] A. Genel and J. Lindenstrauss, An example concerning fixed points, Israel. J. Math. 22 (1975), 81-86. [6] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990. [7] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. [8] G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, Èkonom. i Mat. Metody 12 (1976), 747-756. [9] P. Kumam, A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping, Nonlinear Anal. Hybrid Syst. 2(4) (2008), 1245-1255. [10] P. Kumam, A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping, J. Appl. Math. Comput. 29(1-2) (2009), 263-280. [11] G. Marino and H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), 336-346. [12] A. Moudafi and M. Théra, Proximal and dynamical approaches to equilibrium problems, Lecture Notes in Econom. and Math. Systems, 477, Springer-Verlag, New York, 1999, pp. 187-201. [13] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. [14] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506-515. [15] W. Takahashi, Y. Takeuchi and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008), 276-286. [16] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), 417-428. [17] J.-C. Yao and O. Chadli, Pseudomonotone complementarity problems and variational inequalities, Handbook of Generalized Convexity and Monotonicity, J. P. Crouzeix, N. Haddjissas and S. Schaible, eds., Springer, New York, 2005, pp. 501-558. [18] L. C. Zeng, S. Schaible and J. C. Yao, Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl. 124 (2005), 725-738. [19] L.-C. Zeng and J.-C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math. 10 (2006), 1293-1303.

Keywords and phrases: hybrid extragradient method, nonexpansive mapping, k-Lipschitz continuous mapping, variational inequality problem, equilibrium problem, fixed points.