## Taiwanese Journal of Mathematics

### SOME NEW ITERATIVE ALGORITHMS FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS WITH STRICT PSEUDO-CONTRACTIONS AND MONOTONE MAPPINGS

#### Abstract

In this paper, we propose some parallel and cyclic algorithms based on the extragradient method (nonextragradient method) for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a finite family of strict pseudo-contractions and the set of the variational inequality for a monotone, Lipschitz continuous mapping (an inverse strongly monotone mapping). We obtain some weak and strong convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, improve and unify some well-known results in the literature.

#### Article information

Source
Taiwanese J. Math., Volume 13, Number 5 (2009), 1537-1582.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405558

Digital Object Identifier
doi:10.11650/twjm/1500405558

Mathematical Reviews number (MathSciNet)
MR2554475

Zentralblatt MATH identifier
1193.47068

#### Citation

Peng, Jian-Wen; Yao, Jen-Chih. SOME NEW ITERATIVE ALGORITHMS FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS WITH STRICT PSEUDO-CONTRACTIONS AND MONOTONE MAPPINGS. Taiwanese J. Math. 13 (2009), no. 5, 1537--1582. doi:10.11650/twjm/1500405558. https://projecteuclid.org/euclid.twjm/1500405558

#### References

• [1.] J. W. Peng and J. C. Yao, A New Hybrid-extragradient Method For Generalized Mixed Equilibrium Problems and Fixed Point Problems and variational inequality problems, Taiwan. J. Math., 2008, (to appear).
• \item[2.] L. C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, Journal of Computational and Applied Mathematics, 214 (2008), 186-201.
• [3.] G. Bigi, M. Castellani and G. Kassay, A dual view of equilibrium problems, J. Math. Anal. Appl., 342 (2008), 17-26.
• [4.] S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Analysis, 69 (2008), 1025-1033.
• [5.] S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program, 78 (1997), 29-41.
• [6.] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.
• [7.] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20 (1967), 197-228.
• [8.] K. Goebel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.
• [9.] P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.
• [10.] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2006), 506-515.
• [11.] J. W. Peng and J. C. Yao, A Modified CQ Method For Equilibrium Problems and Fixed Point Problems and variational inequality problems, Fixed Point Theorey, 9 (2008), 515-531.
• [12.] A. Tada and W. Takahashi, Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem, J. Optim. Theory Appl., 133 (2007), 359-370.
• [13.] V. Colao, G. Marino and H. K. Xu, An iterative method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl., 344 (2008), 340-352.
• [14.] L. C. Ceng, S. AI-Homidan, Q. H. Ansari and J. C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, Journal of Computational and Applied Mathematics, (2008), doi:10.1016/j.cam.2008. 03.032.
• [15.] S. Plubtieng and R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation, 197 (2008), 548-558.
• [16.] S. S. Chang, H. W. Joseph Lee and C. K. Chan, A new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization, Nonlinear Analysis, (2008), doi:10.1016/j.na.2008.04.035.
• [17.] 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.
• [18.] H. Y. Zhou, Convergence theorems of fixed points for $\kappa$-strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis, 69 (2008), 456-462.
• [19.] G. L. Acedoa and H. K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis, 67 (2007), 2258-2271.
• [20.] Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 561-597.
• [21.] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75-88.
• [22.] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.
• [23.] C. Martinez-Yanes and H.-K. Xu, Strong convergence of the $CQ$ method for fixed point iteration processes, Nonlinear Analysis, 64 (2006), 2400-2411.
• [24.] L. C. Zeng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan. J. Math., 10 (2006), 1293-1303.
• [25.] N. Nadezhkina and W. Takahashi, Strong Convergence Theorem by a Hybrid Method for Nonexpansive Mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16(4) (2006), 1230-1241.
• [26.] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.
• [27.] Y. Yao, Y. C. Liou and J. C. Yao, Convergence Theorem for Equilibrium Problems and Fixed Point Problems of Infinite Family of Nonexpansive Mappings, Fixed Point Theory and Applications Volume 2007, Article ID 64363, 12 pages.
• [28.] L. C. Ceng and J. C. Yao, Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Appl. Math. and Compu., (2007), doi:10.1016/j.amc.2007.09.011.
• [29.] O. Chadli, I. V. Konnov and J. C. Yao, Descent methods for equilibrium problems in a Banach space, Computers and Mathematics with Applications, 48 (2004), 609-616.
• [30.] L. C. Ceng, A. Petrusel, C. Lee and M. M. Wong, Two Extragradient approximation methods for variational inequalities and fixed point problems of strict pseudo-contractions, Taiwan. J. Math., 13 (2009), 607-632.
• [31.] J. W. Peng and J. C. Yao, Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Mathematical and Computer modelling, 49 (2009), 1816-1828.