A monotone projection algorithm for solving fixed points of nonlinear mappings and equilibrium problems
Authors
Mingliang Zhang
 School of Mathematics and Statistics, Henan University, Kaifeng 475000, China.
Sun Young Cho
 Department of Mathematics, Gyeongsang National University, Jinju 660701, Korea.
Abstract
In this paper, fixed points of asymptotically quasi\(\phi\)nonexpansive mappings in the intermediate sense
and equilibrium problems are investigated based on a monotone projection algorithm. Strong convergence
theorems are established in the framework of reflexive Banach spaces.
Share and Cite
ISRP Style
Mingliang Zhang, Sun Young Cho, A monotone projection algorithm for solving fixed points of nonlinear mappings and equilibrium problems, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 4, 14531462
AMA Style
Zhang Mingliang, Cho Sun Young, A monotone projection algorithm for solving fixed points of nonlinear mappings and equilibrium problems. J. Nonlinear Sci. Appl. (2016); 9(4):14531462
Chicago/Turabian Style
Zhang, Mingliang, Cho, Sun Young. "A monotone projection algorithm for solving fixed points of nonlinear mappings and equilibrium problems." Journal of Nonlinear Sciences and Applications, 9, no. 4 (2016): 14531462
Keywords
 Asymptotically quasi\(\phi\)nonexpansive mapping
 equilibrium problem
 fixed point
 generalized projection.
MSC
References

[1]
R. P. Agarwal, Y. J. Cho, X. Qin, Generalized projection algorithms for nonlinear operators, Numer. Funct. Anal. Optim., 28 (2007), 11971215.

[2]
Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A. G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York (1996)

[3]
B. A. Bin Dehaish, X. Qin, A. Latif, H. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 13211336.

[4]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123145.

[5]
D. Butnariu, S. Reich, A. J. Zaslavski, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim., 24 (2003), 489508.

[6]
Y. Censor, S. Reich, Iterations of paracontractionsand firmly nonexpansive operators with applications to feasibility and optimization, Optimization, 37 (1996), 323339.

[7]
S. Y. Cho, X. Qin, On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems, Appl. Math. Comput., 235 (2014), 430438.

[8]
S. Y. Cho, X. Qin, S. M. Kang , Iterative processes for common fixed points of two different families of mappings with applications, J. Global Optim., 57 (2013), 14291446.

[9]
S. Y. Cho, X. Qin, L. Wang , Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 15 pages.

[10]
B. S. Choudhury, S. Kundu, A viscosity type iteration by weak contraction for approximating solutions of generalized equilibrium problem, J. Nonlinear Sci. Appl., 5 (2012), 243251.

[11]
K. Fan, A minimax inequality and applications, III , (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, (1972), 103113.

[12]
J. Gwinner, F. Raciti, Random equilibrium problems on networks, Math. Comput. Modelling, 43 (2006), 880891.

[13]
R. H. He, Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FCspaces, Adv. Fixed Point Theory, 2 (2012), 4757.

[14]
J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, Appl. Math. Comput., 213 (2009), 498505.

[15]
J. K. Kim , Strong convergence theorems byhybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi\(\phi\)nonexpansive mappings, Fixed Point Theory Appl., 2011 (2011), 15 pages.

[16]
J. K. Kim, P. N. Anh, Y. M. Nam, Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems, J. Korean Math. Soc., 49 (2012), 187200.

[17]
B. Liu, C. Zhang, Strong convergence theorems for equilibrium problems and quasi\(\phi\)nonexpansive mappings , Nonlinear Funct. Anal. Appl., 16 (2011), 365385.

[18]
X. Qin, Y. J. Cho, S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math., 225 (2009), 2030.

[19]
X. Qin, Y. J. Cho, S. M. Kang, On hybrid projection methods for asymptotically quasi\(\phi\)nonexpansive mappings, Appl. Math. Comput., 215 (2010), 38743883.

[20]
X. Qin, L. Wang, On asymptotically quasi\(\phi\)nonexpansive mappings in the intermediate sense, Abst. Appl. Anal., 2012 (2012), 13 pages.

[21]
S. Reich, A weak convergence theorem for the alternating method with Bregman distance, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., Dekker, New York, 178 (1996), 313318.

[22]
W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70 (2009), 4557.

[23]
Z. M. Wang, X. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 2014 (2014), 25 pages.

[24]
Y. Yao, Y. J. Cho, Y. C. Liou, Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems, European J. Oper. Res., 212 (2011), 242250.

[25]
Q. Yu, D. Fang, W. Du , Solving the logitbased stochastic user equilibrium problem with elastic demand based on the extended traffic network model, European J. Oper. Res., 239 (2014), 112118.

[26]
L. Zhang, H. Tong, An iterative method for nonexpansive semigroups, variational inclusions and generalized equilibrium problems, Adv. Fixed Point Theory, 4 (2014), 325343.

[27]
J. Zhao, Strong convergence theorems for equilibrium problems, fixed point problems of asymptotically nonexpansive mappings and a general system of variational inequalities, Nonlinear Funct. Anal. Appl., 16 (2011), 447464.

[28]
L. C. Zhao, S. S. Chang , Strong convergence theorems for equilibrium problems and fixed point problems of strict pseudocontraction mappings, J. Nonlinear Sci. Appl., 2 (2009), 7891.