Weak convergence theorems for inverse-strongly skew-monotone operators and generalized mixed equilibrium problems in Banach spaces

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作者：Junmin Chen (1)
Tiegang Fan (1) (2)

1. College of Mathematics and Computer ; Hebei University ; Baoding ; 071002 ; China
2. College of Mechanical and Electrical Engineering ; Hebei Agricultural University ; Baoding ; 071001 ; China
关键词：47H05 ; 47H09 ; 47J25 ; generalized nonexpansive type mapping ; generalized mixed equilibrium problem ; maximal monotone operator ; inverse ; strongly skew ; monotone operator
刊名：Fixed Point Theory and Applications
出版年：2015
出版时间：December 2015
年：2015
卷：2015
期：1
全文大小：1,201 KB
参考文献：1. Iiduka, H, Takahashi, W, Toyoda, M: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 14strong>, 417-429 (2004)
2. Iiduka, H, Takahashi, W: Weak convergence of a projection algorithm for variational inequalities in a Banach spaces. J.聽Math. Anal. Appl. 339strong>, 668-679 (2008) CrossRef
3. Plubtieng, S, Sriprad, W: Existence and approximation of solution of the variational inequality problem with a skew monotone operator defined on the dual spaces of Banach spaces. J. Nonlinear Anal. Optim. 1strong>, 23-33 (2010)
4. Ibaraki, T, Takahashi, W: A new projection and convergence theorems for the projections in Banach spaces. J. Approx. Theory 149strong>, 1-14 (2007) CrossRef
5. Takahashi, W, Zembayashi, K: A strong convergence theorem for the equilibrium problem with a bifunction defined on the dual space of a Banach space. In: Proceedings of the 8th International Conference on Fixed Point Theory and Its Applications, pp. 197-209. Yokohama Publishers, Yokohama (2008)
6. Takahashi, S, Takahashi, W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69strong>, 1025-1033 (2008) CrossRef
7. Peng, JW, Yao, JC: Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping. J. Glob. Optim. 46strong>, 331-345 (2010) CrossRef
8. Zeng, LC, Wu, SY, Yao, JC: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwan. J. Math. 10strong>, 1497-1514 (2006)
9. Peng, JW, Yao, JC: A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems. Taiwan. J. Math. 12strong>, 1401-1433 (2008)
10. Peng, JW, Yao, JC: A new extragradient method for mixed equilibrium problems, fixed point problems and variational inequality problems. Math. Comput. Model. 49strong>, 1816-1828 (2009) CrossRef
11. Zeng, LC, Ansari, QH, Yao, JC: Viscosity approximation methods for generalized equilibrium problems and fixed point problems. J. Glob. Optim. 43strong>, 487-502 (2009) CrossRef
12. Qin, X, Cho, SY, Kang, SM: Strong convergence of shrinking projection methods for quasi- / 蠒-nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 234strong>, 750-760 (2010) CrossRef
13. Saewan, S, Cho, YJ, Kumam, P: Weak and strong convergence theorems for mixed equilibrium problems in Banach spaces. Optim. Lett. 8strong>, 501-518 (2014) CrossRef
14. Cho, YJ, Argyros, IK, Petrot, N: Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Comput. Math. Appl. 50strong>, 2292-2301 (2010) CrossRef
15. Qin, X, Chang, SS, Cho, YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal., Real World Appl. 11strong>, 2963-2972 (2010) CrossRef
16. Qin, X, Cho, YJ, Kang, SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225strong>, 20-30 (2009) CrossRef
17. Takahashi, W, Zembayashi, K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70strong>, 45-57 (2009) CrossRef
18. Takahashi, W, Yao, JC: Nonlinear operators of monotone type and convergence theorems with equilibrium problems in Banach spaces. Taiwan. J. Math. 15strong>, 787-818 (2011)
19. Yao, Y, Cho, YJ, Chen, R: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Anal. 71strong>, 3363-3373 (2009) CrossRef
20. Yao, Y, Cho, YJ, Liou, Y: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 212strong>, 242-250 (2011) CrossRef
21. Cai, G, Bu, S: Weak convergence theorems for general equilibrium problems and variational inequality problems and fixed point problems in Banach spaces. Acta Math. Sci. Ser. B 33strong>, 303-320 (2013) CrossRef
22. Ceng, LC, Petru, A, Yao, JC: Composite viscosity approximation methods for equilibrium problem, variational inequality and common fixed points. J. Nonlinear Convex Anal. 15strong>, 219-240 (2014)
23. Al-Mazrooei, AE, Latif, A, Yao, JC: Solving generalized mixed equilibria, variational inequalities and constrained convex minimization. Abstr. Appl. Anal. 2014strong>, Article ID 587865 (2014)
24. Ceng, LC, Yao, JC: On the triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusion and system of generalized equilibria. Tamkang J. Math. 45strong>(3), 297-334 (2014) CrossRef
25. Alber, YI: Metric and generalized projection in Banach space: properties and applications. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp.聽15-50. Dekker, New York (1996)
26. Su, Y, Xu, HK, Wang, Z: Strong convergence theorem for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Anal. 71strong>, 5616-5628 (2009) CrossRef
27. Xu, ZB, Roach, GF: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157strong>, 189-210 (1991) CrossRef
28. Kamimura, S, Takahashi, W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13strong>, 938-945 (2002) CrossRef
29. Kohsaha, F, Takahashi, W: Generalized nonexpansive retractions and a proximal-type algorithm in Banach spaces. J.聽Nonlinear Convex Anal. 8strong>, 197-209 (2007)
30. Rockafellar, RT: Maximal monotone operators and proximal point algorithm. SIAM J. Control Optim. 14strong>, 887-898 (1976)
31. Cho, YJ, Zhou, HY, Guo, G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 47strong>, 707-717 (2004) CrossRef
32. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by Ishikawa iteration process. J. Math. Anal. Appl. 178strong>, 301-308 (1993) CrossRef
33. Phuangphoo, P, Kumam, P: Existence and modification of Halpern-Mann iterations for fixed point and generalized mixed equilibrium problems with a bifunction defined on the dual space. J. Appl. Math. 2013strong>, Article ID 753096 (2013) CrossRef
刊物主题：Analysis; Mathematics, general; Applications of Mathematics; Differential Geometry; Topology; Mathematical and Computational Biology;
出版者：Springer International Publishing
ISSN：1687-1812

文摘

In this paper, we consider an iterative algorithm for finding the common element of the set of solutions for the generalized mixed equilibrium problems, the common fixed points set of two generalized nonexpansive type mappings, and the set of solutions of the variational inequality for an inverse-strongly skew-monotone operator in Banach spaces. Under mild conditions, the weak convergence theorem is established by using the sunny generalized nonexpansive retraction in Banach spaces. Our results refine, supplement, and extend the corresponding results in (Saewan et al. in Optim. Lett. 8:501-518, 2014), and other results announced by many other authors.

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