Some results on asymptotically quasi-phi-nonexpansive mappings in the intermediate sense and Ky Fan inequalities
-
1551
Downloads
-
2168
Views
Authors
Hongwei Liang
- School of Mathematics and Statistics, Henan University, Kaifeng 475000, China.
Mingliang Zhang
- School of Mathematics and Statistics, Henan University, Kaifeng 475000, China.
Abstract
In this paper, we study asymptotically quasi-\(\phi\)- nonexpansive mappings in the intermediate sense and Ky
Fan inequalities. A convergence theorem is established in a strictly convex and uniformly smooth Banach
space. The results presented in the paper improve and extend some recent results.
Share and Cite
ISRP Style
Hongwei Liang, Mingliang Zhang, Some results on asymptotically quasi-phi-nonexpansive mappings in the intermediate sense and Ky Fan inequalities, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 4, 1675--1684
AMA Style
Liang Hongwei, Zhang Mingliang, Some results on asymptotically quasi-phi-nonexpansive mappings in the intermediate sense and Ky Fan inequalities. J. Nonlinear Sci. Appl. (2016); 9(4):1675--1684
Chicago/Turabian Style
Liang, Hongwei, Zhang, Mingliang. "Some results on asymptotically quasi-phi-nonexpansive mappings in the intermediate sense and Ky Fan inequalities." Journal of Nonlinear Sciences and Applications, 9, no. 4 (2016): 1675--1684
Keywords
- Asymptotically nonexpansive mapping
- quasi-\(\phi\)-nonexpansive mapping
- fixed point
- convergence theorem.
MSC
References
-
[1]
R. P. Agarwal, Y. J. Cho, X. Qin , Generalized projection algorithms for nonlinear operators, Numer. Funct. Anal. Optim., 28 (2007), 1197-1215.
-
[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), 1321-1336.
-
[4]
E. Blum, W. Oettli , From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.
-
[5]
R. E. Bruck, T. Kuczumow, S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. Math., 65 (1993), 169-179.
-
[6]
D. Butnariu, S. Reich, A. J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal., 7 (2001), 151-174.
-
[7]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120
-
[8]
G. Cai, S. Bu , Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces, J. Comput. Appl. Math., 247 (2013), 34-52.
-
[9]
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), 430-438.
-
[10]
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.
-
[11]
W. Cholamjiak, P. Cholamjiak, S. Suantai, Convergence of iterative schemes for solving fixed point problems for multi-valued nonself mappings and equilibrium problems, J. Nonlinear Sci. Appl., 8 (2015), 1245-1256.
-
[12]
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), 243-251.
-
[13]
I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht (1990)
-
[14]
S. Dafermos, A. Nagurney, A network formulation of market equilibrium problems and variational inequalities, Oper. Res. Lett., 3 (1984), 247-250.
-
[15]
Y. Hao, On generalized quasi-\(\phi\)-nonexpansive mappings and their projection algorithms, Fixed Point Theory Appl., 2013 (2013), 13 pages.
-
[16]
Y. Hao, Some results on a modified Mann iterative scheme in a reflexive Banach space, Fixed Point Theory Appl., 2013 (2013), 14 pages.
-
[17]
R. H. He, Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces, Adv. Fixed Point Theory, 2 (2012), 47-57.
-
[18]
H. Iiduka, Fixed point optimization algorithm and its application to network bandwidth allocation, J. Comput. Appl. Math., 236 (2012), 1733-1742.
-
[19]
J. K. Kim, Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-\(\phi\)-nonexpansive mappings, Fixed Point Theory Appl., 2011 (2011), 15 pages.
-
[20]
B. Liu, C. Zhang, Strong convergence theorems for equilibrium problems and quasi-\(\phi\)-nonexpansive mappings, Nonlinear Funct. Anal. Appl., 16 (2011), 365-385.
-
[21]
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), 20-30.
-
[22]
X. Qin, S. Y. Cho, S. M. Kang, On hybrid projection methods for asymptotically quasi-\(\phi\)-nonexpansive mappings, Appl. Math. Comput., 215 (2010), 3874-3883.
-
[23]
X. Qin, S. Y. Cho, L. Wang, Algorithms for treating equilibrium and fixed point problems, Fixed Point Theory Appl., 2013 (2013), 15 pages.
-
[24]
X. Qin, L. Wang, On asymptotically quasi-\(\phi\)-nonexpansive mappings in the intermediate sense, Abst. Appl. Anal., 2012 (2012), 14 pages.
-
[25]
J. Shen, L. P. Pang, An approximate bundle method for solving variational inequalities, Commun. Optim. Theory, 1 (2012), 1-18.
-
[26]
T. Takahashi, Nonlinear Functional Analysis, Yokohama-Publishers, Tokoyo (2000)
-
[27]
N. T. T. Thuy, Convergence rate of the Tikhonov regularization for ill-posed mixed variational inequalities with inverse-strongly monotone perturbations , Nonlinear Funct. Anal. Appl., 5 (2010), 467-479.
-
[28]
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.
-
[29]
H. Zegeye, N. Shahzad , Strong convergence theorem for a common point of solution of variational inequality and fixed point problem, Adv. Fixed Point Theory, 2 (2012), 374-397.
-
[30]
M. Zhang, Iterative algorithms for a system of generalized variational inequalities in Hilbert spaces, Fixed Point Theory Appl., 2012 (2012), 14 pages.
-
[31]
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), 447-464.