Strong convergence of a general iterative algorithm for asymptotically nonexpansive semigroups in Banach spaces
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Authors
Lishan Liu
- School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China.
- Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia.
Chun Liu
- School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China.
Fang Wang
- School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China.
Yonghong Wu
- Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia.
Abstract
In this paper, we study a general iterative process strongly converging to a common fixed point of an
asymptotically nonexpansive semigroup \(\{T(t) : t \in \mathbb{R }^+\}\) in the framework of reflexive and strictly convex
spaces with a uniformly Gáteaux differentiable norm. The process also solves some variational inequalities.
Our results generalize and extend many existing results in the research field.
Share and Cite
ISRP Style
Lishan Liu, Chun Liu, Fang Wang, Yonghong Wu, Strong convergence of a general iterative algorithm for asymptotically nonexpansive semigroups in Banach spaces, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 10, 5695--5711
AMA Style
Liu Lishan, Liu Chun, Wang Fang, Wu Yonghong, Strong convergence of a general iterative algorithm for asymptotically nonexpansive semigroups in Banach spaces. J. Nonlinear Sci. Appl. (2016); 9(10):5695--5711
Chicago/Turabian Style
Liu, Lishan, Liu, Chun, Wang, Fang, Wu, Yonghong. "Strong convergence of a general iterative algorithm for asymptotically nonexpansive semigroups in Banach spaces." Journal of Nonlinear Sciences and Applications, 9, no. 10 (2016): 5695--5711
Keywords
- Asymptotically nonexpansive semigroups
- variational inequality
- strong convergence
- reflexive and strictly convex Banach spaces
- fixed point.
MSC
References
-
[1]
K. Aoyama, F. Kohsaka, Viscosity approximation process for a sequence of quasinonexpansive mappings, Fixed Point Theory and Appl., 2014 (2014), 11 pages
-
[2]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103--120
-
[3]
G. Cai, S. Q. Bu, A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems, Comput. Math. Appl., 62 (2011), 440--454
-
[4]
G. Cai, S. Q. Bu, Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces, Math. Comput. Modelling, 55 (2012), 538--546
-
[5]
G. Cai, S. Q. Bu, A viscosity scheme for mixed equilibrium problems, variational inequality problems and fixed point problems, Math. Comput. Modelling, 57 (2013), 1212--1226
-
[6]
K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171--174
-
[7]
K. S. Ha, J. S. Jung, On generators and nonlinear semigroups in Banach spaces, J. Korean Math. Soc., 25 (1988), 245--257
-
[8]
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957--961
-
[9]
I. Inchan, S. Plubtieng, Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. Hybrid Syst., 2 (2008), 1125--1135
-
[10]
A. R. Khan, H. Fukhar-ud-din, A. Kalsoom, B. S. Lee, Convergence of a general algorithm of asymptotically nonexpansive maps in uniformly convex hyperbolic spaces, Appl. Math. Comput., 238 (2014), 547--556
-
[11]
X. N. Li, J. S. Gu, Strong convergence of modified Ishikawa iteration for a nonexpansive semigroup in Banach spaces, Nonlinear Anal., 73 (2010), 1058--1092
-
[12]
L. S. Liu, Fixed points of local strictly pseudo-contractive mappings using Mann and Ishikawa iteration with errors, Indian J. Pure Appl. Math., 26 (1995), 649--659
-
[13]
L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114--125
-
[14]
L. S. Liu, Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m-accretive operators in Banach spaces, Nonlinear Anal., 34 (1998), 307--317
-
[15]
J. Lou, L.-J. Zhang, Z. He, Viscosity approximation methods for asymptotically nonexpansive mappings, Appl. Math. Comput., 203 (2008), 171--177
-
[16]
G. Marino, V. Colao, X. L. Qin, S. M. Kang, Strong convergence of the modified Mann iterative method for strict pseudo-contractions, Comput. Math. Appl., 57 (2009), 455--465
-
[17]
G. Marino, H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43--52
-
[18]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46--55
-
[19]
C. I. Podilchuk, R. J. Mammone, Image recovery by convex projections using a least-squares constrain, J. Optical Soc. Amer. A, 7 (1990), 517--521
-
[20]
X. L. Qin, M. J. Shang, S. M. Kang, Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 70 (2009), 1257--1264
-
[21]
G. S. Saluja, M. Postolache, A. Kurdi, Convergence of three-step iterations for nearly asymptotically nonexpansive mappings in CAT(k) spaces, J. Inequal. Appl., 2015 (2015), 18 pages
-
[22]
M. I. Sezan, H. Stark, Applications of convex projection theory to image recovery in tomography and related areas, H. Stark (Ed.), Image Recovery: Theory and Applications, Academic, 1987 (1987), 415--462
-
[23]
Y. S. Song, S. M. Xu, Strong convergence theorems for nonexpansive semigroup in Banach spaces, J. Math. Anal. Appl., 338 (2008), 152--161
-
[24]
Y. F. Su, S. H. Li, Strong convergence theorems on two iterative method for non-expansive mappings, Appl. Math. Comput., 181 (2006), 331--341
-
[25]
B. S. Thakur, R. Dewangan, M. Postolache, Strong convergence of new iteration process for a strongly continuous semigroup of asymptotically pseudocontractive mappings, Numer. Funct. Anal. Optim., 34 (2013), 1418--1431
-
[26]
D. Thakur, B. S. Thakur, M. Postolache, New iteration scheme for numerical reckoning fixed points of nonexpansive mappings, J. Inequal. Appl., 2014 (2014), 15 pages
-
[27]
B. S. Thakur, D. Thakur, M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat, ((in press)),
-
[28]
S. Thianwan, Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space, J. Comput. Appl. Math., 224 (2009), 688--695
-
[29]
Y. X. Tian, S. S. Chang, J. L. Huang, On the approximation problem of common fixed points for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces, Comput. Math. Appl., 53 (2007), 1847--1853
-
[30]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240--256
-
[31]
H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279--291
-
[32]
L.-P. Yang, The general iterative scheme for semigroups of nonexpansive mappings and variational inequalities with applications, Math. Comput. Modelling, 57 (2013), 1289--1297
-
[33]
L.-P. Yang, W.-M. Kong, A general iterative algorithm for semigroups of nonexpansive mappings with generalized contractive mapping, Appl. Math. Comput., 222 (2013), 671--679
-
[34]
Y. H. Yao, M. Postolache, Iterative methods for pseudomonotone variational inequalities and fixed-point problems, J. Optim. Theory Appl., 155 (2012), 273--287
-
[35]
Y. H. Yao, M. Postolache, Y.-C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory and Appl., 2013 (2013), 12 pages
-
[36]
Y. H. Yao, M. Postolache, Y.-C. Liou, Variant extragradient-type method for monotone variational inequalities, Fixed Point Theory and Appl., 2013 (2013), 15 pages
-
[37]
D. C. Youla, Mathematical theory of image restoration by the method of convex projections, H. Stark (Ed.), Image Recovery: Theory and Applications, Academic, New York, 1987 (1987), 29--77
-
[38]
D. C. Youla, On deterministic convergence of iterations of relaxed projection operators, J. Vis. Commun. Image Represent., 1 (1990), 12--20
-
[39]
H. Zegeye, N. Shahzad, Convergence theorems for strongly continuous semi-groups of asymptotically nonexpansive mappings, Nonlinear Anal., 71 (2009), 2308--2315
-
[40]
H. Zegeye, N. Shahzad, Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), 4007--4014