The generalized viscosity implicit rule of nonexpansive semigroup in Banach spaces
-
2342
Downloads
-
5069
Views
Authors
Chaichana Jaiboon
- Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin, Nakhon Pathom 73170, Thailand.
Somyot Plubtieng
- Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand.
Phayap Katchang
- Rajamangala University of Technology Lanna Tak, Division of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand, Tak 63000, Thailand.
Abstract
In this research, we focus on a common fixed point problem of a
nonexpansive semigroup with the generalized viscosity methods for
implicit iterative algorithms. Our main objective is to construct
the new strong convergence theorems under certain appropriate
conditions in uniformly convex and uniformly smooth Banach spaces.
Specifically, the main results make a contribution to the implicit
midpoint theorems. The findings for theorems in Hilbert spaces and
the other forms of a nonexpansive semigroup can be used in several
practical purposes. Finally, a numerical example in 3 dimensions is
provided to support our main results.
Share and Cite
ISRP Style
Chaichana Jaiboon, Somyot Plubtieng, Phayap Katchang, The generalized viscosity implicit rule of nonexpansive semigroup in Banach spaces, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 6, 746--761
AMA Style
Jaiboon Chaichana, Plubtieng Somyot, Katchang Phayap, The generalized viscosity implicit rule of nonexpansive semigroup in Banach spaces. J. Nonlinear Sci. Appl. (2018); 11(6):746--761
Chicago/Turabian Style
Jaiboon, Chaichana, Plubtieng, Somyot, Katchang, Phayap. "The generalized viscosity implicit rule of nonexpansive semigroup in Banach spaces." Journal of Nonlinear Sciences and Applications, 11, no. 6 (2018): 746--761
Keywords
- Nonexpansive semigroup
- fixed point
- generalized viscosity
- implicit
- Banach space
MSC
References
-
[1]
S. Atsushiba, W. Takahashi , Strong convergence of Mann’s-type iterations for nonexpansive semigroups in general Banach spaces, Nonlinear Anal., 61 (2005), 881–899.
-
[2]
J.-P. Gossez, E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math., 40 (1972), 565–573.
-
[3]
W.-B. Guan , An iterative method for variational inequality problems, J. Inequal. Appl., 2013 (2013), 10 pages.
-
[4]
Y. Ke, C. Ma , The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), 21 pages.
-
[5]
A. T.-M. Lau, Amenability and fixed point property for semigroup of nonexpansive mapping, Fixed point theory Appl. (Marseille, (1989)), Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow (1991)
-
[6]
A. T.-M. Lau, Y. Zhang , Fixed point properties of semigroups of non-expansive mappings, J. Funct. Anal., 254 (2008), 2534–2554.
-
[7]
L.-C. Lim , On characterizations of Meir-Keeler contractive maps , Nonlinear Anal., 46 (2001), 113–120.
-
[8]
P. Sunthrayuth, P. Kumam, Viscosity approximation methods based on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces, Math. Comput. Modelling, 58 (2013), 1814–1828.
-
[9]
T. Suzuki , Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227–239.
-
[10]
T. Suzuki, Moudafi’s viscosity approximations with MeirKeeler contractions, J. Math. Anal. Appl., 325 (2007), 342–352.
-
[11]
Y. Wang, H.-K. Xu, Hybrid method for a class of accretive variational inequalities involving nonexpansive mappings, J. Inequal. Appl., 2014 (2014), 9 pages.
-
[12]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
-
[13]
H.-K. Xu, M. A. Alghamdi, N. Shahzad, The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), 12 pages.
-
[14]
Q. Yan, G. Cai, P. Luo , Strong convergence theorems for the generalized viscosity implicit rules of nonexpansive mappings in uniformly smooth Banach spaces, J. Nonlinear Sci. Appl., 9 (2016), 4039–4051.