Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mapping in complete CAT(0) spaces
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Authors
Yi Li
- School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China.
Hongbo Liu
- School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China.
Abstract
In this paper, the implicit midpoint rule of asymptotically nonexpansive mapping in CAT(0) spaces is introduced. By the
viscosity approximation method, we prove that the proposed implicit iteration converges strongly to a fixed point of asymptotically
nonexpansive mapping under certain assumptions imposed on the sequence of parameters. The results presented in the
paper improve and extend various results in the existing literature.
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ISRP Style
Yi Li, Hongbo Liu, Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mapping in complete CAT(0) spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 3, 1270--1280
AMA Style
Li Yi, Liu Hongbo, Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mapping in complete CAT(0) spaces. J. Nonlinear Sci. Appl. (2017); 10(3):1270--1280
Chicago/Turabian Style
Li, Yi, Liu, Hongbo. "Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mapping in complete CAT(0) spaces." Journal of Nonlinear Sciences and Applications, 10, no. 3 (2017): 1270--1280
Keywords
- viscosity approximation methods
- complete CAT(0) space
- \(\Delta\)-convergence
- Asymptotically nonexpansive
- implicit iteration.
MSC
References
-
[1]
M. Abbas, B. S. Thakur, D. Thakur, Fixed points of asymptotically nonexpansive mappings in the intermediate sense in CAT(0) spaces, Commun. Korean Math. Soc., 28 (2013), 107–121.
-
[2]
B. Ahmadi Kakavandi, M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal., 73 (2010), 3450–3455.
-
[3]
M. A. Alghamdi, M. A. Alghamdi, N. Shahzad, H.-K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014 ), 9 pages.
-
[4]
H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769–806.
-
[5]
W. Auzinger, R. Frank, Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math., 56 (1989), 469–499.
-
[6]
G. Bader, P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), 373–398.
-
[7]
I. D. Berg, I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218.
-
[8]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123–145.
-
[9]
M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1999)
-
[10]
K. S. Brown, Buildings, Springer-Verlag, New York (1989)
-
[11]
F. Bruhat, J. Tits, Groupes réductifs sur un corps local, (French) Inst. Hautes Etudes Sci. Publ. Math., 41 (1972), 5–251.
-
[12]
P. Chaoha, A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 320 (2006), 983–987.
-
[13]
H. Dehghan, J. Rooin, A characterization of metric projection in CAT (0) spaces, ArXiv, 2013 (2013 ), 3 pages.
-
[14]
S. Dhompongsa, B. Panyanak, On \(\Delta\)-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572– 2579.
-
[15]
K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1984)
-
[16]
W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689–3696.
-
[17]
L. Leuştean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, Nonlinear analysis and optimization I. Nonlinear analysis, Contemp. Math., Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 513 (2010), 193–210.
-
[18]
T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179–182.
-
[19]
X.-D. Liu, S.-S. Chang, Viscosity approximation methods for hierarchical optimization problems in CAT(0) spaces, J. Inequal. Appl., 2013 (2013 ), 14 pages.
-
[20]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
-
[21]
S. Somalia, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (2002), 327–332.
-
[22]
M. van Veldhuizen, Asymptotic expansions of the global error for the implicit midpoint rule (stiff case), Computing, 33 (1984), 185–192.
-
[23]
R.Wangkeeree, P. Preechasilp, Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013 ), 15 pages.
-
[24]
R. Wangkeeree, P. Preechasilp, Viscosity approximation methods for nonexpansive semigroups in CAT(0) spaces, Fixed Point Theory Appl., 2013 (2013 ), 16 pages.
-
[25]
H.-K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659–678.
-
[26]
H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
-
[27]
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.
-
[28]
L. Yang, F. H. Zhao, Strong and \(\Delta\)-convergence theorems for total asymptotically nonexpansive nonself mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013 ), 17 pages.
-
[29]
L.-C. Zhao, S.-S. Chang, C.-F. Wen, Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces, J. Nonlinear Sci. Appl., 9 (2016), 4478–4488.