Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces
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Authors
Liang-cai Zhao
- College of Mathematics, Yibin University, Yibin, Sichuan, 644007, China.
Shih-sen Chang
- Center for General Education, China Medical University, Taichung, 40402, Taiwan.
Lin Wang
- College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China.
Gang Wang
- College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China.
Abstract
The purpose of this paper is to introduce the implicit midpoint rule of nonexpansive mappings in CAT(0) spaces. The strong
convergence of this method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown
that the limit of the sequence generated by the implicit midpoint rule solves an additional variational inequality. Applications
to nonlinear Volterra integral equations and nonlinear variational inclusion problem are included. The results presented in the
paper extend and improve some recent results announced in the current literature.
Share and Cite
ISRP Style
Liang-cai Zhao, Shih-sen Chang, Lin Wang, Gang Wang, Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 2, 386--394
AMA Style
Zhao Liang-cai, Chang Shih-sen, Wang Lin, Wang Gang, Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces. J. Nonlinear Sci. Appl. (2017); 10(2):386--394
Chicago/Turabian Style
Zhao, Liang-cai, Chang, Shih-sen, Wang, Lin, Wang, Gang. "Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces." Journal of Nonlinear Sciences and Applications, 10, no. 2 (2017): 386--394
Keywords
- Viscosity
- implicit midpoint rule
- nonexpansive mapping
- projection
- variational inequality
- CAT(0) space.
MSC
References
-
[1]
B. Ahmadi Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc., 141 (2013), 1029– 1039.
-
[2]
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.
-
[3]
H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769–806.
-
[4]
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.
-
[5]
G. Bader, P. Deuflhard , A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), 373–398.
-
[6]
S. Banach, Metric spaces of nonpositive curvature, Springer, New York (1999)
-
[7]
I. D. Berg, I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218.
-
[8]
M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1999)
-
[9]
K. S. Brown, Buildings, Springer-Verlag, New York (1989)
-
[10]
S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013 ), 14 pages.
-
[11]
H. Dehghan, J. Rooin, A characterization of metric projection in CAT(0) spaces, In: International Conference on Functional Equation, Geometric Functions and Applications (ICFGA 2012), 10-12th May 2012, pp. 41-43. Payame Noor University, Tabriz (2012)
-
[12]
S. Dhompongsa, W. A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 35–45.
-
[13]
S. Dhompongsa, B. Panyanak, On \(\Delta\)-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572– 2579.
-
[14]
W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689–3696.
-
[15]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
-
[16]
S. Saejung, Halpern’s iteration in CAT(0) spaces, Fixed Point Theory Appl., 2010 (2010 ), 13 pages.
-
[17]
C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., 1 (1993), 1–10.
-
[18]
S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (2002), 327–332.
-
[19]
R.Wangkeeree, P. Preechasilp, Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013 ), 15 pages.
-
[20]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
-
[21]
H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
-
[22]
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.
-
[23]
Y.-H. Yao, N. Shahzad, N.-C. Liou, Modified semi-implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015 ), 15 pages.
-
[24]
S.-S. Zhang, Integral equations, Chongqing press, Chongqing (1984)
-
[25]
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.
