Strong convergence of Halpern method for firmly type nonexpansive mappings
-
1980
Downloads
-
3443
Views
Authors
Yonghong Yao
- Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Sichuan 610054, China.
Mihai Postolache
- China Medical University, Taichung, Taiwan.
- University Politehnica of Bucharest, Bucharest, Romania.
Naseer Shahzad
- Department of Mathematics, King Abdulaziz University, P. O. B. 80203, Jeddah 21589, Saudi Arabia.
Abstract
In this paper, Halpern method is applied to find fixed points of a
class of firmly type nonexpansive mappings. A strong convergence
result is obtained under the control conditions (C1) and (C2).
Our conclusion obtained in this paper gives the affirmative answer
of the Halpern open problem for this class of mapping.
Share and Cite
ISRP Style
Yonghong Yao, Mihai Postolache, Naseer Shahzad, Strong convergence of Halpern method for firmly type nonexpansive mappings, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 11, 5932--5938
AMA Style
Yao Yonghong, Postolache Mihai, Shahzad Naseer, Strong convergence of Halpern method for firmly type nonexpansive mappings. J. Nonlinear Sci. Appl. (2017); 10(11):5932--5938
Chicago/Turabian Style
Yao, Yonghong, Postolache, Mihai, Shahzad, Naseer. "Strong convergence of Halpern method for firmly type nonexpansive mappings." Journal of Nonlinear Sciences and Applications, 10, no. 11 (2017): 5932--5938
Keywords
- Fixed point
- Halpern method
- firmly type nonexpansive mappings
- strong convergence
MSC
References
-
[1]
F. E. Browder, Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces, Arch. Rational Mech. Anal., 24 (1967), 82–90.
-
[2]
L. C. Ceng, N. C. Wong, J. C. Yao, Strong and weak convergence theorems for an infinite family of nonexpansive mappings and applications, Fixed Point Theory Appl., 2012 (2012), 21 pages.
-
[3]
Y.-L. Cui, X. Liu , Notes on Browder’s and Halpern’s methods for nonexpansive maps, Fixed Point Theory, 10 (2009), 89–98.
-
[4]
A. Genel, J. Lindenstrass , An example concerning fixed points, Israel J. Math., 22 (1975), 81–86.
-
[5]
K. Goebel, W. A. Kirk , Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)
-
[6]
B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 73 (1967), 957–961.
-
[7]
P.-L. Lions , Approximation de points fixes de contractions, R. Acad. Sci. Paris Sr. A-B, 284 (1977), 1357–1359.
-
[8]
P.-E. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469–479.
-
[9]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
-
[10]
Z. Opial, Weak convergence of the sequence of successive approximations of nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597.
-
[11]
N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces , Proc. Amer. Math. Soc., 125 (1997), 3641–3645.
-
[12]
Y.-S. Song, X.-K. Chai, Halpern iteration for firmly type nonexpansive mappings, Nonlinear Anal., 71 (2009), 4500–4506.
-
[13]
T. Suzuki , Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 10 pages.
-
[14]
R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math., 58 (1992), 486–491.
-
[15]
H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109–113.
-
[16]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
-
[17]
H.-K. Xu , Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
-
[18]
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.
-
[19]
Y.-H. Yao, R. P. Agarwal, M. Postolache, Y.-C. Liou, Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., 2014 (2014), 14 pages.
-
[20]
Y.-H. Yao, R.-D. Chen, J.-C. Yao, Strong convergence and certain control conditions for modified Mann iteration, Nonlinear Anal., 68 (2008), 1687–1693.
-
[21]
Y.-H. Yao, Y.-C. Liou, T.-L. Lee, N.-C. Wong, An iterative algorithm based on the implicit midpoint rule for nonexpansive mappings, J. Nonlinear Convex Anal., 17 (2016), 655–668.
-
[22]
Y.-H. Yao, M. Postolache, Iterative methods for pseudomonotone variational inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2012), 273–287.
-
[23]
Y.-H. Yao, M. Postolache, S. M. Kang , Strong convergence of approximated iterations for asymptotically pseudocontractive mappings, Fixed Point Theory Appl., 2014 (2014), 13 pages.
-
[24]
Y.-H. Yao, N. Shahzad, Y.-C. Liou, Modified semi-implicit midpoint rule for nonexpansive mappings , Fixed Point Theory Appl., 2015 (2015), 15 pages.
