Strong and weak convergence of Mann iteration of monotone \(\alpha\)-nonexpansive mappings in uniformly convex Banach spaces
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Authors
Yuchun Zheng
- College of Statistics and Mathematics, Yunnan University of Finance and Economics, Longquan Road, Kunming, 650221, P. R. China.
- School of Mathematics and Information Science, Henan Normal University, XinXiang HeNan, 453007, P. R. China.
Lin Wang
- College of Statistics and Mathematics, Yunnan University of Finance and Economics, Longquan Road, Kunming, 650221, P. R. China.
Abstract
In this paper, the demiclosed principle of monotone \(\alpha\)-nonexpansive mapping is showed in a uniformly convex Banach space with the partial order ``\(\leq\)". With the help of such a demiclosed principle, the strong convergence of Mann iteration of monotone \(\alpha\)-nonexpansive mapping \(T\) are proved without some compact conditions such as semi-compactness of \(T\), and the weakly convergent conclusions of such an iteration are studied without the conditions such as Opial's condition. These convergent theorems are obtained under the iterative coefficient satisfying the condition, \[\sum\limits_{k=1}^{+\infty}\min\{\alpha_k,(1-\alpha_k)\}=+\infty,\]
which contains \(\alpha_k=\frac1{k+1}\) as a special case
Share and Cite
ISRP Style
Yuchun Zheng, Lin Wang, Strong and weak convergence of Mann iteration of monotone \(\alpha\)-nonexpansive mappings in uniformly convex Banach spaces, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 9, 1085--1095
AMA Style
Zheng Yuchun, Wang Lin, Strong and weak convergence of Mann iteration of monotone \(\alpha\)-nonexpansive mappings in uniformly convex Banach spaces. J. Nonlinear Sci. Appl. (2018); 11(9):1085--1095
Chicago/Turabian Style
Zheng, Yuchun, Wang, Lin. "Strong and weak convergence of Mann iteration of monotone \(\alpha\)-nonexpansive mappings in uniformly convex Banach spaces." Journal of Nonlinear Sciences and Applications, 11, no. 9 (2018): 1085--1095
Keywords
- Ordered Banach space
- fixed point
- monotone \(\alpha\)-nonexpansive mapping
- strong convergence
MSC
- 47H06
- 47J05
- 47J25
- 47H10
- 47H17
- 49J40
- 65J15
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