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
References
-
[1]
R. P. Agarwal, D. O’Regan, D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Springer, New York (2009)
-
[2]
M. Bachar, M. A Khamsi, On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces, Fixed Point Theory Appl., 2015 (2015), 11 pages.
-
[3]
V. Berinde, A convergence theorem for Mann iteration in the class of Zamfirescu operators, An. Univ. Vest Timi. Ser. Mat.-Inform., 45 (2007), 33–41.
-
[4]
B. A. B. Dehaish, M. A. Khamsi, Mann iteration process for monotone nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015), 7 pages.
-
[5]
K. Deimling, Nonlinear Functional Analysis, Dover Publications Inc., New York (2010)
-
[6]
A. B. George, C. A. Nse, A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in Hilbert Spaces, Appl. Math., 5 (2014), 2195–2198.
-
[7]
S. George, P. Shaini, Convergence theorems for the class of Zamfirescu operators, Int. Math. Forum, 7 (2012), 1785–1792.
-
[8]
K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge (1990)
-
[9]
F. Gu, J. Lu, Stability of Mann and Ishikawa iterative processes with errors for a class of nonlinear variational inclusions problem, Math. Commun., 9 (2004), 149–159.
-
[10]
J. S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 302 (2005), 509–520.
-
[11]
J. K. Kim, Z. Liu, Y. M. Nam, S. A. Chun, Strong convergence theorems and stability problems of Mann and Ishikawa iterative sequences for strictly hemi-contractive mappings, J. Nonlinear Convex Anal., 5 (2004), 285–294.
-
[12]
W. A. Kirk, B. Sims, Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht (2001)
-
[13]
F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim., 19 (2008), 824–835.
-
[14]
L.-J. Lin, S.-Y. Wang, Fixed point theorems for (a, b)-monotone mapping mapping in Hilbert spaces, Fixed Point Theory Appl., 2012 (2012), 14 pages.
-
[15]
L. S. Liu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114–125.
-
[16]
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510.
-
[17]
E. Naraghirad, N.-C. Wong, J.-C. Yao, Approximating fixed points of \(\alpha\)-nonexpansive mappings in uniformly convex Banach spaces and CAT(0) spaces, Fixed Point Theory Appl., 2013 (2013), 20 pages.
-
[18]
G. A. Okeke, J. K. Kim, Convergence and summable almost T-stability of the random Picard-Mann hybrid iterative process, J. Inequal. Appl., 2015 (2015), 14 pages.
-
[19]
Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings in Banach spaces, Bull. Amer. Math. Soc., 73 (1967), 591–597.
-
[20]
A. Pazy, Asymptotic behavior of contractions in Hilbert spaces, Israel J. Math., 9 (1971), 335–340.
-
[21]
Y. S. Song, R. D. Chen, Pazy’s fixed point theorem with respect to the partial order in uniformly convex Banach spaces, preprint arXiv, (2016), 17 pages.
-
[22]
Y. S. Song, R. D. Chen, Weak and strong convergence of Mann’s-type iterations for a countable family of nonexpansive mappings, J. Korean Math. Soc., 45 (2008), 1393–1404.
-
[23]
Y. S. Song, P. Kumam, Y. J. Cho, Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces, Fixed Point Theory Appl., 2016 (2016), 11 pages.
-
[24]
Y. S. Song, K. Promluang, P. Kumam, Y. J. Cho, Some convergence theorems of the Mann iteration for monotone \(\alpha\)- nonexpansive mappings, Appl. Math. Comput., 287/288 (2016), 74–82.
-
[25]
Y. S. Song, H. J. Wang, Strong convergence for the modified Mann’s iteration of \(\lambda\)-strict pseudocontraction, Appl. Math. Comput., 237 (2014), 405–410.
-
[26]
J. Sun, Nonlinear Functional Analysis and Its Application, Science Publishing House, Beijing (2008)
-
[27]
T. Suzuki, Strong convergence of Krasnoselskii and Mann’s sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227–239.
-
[28]
W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal., 11 (2010), 79–88.
-
[29]
W. Takahashi, Nonlinear Functional Analysis–Fixed Point Theory and its Applications, Yokohama Publishers inc, Yokohama (2000)
-
[30]
W. Takahashi, J. C. Yao, Fixed point theorems and ergodic theorems for nonlinear mappings in a Hilbert space, Taiwan. J. Math., 15 (2011), 457–472.
-
[31]
H. Zhang, Y. Su, Convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces, Nonlinear Anal., 71 (2009), 4572–4580.
-
[32]
H. Y. Zhou, Convergence theorems for \(\lambda\)-strict pseudo-contractions in 2-uniformly smooth Banach spaces, Nonlinear Anal., 69 (2008), 3160–3173.
-
[33]
H. Y. Zhou, M. H. Zhang, H. Y. Zhou, A convergence theorem on Mann iteration for strictly pseudo-contraction mappings in Hilbert spaces (Chinese), J. Hebei Univ. Nat. Sci., 26 (2006), 348–349.