On strong and \(\Delta\)-convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in \(CAT(\kappa)\) spaces
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Authors
Plern Saipara
- Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
Parin Chaipunya
- Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
Yeol Je Cho
- Department of Mathematics Education, Gyeongsang Natoinal University, Jinju 660-701, Korea.
Poom Kumam
- Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
- Theoretical and Computational Science (TaCS) Center, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
Abstract
In this paper, we obtain strong and \(\Delta\)-convergence theorems of modified S-iteration for total asymptotically nonexpansive mappings in \(CAT(\kappa)\) spaces with \(\kappa > 0\). Our results extend and improve the corresponding recent results announced by Panyanak [B. Panyanak, J. Inequal. Appl., 2014 (2014), 13 pages]
and many authors.
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ISRP Style
Plern Saipara, Parin Chaipunya, Yeol Je Cho, Poom Kumam, On strong and \(\Delta\)-convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in \(CAT(\kappa)\) spaces, Journal of Nonlinear Sciences and Applications, 8 (2015), no. 6, 965--975
AMA Style
Saipara Plern, Chaipunya Parin, Cho Yeol Je, Kumam Poom, On strong and \(\Delta\)-convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in \(CAT(\kappa)\) spaces. J. Nonlinear Sci. Appl. (2015); 8(6):965--975
Chicago/Turabian Style
Saipara, Plern, Chaipunya, Parin, Cho, Yeol Je, Kumam, Poom. "On strong and \(\Delta\)-convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in \(CAT(\kappa)\) spaces." Journal of Nonlinear Sciences and Applications, 8, no. 6 (2015): 965--975
Keywords
- fixed point
- total asymptotically nonexpansive mapping
- \(\Delta\)- convergence
- \(CAT(\kappa)\) space
- S-iteration.
MSC
References
-
[1]
A. Abkar, M. Eslamian, Common fixed point results in CAT(0) spaces, Nonlinear Anal., 74 (2011), 1835-1840.
-
[2]
R. P. Agarwal, D. O'Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex. Anal., 8 (2007), 61-79.
-
[3]
Y. I. Alber, C. E. Chidume, H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2006 (2006), 20 pages.
-
[4]
M. Basarir, A. Sahin , On the strong and \(\Delta\)-convergence for total asymptotically nonexpansive mappings on a CAT(0) space , Carpathian Math. Publ., 5 (2013), 170-179.
-
[5]
M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin (1999)
-
[6]
F. Bruhat, J. Tits, Groupes reductifs sur un corps local, I. Donnees radicielles valuees, Inst. Hautes Etudes Sci. Publ. Math., 41 (1972), 5-251.
-
[7]
S. S. Chang, L. Wang, H. W. Joseph Lee, C. K. Chan, L. Yang , Demiclosed principle and \(\Delta\)-convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces, Appl. Math. Comput., 219 (2012), 2611-2617.
-
[8]
S. S. Chang, L. Wang, H. W. Joseph Lee, C. K. Chan, Strong and \(\Delta\)-convergence for mixed type total asymptotically nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl., 122 (2013), 16 pages.
-
[9]
P. Chaoha, A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 320 (2006), 983-987.
-
[10]
Y. J. Cho, L. Ciric, S. Wang, Convergence theorems for nonexpansive semigroups in CAT(0) spaces, Nonlinear Anal., 74 (2011), 6050-6059.
-
[11]
S. Dhompongsa, B. Panyanak, On \(\Delta\)-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572-2579.
-
[12]
S. Dhompongsa, A. Kaewkhao, B. Panyanak, Lim's theorems for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl., 312 (2005), 478-487.
-
[13]
R. Espinola, A. Fernandez-Leon, CAT(\(\kappa\))-spaces, weak convergence and fixed points, J. Math. Anal. Appl., 353 (2009), 410-427.
-
[14]
K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174.
-
[15]
J. S. He, D. H. Fang, G. L'opez, C. Li , Mann's algorithm for nonexpansive mappings in CAT(\(\kappa\)) spaces, Nonlinear Anal., 75 (2012), 445-452.
-
[16]
T. Jinfang, C. Shih-sen, Viscosity approximation methods for two nonexpansive semigroups in CAT(0) spaces, Fixed Point Theory Appl., 2013 (2013), 16 pages.
-
[17]
E. Karapinar, H. Salahifard, S. M. Vaezpour, Demiclosedness principle for total asymptotically nonexpansive mappings in CAT(0) spaces , J. Appl. Math., 2014 (2014), 10 pages.
-
[18]
W. A. Kirk, Geodesic geometry and fixed point theory, in: Seminar of Mathematical Analysis (Malaga/Seville,2002/2003). Colecc. Abierta. Univ. Sevilla Secr. Publ. Seville., 64 (2003), 195-225.
-
[19]
W. A. Kirk, Geodesic geometry and fixed point theory II, in: International Conference on Fixed Point Theory and Applications, Yokohama Publ. Yokohama, (2004), 113-142.
-
[20]
W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces , Nonlinear Anal., 68 (2008), 3689-3696.
-
[21]
P. Kumam, G. S. Saluja, H. K. Nashine, Convergence of modified S-iteration process for two asymptotically nonexpansive mappings in the intermediate sense in CAT(0) spaces, J. Inequal. Appl., 2014 (2014), 15 pages.
-
[22]
L. Leustean, A quadratic rate of asymptotic regularity for CAT(0) spaces, J. Math. Anal. Appl., 325 (2007), 386-399.
-
[23]
T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179-182.
-
[24]
S. Ohta, Convexities of metric spaces Geom, Dedicata, 125 (2007), 225-250.
-
[25]
B. Panyanak , On total asymptotically nonexpansive mappings in CAT(\(\kappa\)) spaces, J. Inequal. Appl., 2014 (2014), 13 pages.
-
[26]
S. Saejung, Halpern's iteration in CAT(0) spaces , Fixed Point Theory Appl., 2010 (2010), 13 pages.
-
[27]
D. R. Sahu, Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces, Comment. Math. Univ. Carolin., 46 (2005), 653-666.
-
[28]
J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153-159.
-
[29]
N. Shahzad, J. Markin, Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces, J. Math. Anal. Appl., 337 (2008), 1457-1464.
-
[30]
C. Shih-sen, W. Lin, W. J. L. Heung, C. Chi-kin, Strong and \(\Delta\)-convergence for mixed type total asymptotically nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl., 122 (2013), 16 pages.
-
[31]
J. F. Tang, S. S. Chang, H. W. Joseph Lee, C. K. Chan, Iterative algorithm and \(\Delta\)-convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces, Abst. Appl. Anal., 2012 (2012), 11 pages.
-
[32]
K. K. Tan, H. K. Xu, Fixed point iteration process for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 122 (1994), 733-739.
-
[33]
L. L. Wan, \(\Delta\)-convergence for mixed-type total asymptotically nonexpansive mappings in hyperbolic spaces, J. Inequal. Appl., 2013 (2013), 8 pages.
-
[34]
L. Wang, S. S. Chang, Z. Ma , Convergence theorems for total asymptotically nonexpansive non-self mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013), 10 pages.
-
[35]
L. Yang, F. H. Zhao, Strong and \(\Delta\)-convergence theorems for total asymptotically nonexpansive non-self mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013), 17 pages.
-
[36]
L. C. Zhao, S. S. Chang, J. K. Kim, Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces, Fixed Point Theory Appl., 2013 (2013), 11 pages.
-
[37]
L. C. Zhao, S. S. Chang, X. R. Wang, Convergence theorems for total asymptotically nonexpansive mappings in hyperbolic spaces, J. Appl. Math., 2013 (2013), 5 pages.