# Convergence analysis of new modified iterative approximating processes for two finite families of total asymptotically nonexpansive nonself mappings in hyperbolic spaces

Volume 10, Issue 4, pp 1407--1423
Publication Date: April 20, 2017 Submission Date: October 21, 2016
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### Authors

Ting-jian Xiong - Department of Mathematics, Sichuan University of Science & Engineering, 643000 Zigong, Sichuan, P. R. China. Heng-you Lan - Department of Mathematics, Sichuan University of Science & Engineering, 643000 Zigong, Sichuan, P. R. China.

### Abstract

In this paper, we introduce and study a class of new modified iterative approximation processes for two finite families of total asymptotically nonexpansive nonself mappings in hyperbolic spaces. By using generalization of Schu’s lemma and Tan-Xu’s inequality, some important related properties of this modified iterative approximation are proposed and analyzed. Further, based on the related properties, we prove $\Delta$-convergence and strong convergence of the modified iterative approximating process in hyperbolic spaces. Because a total asymptotically nonexpansive nonself mapping in hyperbolic spaces includes asymptotically nonexpansive mapping, (generalized) nonexpansive mapping of all normed linear spaces, Hadamard manifolds and CAT(0) spaces as special cases, the results presented in this paper improve and generalize the corresponding results in the literature.

### Share and Cite

##### ISRP Style

Ting-jian Xiong, Heng-you Lan, Convergence analysis of new modified iterative approximating processes for two finite families of total asymptotically nonexpansive nonself mappings in hyperbolic spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 4, 1407--1423

##### AMA Style

Xiong Ting-jian, Lan Heng-you, Convergence analysis of new modified iterative approximating processes for two finite families of total asymptotically nonexpansive nonself mappings in hyperbolic spaces. J. Nonlinear Sci. Appl. (2017); 10(4):1407--1423

##### Chicago/Turabian Style

Xiong, Ting-jian, Lan, Heng-you. "Convergence analysis of new modified iterative approximating processes for two finite families of total asymptotically nonexpansive nonself mappings in hyperbolic spaces." Journal of Nonlinear Sciences and Applications, 10, no. 4 (2017): 1407--1423

### Keywords

• Convergence analysis
• new modified iterative approximating process
• $\Delta$-convergence and strong convergence
• total asymptotically nonexpansive nonself mapping
• hyperbolic space.

•  47H09
•  47H10
•  54E70

### References

• [1] 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.

• [2] B. Ali, Convergence theorems for finite families of total asymptotically nonexpansive mappings in hyperbolic spaces, Fixed Point Theory Appl., 2016 (2016 ), 13 pages.

• [3] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1999)

• [4] H. Fukhar-ud-din, M. A. A. Khan, Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces, Fixed Point Theory Appl., 2013 (2013), 18 pages.

• [5] S.H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl., 2013 (2013 ), 10 pages.

• [6] S. H. Khan, H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal., 61 (2005), 1295–1301.

• [7] A. R. Khan, H. Fukhar-ud-din, M. A. A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl., 2012 (2012 ), 12 pages.

• [8] A. R. Khan, M. A. Khamsi, H. Fukhar-ud-din, Strong convergence of a general iteration scheme in CAT(0) spaces, Nonlinear Anal., 74 (2011), 783–791.

• [9] U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc., 357 (2005), 89–128.

• [10] W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689–3696.

• [11] P. K. F. Kuhfittig, Common fixed points of nonexpansive mappings by iteration, Pacific J. Math., 97 (1981), 137–139.

• [12] 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.

• [13] L. Leuştean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, Nonlinear analysis and optimization I,/ Nonlinear analysis, Contemp. Math., Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 513 (2010), 193–209.

• [14] Y. Li, L. H. Bo, $\Delta$-convergence analysis of improved Kuhfittig iterative for asymptotically nonexpansive nonself-mappings in W-hyperbolic spaces, J. Inequal. Appl., 2014 (2014 ), 9 pages.

• [15] T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179–182.

• [16] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229.

• [17] S. Plubtieng, K. Ungchittrakool, R. Wangkeeree, Implicit iterations of two finite families for nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 28 (2007), 737–749.

• [18] S. Reich, I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., 15 (1990), 537–558.

• [19] A. Şahin, M. Başarır, On the strong and $\Delta$-convergence of SP-iteration on CAT(0) space, J. Inequal. Appl., 2013 (2013 ), 10 pages.

• [20] A. Şahin, M. Başarır, Some convergence results for modified SP-iteration scheme in hyperbolic spaces, Fixed Point Theory Appl., 2014 (2014 ), 11 pages.

• [21] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153–159.

• [22] H. F. Senter, W. G. Dotson Jr., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380.

• [23] T. Shimizu, W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal., 8 (1996), 197–203.

• [24] S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2005), 506–517.

• [25] W. Takahashi, A convexity in metric space and nonexpansive mappings, I., Kōdai Math. Sem. Rep., 22 (1970), 142–149.

• [26] K.-K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301–308.

• [27] B. S. Thakur, D. Thakur, M. Postolache, Modified Picard-Mann hybrid iteration process for total asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015 ), 11 pages.

• [28] L.-L. Wan, $\Delta$-convergence for mixed-type total asymptotically nonexpansive mappings in hyperbolic spaces, J. Inequal. Appl., 2013 (2013), 8 pages.

• [29] L.-L. Wan, Demiclosed principle and convergence theorems for total asymptotically nonexpansive nonself mappings in hyperbolic spaces, Fixed Point Theory Appl., 2015 (2015 ), 10 pages.

• [30] L.Wang, S.-S. Chang, Z.-L. Ma, Convergence theorems for total asymptotically nonexpansive non-self mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013 ), 10 pages.

• [31] T.-J. Xiong, H.-Y. Lan, Convergence analysis of new iterative approximating schemes with errors for total asymptotically nonexpansive mappings in hyperbolic spaces, J. Comput. Anal. Appl., 20 (2016), 902–913.

• [32] L. Yang, F. H. Zhao, Strong and $\Delta$-convergence theorems for total asymptotically nonexpansive nonself mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013 ), 17 pages.

• [33] I. Yildirim, M. Özdemir, Approximating common fixed points of asymptotically quasi-nonexpansive mappings by a new iterative process, Arab. J. Sci. Eng., 36 (2011), 393–403.