A modified iteration for total asymptotically nonexpansive mappings in Hadamard spaces

Izhar Uddin; Sabiya Khatoon; Nabil Mlaiki; Thabet Abdeljawad; Izhar Uddin; Sabiya Khatoon; Nabil Mlaiki; Thabet Abdeljawad

doi:10.3934/math.2021279

AIMS Mathematics

2021, Volume 6, Issue 5: 4758-4770. doi: 10.3934/math.2021279

Previous Article Next Article

Research article Special Issues

A modified iteration for total asymptotically nonexpansive mappings in Hadamard spaces

1.
Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India
2.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh-11586, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung-40402, Taiwan
4.
Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

Received: 15 October 2020 Accepted: 08 February 2021 Published: 25 February 2021
MSC : 47H10, 47H09, 47E10

The motive of this paper is to study the convergence analysis of a modified iteration procedure for total asymptotically nonexpansive mapping under some suitable conditions in the setting of CAT(0) spaces. By using MATLAB R2018a, we also illustrate numerical experiment to compare the rate of convergence of the new iteration process with some existing iteration processes.
- CAT(0) space,
- total asymptotically nonexpansive mappings,
- weak and strong convergence
Citation: Izhar Uddin, Sabiya Khatoon, Nabil Mlaiki, Thabet Abdeljawad. A modified iteration for total asymptotically nonexpansive mappings in Hadamard spaces[J]. AIMS Mathematics, 2021, 6(5): 4758-4770. doi: 10.3934/math.2021279

Related Papers:

Abstract

The motive of this paper is to study the convergence analysis of a modified iteration procedure for total asymptotically nonexpansive mapping under some suitable conditions in the setting of CAT(0) spaces. By using MATLAB R2018a, we also illustrate numerical experiment to compare the rate of convergence of the new iteration process with some existing iteration processes.

References

[1]	M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik, 66 (2014), 223–234.
[2]	R. P. Agarwal, D. O'Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex A., 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., 10673 (2006).
[4]	P. Cholamjiak, The modified proximal point algorithm in CAT(0) spaces, Optim. Lett., 9 (2015), 1401–1410. doi: 10.1007/s11590-014-0841-8
[5]	S. Dhompongsa, W. A. Kirk, B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal.: Theory Methods Appl., 65 (2006), 762–772. doi: 10.1016/j.na.2005.09.044
[6]	S. Dhompongsa, B. Panyanak, On $\Delta$-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572–2579. doi: 10.1016/j.camwa.2008.05.036
[7]	S. Dhompongsa, W. A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 35–45.
[8]	B. Halpern, Fixed points of nonexpanding maps, B. Am. Math. Soc., 73 (1967), 957–961. doi: 10.1090/S0002-9904-1967-11864-0
[9]	S. Husain, N. Singh, $\Delta$-convergence for proximal point algorithm and fixed point problem in CAT(0) spaces, Fixed Point Theory Appl., 8 (2019). Available from: https://doi.org/10.1186/s13663-019-0658-3.
[10]	S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. doi: 10.1090/S0002-9939-1974-0336469-5
[11]	E. Karapinar, H. Salahifard, S. M. Vaezpour, Demiclosedness principle for total asymptoticallty nonexpansive mappings in CAT(0) spaces, J. Appl. Math., (38150) (2014).
[12]	C. Garodia, I. Uddin, A new fixed point algorithm for finding the solution of a delay differential equation, AIMS Mathematics, 5 (2020), 3182–3200. doi: 10.3934/math.2020205
[13]	C. Garodia, I. Uddin, S. H. Khan, Approximating common fixed points by a new faster iteration process, Filomat, 34 (2020), 2047–2060. doi: 10.2298/FIL2006047G
[14]	I. Uddin, S. Khatoon, V. Colao, Approximating fixed points of generalized alpha-Reich-Suzuki nonexpansive mappings in CAT(0) space, J. Nonlinear Convex Anal., 21 (2020).
[15]	M. A. A. Khan, P. Cholamjiak, A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces, J. Fixed Point Theory Appl., 22 (2020).
[16]	W. A. Kirk, A Fixed point theorem for mappings which do not increase distances, Math. Monthly, 72 (1965), 1004–1006. doi: 10.2307/2313345
[17]	W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689–3696. doi: 10.1016/j.na.2007.04.011
[18]	W. Kumam, N. Pakkaranang, P. Kumam, P. Cholamjiak, Convergence analysis of modified Picard-S hybrid iterative algorithms for total asymptotically nonexpansive mappings in Hadamard spaces, Int. J. Comput. Math., 97 (2020), 175–188. doi: 10.1080/00207160.2018.1476685
[19]	T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179–182. doi: 10.1090/S0002-9939-1976-0423139-X
[20]	W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. doi: 10.1090/S0002-9939-1953-0054846-3
[21]	M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. doi: 10.1006/jmaa.2000.7042
[22]	L. Qihou, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl., 259 (2001), 18–24. doi: 10.1006/jmaa.2000.7353
[23]	J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc., 43 (1991), 153–159. doi: 10.1017/S0004972700028884
[24]	H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Am. Math. Soc., 44 (1974), 370–385.
[25]	S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 331 (2005), 506–517.
[26]	R. Suparatulatorn, P. Cholamjiak, S. Suantai, On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces, Optim. Meth. Soft., 32 (2017), 182–192. doi: 10.1080/10556788.2016.1219908
[27]	S. Shahzad, R. Al-Dubiban, Approximating common fixed points of nonexpansive mappings in Banach spaces, Georgian Math. J., 13 (2006), 529–537.
[28]	B. S. Thakur, D. Thakur, M. Postolache, Modified Picard-Mann hybrid iteration process for total asymptotically nonexpansive mappings, Fixed Point Theory Appl., 140 (2015).

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)