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Natural Sciences Issue (Vietnamese)
Date Published:
03/05/2026
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Nguyen, T. H., & Lê, T. P. (2026). Convergence of three-step iteration for asymptotically nonexpansive mappings in CAT(0) spaces. Dong Thap University Journal of Science, Online First. https://doi.org/10.52714/dthu.ns.3028.1888
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Convergence of three-step iteration for asymptotically nonexpansive mappings in CAT(0) spaces
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Abstract
In this paper, we introduce a three-step iteration scheme for approximating common fixed points of three asymptotically nonexpansive mappings in CAT(0) spaces. We also establish and prove some $Delta$-convergence and strong convergence results of this iterative sequence to common fixed points of three asymptotically nonexpansive mappings in complete CAT(0) spaces. In addition, we give an example to illustrate the convergence results obtained.
Keywords
asymptotically nonexpansive mapping, three-step iteration scheme, common fixed poin, CAT(0) space
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
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Chang, S. S., Wang, L., Lee, H. J., Chan, C. K., & Yang, L. (2012). Demiclosed priciple and $Delta$-convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces, Appl. Math. Comput., 219(5), 2611-2617.
https://doi.org/10.1016/j.amc.2012.08.095
Dhompongsa, S., & Panyanak, B. (2008). On $Delta$-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (10), 2572-2579.
https://doi.org/10.1016/j.camwa.2008.05.036
Dhompongsa, S., Kirk, W. A., & Panyanak, B. (2007). Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear and Convex Anal., 8, 35-45.
Kirk, W. A., & Panyanak, B. (2008). A concept of convergence in geodesic spaces, Nonlinear Anal., 68, 3689-3696. https://doi.org/10.1016/j.na.2007.04.011
Kirk, W. A. (2003). Geodesic geometry and fixed point theory in: seminar of mathematical analysis (Malaga/Seville, 2002/2003), colecc. Abierta, univ. Sevilla secr. Publ. Seville, 64, 195-225.
Kirk, W. A. (2004). Geodesic geometry and fixed point theory II. In International Conferenceon Fixed Point Theory and Applications, 113-142, Yokohama Publ., Yokohama.
Khan, S. H., & Abbas, M. (2011). Strong and $Delta$-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl. ,61,109-116.
https://doi.org/10.1016/j.camwa.2010.10.037
Lim, T. C. (1976). Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60, 179–182. https://doi.org/10.1090/S0002-9939-1976-0423139-X
Nanjara, B., & Panyanak, B. (2010). Demiclosedness principle for asymptotically nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl., 1-14. https://doi:10.1155/2010/268780
Niwongsa, Y., & Panyanak, B. (2010). Noor iterations for asymptotically nonexpansive mappings in CAT(0) spaces. Int. J. Math. Anal., 4(13), 645-656.
Sahin, A., & Basarir, M. (2013). On the strong convergrnce of a modified S-iteration process for asymptotically quasi-nonexpansive mapping in CAT(0) space, Fixed Point Theory Appl., 1-10. https://doi:10.1186/1687-1812-2013-12
Sridarat, P., Suparaturatorn, R., Suantai, S., & Cho, Y. J. (2019). Convergence analysis of SP-iteration for G-nonexpansive mappings with directed graphs, Bull. Malays. Math. Sci. Soc., 42(5), 2361-2380. https://doi.org/10.1007/s40840-018-0606-0
Yambangwai, D., & Thianwan, T. (2021). ∆-Convergence and strong convergence for asymptotically nonexpansive mappings on a CAT(0) space, Thai J. Math., 19(3), 813-826.
Yambangwai, D., Aunruean, S., & Thianwan, T. (2020). A new modified three-step iteration method for G-nonexpansive mappings in Banach spaces with a graph, Numerical Algorithms, 84(2), 537-565. https://doi.org/10.1007/s11075-019-00768-w
Zhou, H., Agarwal, R. P., Cho, Y. J., & Kim, Y. S. (2002). Nonexpansive mappings and iterative methods in uniformly convex Banach spaces, Georgian Math. J. 9, 591-600. https://doi.org/10.1515/GMJ.2002.591
Chang, S. S., Wang, L., Lee, H. J., Chan, C. K., & Yang, L. (2012). Demiclosed priciple and $Delta$-convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces, Appl. Math. Comput., 219(5), 2611-2617.
https://doi.org/10.1016/j.amc.2012.08.095
Dhompongsa, S., & Panyanak, B. (2008). On $Delta$-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (10), 2572-2579.
https://doi.org/10.1016/j.camwa.2008.05.036
Dhompongsa, S., Kirk, W. A., & Panyanak, B. (2007). Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear and Convex Anal., 8, 35-45.
Kirk, W. A., & Panyanak, B. (2008). A concept of convergence in geodesic spaces, Nonlinear Anal., 68, 3689-3696. https://doi.org/10.1016/j.na.2007.04.011
Kirk, W. A. (2003). Geodesic geometry and fixed point theory in: seminar of mathematical analysis (Malaga/Seville, 2002/2003), colecc. Abierta, univ. Sevilla secr. Publ. Seville, 64, 195-225.
Kirk, W. A. (2004). Geodesic geometry and fixed point theory II. In International Conferenceon Fixed Point Theory and Applications, 113-142, Yokohama Publ., Yokohama.
Khan, S. H., & Abbas, M. (2011). Strong and $Delta$-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl. ,61,109-116.
https://doi.org/10.1016/j.camwa.2010.10.037
Lim, T. C. (1976). Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60, 179–182. https://doi.org/10.1090/S0002-9939-1976-0423139-X
Nanjara, B., & Panyanak, B. (2010). Demiclosedness principle for asymptotically nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl., 1-14. https://doi:10.1155/2010/268780
Niwongsa, Y., & Panyanak, B. (2010). Noor iterations for asymptotically nonexpansive mappings in CAT(0) spaces. Int. J. Math. Anal., 4(13), 645-656.
Sahin, A., & Basarir, M. (2013). On the strong convergrnce of a modified S-iteration process for asymptotically quasi-nonexpansive mapping in CAT(0) space, Fixed Point Theory Appl., 1-10. https://doi:10.1186/1687-1812-2013-12
Sridarat, P., Suparaturatorn, R., Suantai, S., & Cho, Y. J. (2019). Convergence analysis of SP-iteration for G-nonexpansive mappings with directed graphs, Bull. Malays. Math. Sci. Soc., 42(5), 2361-2380. https://doi.org/10.1007/s40840-018-0606-0
Yambangwai, D., & Thianwan, T. (2021). ∆-Convergence and strong convergence for asymptotically nonexpansive mappings on a CAT(0) space, Thai J. Math., 19(3), 813-826.
Yambangwai, D., Aunruean, S., & Thianwan, T. (2020). A new modified three-step iteration method for G-nonexpansive mappings in Banach spaces with a graph, Numerical Algorithms, 84(2), 537-565. https://doi.org/10.1007/s11075-019-00768-w
Zhou, H., Agarwal, R. P., Cho, Y. J., & Kim, Y. S. (2002). Nonexpansive mappings and iterative methods in uniformly convex Banach spaces, Georgian Math. J. 9, 591-600. https://doi.org/10.1515/GMJ.2002.591
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