Abstract
In this paper, we prove some strong and \(\triangle \)-convergence theorems of the \(K^{*}\) iteration process for two different classes of generalized nonexpansive mappings in CAT(0) spaces.
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References
Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Springer, Berlin, Heidelberg (1999)
Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory, vol. 8, pp. 75–263. Springer Verlag, MSRI Publ. (1987)
Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Math, vol. 33. Amer. Math. Soc., Providence, RI (2001)
Bartolini, I., Ciaccia, P., Patella, M.: String matching with metric trees using an approximate distance. In: SPIR Lecture Notes in Computer Science, vol. 2476. Springer, Berlin (1999)
Semple, C.: Phylogenetics. Oxford Lecture Series in Mathematics and Its Application. Oxford University Press, Oxford (2003)
Espinola, R., Kirk, W.A.: Fixed point theorems in \(\mathbb{R} \)-trees with applications to graph theory. Topology Appl. 153(7), 1046–1055 (2006)
Kirk, W.A.: Geodesic geometry and fixed point theory. In Seminar of Math. Anal. (Malaga/Seville, 2002/2003), In: Colecc. Abierta, vol. 6. Univ. Sevilla Secr. Publ. Seville 195–225 (2003)
Kirk, W.A.: Geodesic geometry and fixed point theory II. In: International Conference on Fixed Point Theo. Appl., Yokohama Publ., Yokohama 113–142 (2004)
Dhompongsa, S., Kaewkhao, A., Panyanak, B.: Lim’s theorems for multi-valued mappings in CAT\((0)\) spaces. J. Math. Anal. Appl. 312(2), 478–487 (2005)
Dhompongsa, S., Panyanak, B.: On \(\triangle \)-convergence theorems in CAT\((0)\) spaces. Comput. Math. Appl. 56, 2572–2579 (2008)
Laowang, W., Panyanak, B.: Approximating fixed points of nonexpansive nonself mappings in CAT\((0)\) spaces. Fixed Point Theory Appl. 2010, Article ID 367274 (2010)
Khan, S.H., Abbas, M.: Strong and \(\triangle \)-convergence of some iterative schemes in CAT\((0)\) spaces. Comput. Math. Appl. 61, 109–116 (2011)
Şahin, A., Başarır, M.: On the strong convergence of a modified \(S\)-iteration process for asymptotically quasi-nonexpansive mappings in a CAT\((0)\) space. Fixed Point Theory Appl. 2013, Article ID 12 (2013)
Thakur, B.S., Thakur, D., Postolache, M.: Modified Picard-Mann hybrid iteration process for total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2015, Article ID 140 (2015)
Ullah, K., Iqbal, K., Arshad, M.: Some convergence results using \( K\)-iteration process in CAT\((0)\) spaces. Fixed Point Theory Appl. 2018, Article ID 11 (2018)
Alber, Y.I., Chidume, C.E., Zegeye, H.: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2006, Article ID 10673 (2006)
Goebel, K., Kirk, W.A.: Iteration processes for nonexpansive mappings. Contemp. Math. 21, 115–123 (1983)
Panyanak, B.: On total asymptotically nonexpansive mappings in CAT \((\kappa )\) spaces. J. Inequal. Appl. 2014, Article ID 336 (2014)
Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340, 1088–1095 (2008)
Garcia-Falset, J., Liorens-Fuster, E., Suzuki, T.: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 375(1), 185–195 (2011)
Ullah, K., Arshad, M.: New three-step iteration process and fixed point approximation in Banach spaces. J. Linear Topol. Algebra 7(2), 87–100 (2018)
Gürsoy, F., Karakaya, V.: A Picard-S hybrid type iteration method for solving a differantial equation with retarded argument (2014). arXiv:1403.2546v2
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61–79 (2007)
Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)
Chang, S.S., Wang, L., Joseph Lee, H.W., Chan, C.K., Yang, L.: Demiclosed principle and \(\triangle \)-convergence theorems for total asymptotically nonexpansive mappings in CAT\((0)\) spaces. Appl. Math. Comput. 219, 2611–2617 (2012)
Dhompongsa, S., Kirk, W.A., Sims, B.: Fixed points of uniformly lipschitzian mappings. Nonlinear Anal. 65, 762–772 (2006)
Lim, T.C.: Remarks on some fixed point theorems. Proc. Amer. Math. Soc. 60, 179–182 (1976)
Kirk, W.A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008)
Dhompongsa, S., Kirk, W.A., Panyanak, B.: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 8, 35–45 (2007)
Liu, Q.: Iterative sequences for asymptotically quasi-nonexpansive mappings with eror member. J. Math. Anal. 259, 18–24 (2001)
Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375–380 (1974)
Şahin, A., Başarır, M.: On the strong and \(\triangle \) -convergence theorems for nonself mappings on a CAT\((0)\) space. In: Proceedings of the 10th IC-FPTA, 227 - 240, July 9-18, Cluj-Napoca, Romania (2012)
Başarır, M., Şahin, A.: On the strong and \(\triangle \)-convergence for total asymptotically nonexpansive mappings on a CAT\((0)\) space. Carpathian Math. Publ. 5(2), 170–179 (2013)
Kim, G.E.: Strong convergence to fixed point of a total asymptotically nonexpansive mapping. Fixed Point Theory Appl. 2013, Article ID 302 (2013)
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Şahin, A., Başarır, M. (2021). Some Convergence Results of the \(K^{*}\) Iteration Process in CAT(0) Spaces. In: Cho, Y.J., Jleli, M., Mursaleen, M., Samet, B., Vetro, C. (eds) Advances in Metric Fixed Point Theory and Applications. Springer, Singapore. https://doi.org/10.1007/978-981-33-6647-3_2
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DOI: https://doi.org/10.1007/978-981-33-6647-3_2
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