Document Type : Research Paper


Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India.


In this paper, we obtain a new modified iteration process in the setting of CAT(0) spaces involving generalized $\alpha$-nonexpansive mapping. We prove strong and $\Delta$ convergence results for approximating fixed point via newly defined iteration process. Further, we reconfirm our results by non trivial example and tables.


[1] A.A. Mebawondu and C. Izuchukwu, Some fixed points properties, strong and $Delta$-convergence results for generalized $alpha$-nonexpansive mappings in hyperbolic spaces, Adv. Fixed Point Theory, 8 (2018), pp. 1-20.
[2] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), pp. 957-961.
[3] B.S. Thakur, D. Thakur and M. Postolache, A New iteration scheme for approximating fixed points of nonexpansive mappings, Filomat, 30(10) (2016), pp. 2711-2720.
[4] C. Garodia and I. Uddin, A new iterative method for solving split feasibility problem, Journal of Applied Analysis and Computation, 10(3) (2020), pp. 986-1004.
[5] D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 33, 2001.
[6] D. G"ohde, Zum Prinzip der kontraktiven abbildung, Math. Nachr., 30 (1965), pp. 251-258.
[7] D. Thakur, B.S. Thakur and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Applied Mathematics and Computation, 275 (2016), pp. 147-155.
[8] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), pp. 1041-1044.
[9] H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), pp. 375-380.
[10] H. Piri, B. Daraby, S. Rahrovi and M. Ghasemi, Approximating fixed points of generalized $alpha$-nonexpansive mappings in Banach spaces by new faster iteration process, Numerical Algorithm, 81 (2019), pp. 1129-1148.
[11] K. Goebel and S. Reich, Uniform convexity, Hyperbolic Geometry and Nonexpansive mappings, Monographs
and Textbooks in Pure and Applied Mathematics, 83. Marcel Dekker, Inc., New York, 1984.
[12] K.S. Brown, Buildings, Springer, New York, 1989.
[13] K. Ullah and M. Arshad, Numerical Reckoning Fixed Points for Suzuki's Generalized Nonexpansive Mappings via New Iteration Process, Filomat, 32(1) (2018), pp. 187-196.
[14] K. Ullah and M. Arshad, New iteration process and numerical reckoning fixed points in Banach spaces, U. P. B. Sci. Bull., Series A, 79(4) (2017), pp. 113-122.
[15] K. Ullah and M. Arshad, New three step iteration process and fixed point approximation in Banach spaces, Journal of Linear and Topological Algebra, 7(2) (2018), pp. 87-100.
[16] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Matematicki Vesnik, 66(2) (2014), pp. 223-234.
[17] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin, 1999.
[18] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, Birkhauser Boston, Massachusetts, 152, 1999.
[19] M.A. Noor, New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications, 251(1) (2000), pp. 217-229.
[20] N. Hussain, K. Ullah and M. Arshad, Fixed point approximation for Suzuki generalized nonexpansive mappings via new iteration process, Journal of Nonlinear and Convex Analysis, 19(8) (2018), pp. 1383-1393.
[21] R. Pant and R. Shukla, Approximating Fixed Points of Generalized $alpha$-Nonexpansive Mappings in Banach Spaces, Numerical Functional Analysis and Optimization, 38(2) (2017), pp. 248-266.
[22] R.P. Agarwal, D.O Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Journal of Nonlinear and Convex Analysis, 8(1) (2007), pp. 61-79.
[23] S. Banach, Sur les operations dans les ensembles abstraits et leurs applications, Fundam. Math., 3 (1922), pp. 133-181.
[24] S. Dhompongsa and B. Panyanak, On $Delta$-convergence theorems in CAT(0) spaces, Computers and Mathematics with Appl., 56 (2008), pp. 2572-2579.
[25] S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), pp. 762-772.
[26] S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc., 44 (1974), pp. 147-150.
[27] T. Suzuki, Fixed point theorems and convergence theorems for some generalized non-expansive mapping, J. Math. Anal. Appl., 340 (2008), pp. 1088-1095.
[28] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), pp. 1004-1006.
[29] W.A. Kirk, Geodesic geometry and fixed point theory, in: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Vol. 64 of Coleccion Abierta, University of Seville Secretary of Publications, Seville, Spain, 2003, 195-225.
[30] W.A. Kirk, Geodesic geometry and fixed point theory II, in: International Conference on Fixed point Theory and Applications, Yokohama Publishers, Yokohama, Japan, 2004, 113-142.
[31] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), pp. 3689-3696.
[32] W. Laowang and B. Panyanak, Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces, Fixed Point Theory and Applications, (2010), pp. 1-11.
[33] W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc., 4 (1953), pp. 506-510.