Document Type : Research Paper
Authors
Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India.
Abstract
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.
Keywords
[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.
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.
)
