Some convergence results using a new iterative algorithm in CAT(0) space

Year 2022, Volume: 5 Issue: 3, 263 - 272, 30.09.2022

Anju Panwar Pinki Lamba Santosh Kumar

https://doi.org/10.53006/rna.1097678

Abstract

This paper presents a new iterative algorithm for approximating the invariant points of Suzuki’s generalized nonexpansive maps. Some strong convergence theorems are developed in the context of CAT(0) space. We also included examples to demonstrate the proposed algorithm’s convergence nature. Lastly, the stability of the said iterative algorithm is discussed to validate the results

Keywords

Strong convergence, Stability, CAT(0) spaces, Iterative algorithm

Supporting Institution

None

Project Number

NIL

References

• [1] T. Abdeljawad, K. Ullah, J. Ahmad, M. Sen, and M. N. Khan, Some convergence results for a class of generalized nonexpansive mappings in Banach spaces, Advances in Mathematical Physics, (2021), 2021, 6 pages. Article ID 8837317.
• [2] J. Ahmad, K. Ullah, M. Arshad and Z. Ma, A new iterative method for suzuki mappings in Banach spaces, Journal of Mathematics,(2021), 2021, 7 pages. Article ID 6622931.
• [3] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, In:Graduate studies in Math., Amer. Math. Soc., Providence, Rhode Island, (2001).
• [4] F. Bruhat and J. Tits, Groups rekductifss sur un corps local. I. DonneKes radicielles valueKes, Publ. Math. Inst. Hautes EKtudes Sci., 41,(1972), 5-251.
• [5] M. R. Bridson, and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319 (1999), Springer, Berlin.
• [6] M. Gromov, Hyperbolic Groups. Essays in group theory, Math. Sci. Res. Inst. Publ., 8 (1987), Springer, New York.
• [7] A. Ghiura, Convergence of modi?ed Picard-Mann hybrid iteration process for nearly nonexpansive mappings, International Journal of Mathematics Trends and Technology, 66 (12) (2020), 37-43.
• [8] A. M. Harder, Fixed point theory and stability results for fixed points iteration procedures. Ph. D. Thesis, (1987), University of MissouriRolla.
• [9] A. M. Harder and T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japonica, 33(5) (1988), 693-706.
• [10] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147-150.
• [11] W. A. Kirk, Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville,2002/2003), Colecc. Abierta. University Seville Secretary of Publications, Seville, Spain, 64 (2003), 195-225.
• [12] W. A. Kirk, Geodesic geometry and ?xed point theory II. In International Conference on Fixed point Theory and Apllications, Yokohama Publishers, Yokohama, Japan, 2004: pp.113-142.
• [13] W. R. Mann, Mean value methods in Iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
• [14] A. Pansuwan and W. Sintunavarat, The new hybrid iterative algorithm for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings with numerical experiments, Thai Journal of Mathematics, 19(1) (2021), 157-168.
• [15] T. Suzuki, Fixed points theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340 (2008), 1088-1095.
• [16] B.S. Thakur, B. Thakur and M. Postolache, A new iterative scheme for numerical reckoning ?xed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147-155.
• [17] T. Thianwan, Mixed type algorithms for asymptotically nonexpansive mappings in hyperbolic spaces, European Journal of Pure and Applied Mathematics, 14(3) (2021), 650-665.
• [18] K. Ullah and M. Arshad, New iteration process and numerical reckoning ?xed points in Banach spaces, U.P.B. Sci. Bull., Series A, 79(4) (2017), 113-122.
• [19] K. Ullah and M. Arshad, Numerical reckoning ?xed points for Suzuki's generalized nonexpansive mappings via new iteration process, Filomat, 32(1) (2018), 187-196.
• [20] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc., 113(1991), 727-731.
• [21] B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 267(2) (2002), 444-453.
• [22] Z. Xue, The convergence of ?xed point for a kind of weak contraction. Nonlinear Func. Anal. Appl., 21(3) (2016), 497-500.
• [23] D. Yambangwai and T. Thianwan, ∆-Convergence and strong convergence for asymptotically nonexpansive mappings on a CAT(0) space, Thai Journal of Mathematics, 19(3) (2021), 813-826.