Some New Results for the J-Iterative Scheme in Kohlenbach Hyperbolic Space

Year 2022, Volume: 10 Issue: 2, 210 - 219, 31.10.2022

Aynur Şahin , Metin Basarır

Abstract

In the present paper, we study the J-iterative scheme of Bhutia and Tiwary (J. Linear Topol. Algebra, 8(4), (2019), 237-250) in Kohlenbach hyperbolic space. We prove the weak w^2-stability and data dependence theorems of this iterative scheme for contraction mappings. We also give some △-convergence and strong convergence theorems for generalized α-nonexpansive mappings and finite families of total asymptotically nonexpansive mappings using J-iterative scheme. The results presented here can be viewed as a generalization of several well-known results in CAT(0) space and uniformly convex Banach space.

Keywords

data dependence , fixed point , hyperbolic space , J-iterative scheme , strong convergence , weak w^2-stability , delta-convergence

References

• [1] Y. I. Alber, C. E. Chidume and H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2006:10673, (2006), 20 pages.
• [2] V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, 2007.
• [3] J. D. Bhutia and K. Tiwary, New iteration process for approximating fixed points in Banach spaces, J. Linear Topol. Algebra, 8(4), (2019), 237-250.
• [4] N. Hussain, K. Ullah and M. Arshad, Fixed point approximation for Suzuki generalized nonexpansive mappings via new iteration process, J. Nonlinear Convex Anal., 19(8), (2018), 1383-1393.
• [5] Izhar-ud-din, S. Khatoon, N. Mlaiki and T. Abdeljawad, A modified iteration for total asymptotically nonexpansive mappings in Hadamard spaces, AIMS Math., 6(5), (2021), 4758-4770.
• [6] M. A. A. Khan, H. Fukhar-ud-din and A. Kalsoom, Existence and higher arity iteration for total asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces, Fixed Point Theory Appl., 2016:3, (2016), 18 pages.
• [7] S. Khatoon, Izhar-ud-din and M. Bas¸arır, A modified proximal point algorithm for a nearly asymptotically quasi-nonexpansive mapping with an application, Comp. Appl. Math., 40:250, (2021), 19 pages.
• [8] U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Am. Math. Soc. 357(1), (2004), 89-128.
• [9] A. R. Khan, H. Fukhar-ud-din and M. A. A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl., 2012:54, (2012), 12 pages.
• [10] L. Leustean, A quadratic rate of asymptotic regularity for CAT(0) spaces, J. Math. Anal. Appl., 325(1), (2007), 386-399.
• [11] L. Leustean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces. In A. Leizarowitz, B. S. Mordukhovich, I. Shafrir and A. Zaslavski (eds), Nonlinear Analysis and Optimization I: Nonlinear Analysis, Contemp. Math., Vol. 513, pp. 193-209, Amer. Math. Soc., 2010.
• [12] Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. 259, (2001), 18-24.
• [13] D. Pant and R. Shukla, Approximating fixed points of generalized a-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 38(2), (2017), 248-266.
• [14] H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc., 44, (1974), 375-380.
• [15] T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal., 8, (1996), 197-203.
• [16] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340, (2008), 1088-1095.
• [17] A. S¸ ahin, Some new results of M-iteration process in hyperbolic spaces, Carpathian J. Math., 35(2), (2019), 221-232.
• [18] A. S¸ ahin and M. Bas¸arır, Some convergence results for nearly asymptotically nonexpansive nonself mappings in CAT(k) spaces, Math. Sci. 11, (2017), 79-86.
• [19] A. S¸ ahin and M. Bas¸arır, Some convergence results of the K*-iteration process in CAT(0) space. In Y. J. Cho, M. Jleli, M. Mursaleen, B. Samet and C. Vetro, (eds), Advances in Metric Fixed Point Theory and Applications, pp. 23-40, Springer, Singapore, 2021.
• [20] S¸ . M. S¸ oltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl., 2008:242916, (2008), 7 pages.
• [21] W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Semin. Rep., 22, (1970), 142-149.
• [22] I. Timis¸, On the weak stability of Picard iteration for some contractive type mappings, Annal. Uni. Craiova, Math. Comput. Sci. Series, 37(2), (2010), 106-114.
• [23] K. Ullah and M. Arshad, New iteration process and numerical reckoning fixed point in Banach spaces, U.P.B. Sci. Bull. (Series A), 79(4), (2017), 113-122.
• [24] K. Ullah and M. Arshad, New three-step iteration process and fixed point approximation in Banach spaces, J. Linear Topol. Algebra, 7(2), (2018), 87-100.
• [25] K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat, 32(1), (2018), 187-196.
• [26] L. L. Wan, Demiclosed principle and convergence theorems for total asymptotically nonexpansive nonself mappings in hyperbolic spaces, Fixed Point Theory Appl., 2015:4, (2015), 10 pages.