Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces

Year 2017, Volume: 46 Issue: 3, 373 - 388, 01.06.2017

Faik Gursoy , Abdul Rahim Khan , Hafiz Fukhar-ud-din

Abstract

In this paper, we simplify the iterative scheme introduced by Fukhar-ud-din and Berinde [Iterative Methods for the Class of Quasi-Contractive Type Operators and Comparison of their Rate of Convergence in Convex Metric Spaces, Filomat 30 (1), 223230, 2016] and study convergence and data dependency of the new proposed scheme of a quasi-contractive operator on a hyperbolic space. It is shown that our results provide better convergence rate.

Keywords

Data dependency, strong convergence, iterative scheme, hyperbolic space, rate of convergence

References

• Abbas, M., Khan, S. H. Some
  $\Delta-$convergence theorems in $CAT(0)$ spaces, Hacet. J. Math. Stat. 40 (4), 563569, 2011.
• Agarwal, R. P., O' Regan, D. and Sahu, D. R. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1), 6179, 2007.
• Berinde, V. Iterative approximation of fixed points (Springer-Verlag, Berlin, 2007).
• Bridson, M. and Haeiger, A. Metric spaces of non-positive curvature (Springer-Verlag, Berlin, 1999).
• Chugh, R. and Kumar, V. Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators, Int. J. Comput. Appl. 40 (15), 4146, 2011.
• Dhompongsa, S. and Panyanak, B. On
  $\Delta-$convergence theorems in $CAT(0)$ spaces, Comput. Math. Appl. 56 (10), 25722579, 2008.
• Fukhar-ud-din H. One step iterative scheme for a pair of nonexpansive mappings in a convex metric space, Hacet. J. Math. Stat. 44 (5), 10231031, 2015.
• Fukhar-ud-din, H. and Berinde, V. Iterative Methods for the Class of Quasi-Contractive Type Operators and Comparsion of their Rate of Convergence in Convex Metric Spaces, Filomat 30 (1), 223230, 2016.
• Gürsoy, F., Karakaya, V. and Rhoades, B. Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013 (1), 112, 2013.
• Ishikawa, S. Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1), 147 150, 1974.
• Karakaya, V., Gürsoy, F. and Ertürk, M. Some convergence and data dependence results for various fixed point iterative methods, Kuwait J. Sci. 43 (1), 112128, 2016.
• Khan A. R. On modified Noor iterations for asymptotically nonexpansive mappings, Bull. Belg. Math. Soc. Simon Stevin, 17 (1), 127140, 2010.
• Khan, A. R., Fukhar-ud-din, H. and Khan, M. A. A. An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012 (1), 112, 2012.
• Khan A. R., Gürsoy, F. and Karakaya, V. Jungck-Khan iterative scheme and higher con- vergence rate, Int. J. Comput. Math. 93 (12), 20922105, 2016.
• Khan, A. R., Gürsoy, F. and Kumar, V. Stability and data dependence results for Jungck- Khan iterative scheme, Turkish J. Math. 40 (3), 631640, 2016.
• Khan, A. R., Khamsi, M. A. and Fukhar-ud-din, H. Strong convergence of a general iteration scheme in $CAT(0)-$spaces, Nonlinear Anal. 74 (3), 783791, 2011.
• Khan, A. R., Kumar, V. and Hussain, N. Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput. 231, 521535, 2014.
• Kirk, W. A. Krasnoselskii's iteration process in hyperbolic space, Numer. Funct. Anal. Optim. 4 (4) 371381, 19811982.
• Knopp, K. Theory and Applications of Infinite Series (Berlin, 1931).
• Kohlenbach, U. Some logical metatheorems with applications in functional analysis, Trans. Am. Math. Soc. 357 (1), 89128, 2005.
• Mann, W. R. Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (3), 506-510, 1953.
• Phuengrattana, W. and Suantai, S. Comparison of the rate of convergence of various iter- ative methods for the class of weak contractions in Banach spaces, Thai J. Math. 11 (1), 217226, 2013.
• Singh, S. L., Bhatnagar, C. and Mishra, S. N. Stability of Jungck-type iterative procedures, Int. J. Math. Math. Sci. 2005 (19), 30353043, 2005.
• Soltuz, S. M. and Grosan, T. Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl. 2008, 17, 2008.
• Takahashi, W. A convexity in metric spaces and nonexpansive mappings I, Kodai Math. Sem. Rep. 22 (2), 142149, 1970.
• Talman, L. A. Fixed points for condensing multifunctions in metric spaces with convex structure, Kodai Math. Sem. Rep. 29 (1-2), 6270, 1977.
• Weng, X. Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113 (3), 727731, 1991.
• Xu, B. and Noor, M. A. Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2), 444-453, 2002.