Common Fixed Point Iterations of Generalized Asymptotically Quasi-Nonexpansive Mappings in Hyperbolic Spaces

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DOI: 10.4236/jamp.2014.25021    3,251 Downloads   4,470 Views   Citations

ABSTRACT

We introduce a general iterative method for a finite family of generalized asymptotically quasi- nonexpansive mappings in a hyperbolic space and study its strong convergence. The new iterative method includes multi-step iterative method of Khan et al. [1] as a special case. Our results are new in hyperbolic spaces and generalize many known results in Banach spaces and CAT(0) spaces, simultaneously.

Cite this paper

Khan, A. and Fukhar-ud-din, H. (2014) Common Fixed Point Iterations of Generalized Asymptotically Quasi-Nonexpansive Mappings in Hyperbolic Spaces. Journal of Applied Mathematics and Physics, 2, 170-175. doi: 10.4236/jamp.2014.25021.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Khan, A.R., Domlo, A.A. and Fukhar-ud-din, H. (2008) Common Fixed Points Noor Iteration for a Finite Family of Asymptotically Quasi-Nonexpansive Mappings in Banach Space. Journal of Mathematical Analysis and Applications. 341, 1-11. http://dx.doi.org/10.1016/j.jmaa.2007.06.051
[2] Menger, K. (1928) Untersuchungenüberallgemeine Metrik. Mathematische Annalen, 100, 75-163. http://dx.doi.org/10.1007/BF01448840
[3] Takahashi, W. (1970) A Convexity in Metric Spaces and Nonexpansive Mappings. Kodai. Math Sem. Rep., 22, 142-149. http://dx.doi.org/10.2996/kmj/1138846111
[4] Bridson, M. and Haefliger, A. (1999) Metric Spaces of Non-Positive Curvature. Springer-Verlag, Berlin, Heidelberg, New York. http://dx.doi.org/10.1007/978-3-662-12494-9
[5] Fukhar-ud-din, H. (2013) Strong Convergence of an Ishikawa-type Algorithm inCAT (0) Spaces. Fixed Point Theory and Applications, 2013, 207.
[6] Khan, A.R., Khamsi, M.A. and Fukhar-ud-din, H. (2011) Strong Convergence of a General Iteration Scheme in CAT(0) Spaces, Nonlinear Anal. 74, 783-791. http://dx.doi.org/10.1016/j.na.2010.09.029
[7] Goebel, K. and Reich, S. (1984) Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Series of Monographs and Textbooks in Pure and Applied Mathematics, Dekker, New York.
[8] Khan, A.R., Fukhar-ud-din, H. and Khan, M.A.A. (2012) An Implicit Algorithm for Two Finite Families of Nonexpansive Maps in Hyperbolic Spaces. Fixed Point Theory and Applications, 2012, 54.

  
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