On the convergence of modified $\it S$-iteration process for generalized asymptotically quasi-nonexpansive mappings in CAT(0) spaces
Abstract
$S$-iteration process to converge to fixed point for generalized
asymptotically quasi-nonexpansive mappings in the framework of CAT(0) space. Also we establish some strong convergence theorems of the said iteration process and mapping under semi-compactness and condition (A) which are weaker than completely continuous condition. Our results extend and improve many known results from the existing literature.
Full Text:
PDFReferences
R.P. Agarwal, D. O'Regan, D.R. Sahu, Iterative
construction of fixed points of nearly asymptotically nonexpansive
mappings, J. Nonlinear Convex Anal. textbf{8(1)} (2007), 61-79.
M.R. Bridson and A. Haefliger, Metric spaces
of non-positive curvature, Vol. textbf{319} of Grundlehren der
Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.
K.S. Brown, Buildings, Springer, New York,
NY, USA, 1989.
R. Bruck, T. Kuczumow and S. Reich,
Convergence of iterates of asymptotically nonexpansive mappings in
Banach spaces with the uniform Opial property, Collo. Math.
textbf{65(2)} (1993), 169-179.
F. Bruhat and J. Tits, "Groups reductifs sur
un corps local", Institut des Hautes Etudes Scientifiques.
Publications Mathematiques, Vol. textbf{41}, 5-251, 1972.
S. Dhompongsa and B. Panyanak, On
$triangle$-convergence theorem in CAT(0) spaces, Comput. Math.
Appl. textbf{56}, no.10, pp. 2572-2579, 2008.
J.B. Diaz, F.T. Metcalf, On the structure of the
set of subsequential limit points of successive approximations,
Bull. Amer. Math. Soc. textbf{73} (1967), 516-519.
H. Fukhar-ud-din, S.H. Khan, Convergence of
iterates with errors of asymptotically quasi-nonexpansive and
applications, J. Math. Anal. Appl. textbf{328} (2007), 821-829.
K. Goebel and W.A. Kirk, A fixed point
theorem for asymptotically nonexpansive mappings, Proc. Amer.
Math. Soc. textbf{35} (1972), 171-174.
K. Goebel and S. Reich, Uniform convexity,
hyperbolic geometry, and nonexpansive mappings, Vol. textbf{83}
of Monograph and Textbooks in Pure and Applied Mathematics, Marcel
Dekker Inc., New York, NY, USA, 1984.
S. Imnang and S. Suantai, Common fixed points
of multi-step Noor iterations with errors for a finite family of
generalized asymptotically quasi-nonexpansive mappings, Abstr.
Appl. Anal. Vol. 2009, Article ID 728510, 14 pages, 2009.
M.A. Khamsi and W.A. Kirk, An introduction to
metric spaces and fixed point theory, Pure Appl. Math,
Wiley-Interscience, New York, NY, USA, 2001.
S.H. Khan and M. Abbas, Strong and
$triangle$-convergence of some iterative schemes in CAT(0)
spaces, Comput. Math. Appl. Vol. textbf{61}, no.1, 109-116, 2011.
A.R. Khan, M.A. Khamsi and H. Fukhar-ud-din,
Strong convergence of a general iteration scheme in CAT(0) spaces,
Nonlinear Anal.: Theory, Method and Applications, Vol.
textbf{74}, (2011), no.3, 783-791.
W.A. Kirk, Fixed point theory in CAT(0) spaces
and $mathbb{R}$-trees, Fixed Point and Applications, Vol. 2004,
no.4, 309-316, 2004.
W.A. Kirk, Fixed point theorems for
non-lipschitzian mappings of asymptotically nonexpansive type,
Israel J. Math. textbf{17} (1974), 339-346.
W.A. Kirk, Geodesic geometry and fixed point
theory, in Seminar of Mathematical Analysis (Malaga/Seville,
/2003), Vol. textbf{64} of Coleccion Abierta, 195-225,
University of Seville Secretary of Publications, Seville, Spain,
W.A. Kirk, Geodesic geometry and fixed point
theory II, in International Conference on Fixed point Theory and
Applications, 113-142, Yokohama Publishers, Yokohama, Japan, 2004.
Q.H. Liu, Iterative sequences for
asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl.
(2001), 1-7.
Q.H. Liu, Iterative sequences for
asymptotically quasi-nonexpansive mappings with error member, J.
Math. Anal. Appl. 259(2001), 18-24.
Y. Niwongsa and B. Panyanak, Noor iterations
for asymptotically nonexpansive mappings in CAT(0) spaces, Int. J.
Math. Anal. Vol. textbf{4}, no.13, 2010, 645-656.
D.R. Sahu, J.S. Jung, Fixed point
iteration processes for non-Lipschitzian mappings of
asymptotically quasi-nonexpansive type, Int. J. Math. Math. Sci.
(2003), 2075-2081.
A. Sahin, M. Basarir, On the strong convergrnce of
a modified S-iteration process for asymptotically
quasi-nonexpansive mapping in CAT(0) space, Fixed Point Theory and
Applications 2013, textbf{2013:12}.
G.S. Saluja, Strong convergence theorem for
two asymptotically quasi-nonexpansive mappings with errors in
Banach space, Tamkang J. Math. textbf{38(1)} (2007), 85-92.
J. Schu, Weak and strong convergence to fixed
points of asymptotically nonexpansive mappings, Bull. Austral.
Math. Soc. textbf{43(1)} (1991), 153-159.
H.F. Senter, W.G. Dotson, Approximating fixed
points of nonexpansive mappings, Proc. Amer. Math. Soc.
textbf{44} (1974), 375-380.
N. Shahzad, A. Udomene, Approximating
common fixed points of two asymptotically quasi-nonexpansive
mappings in Banach spaces, Fixed Point Theory and Applications,
Vol. 2006, Article ID 18909, Pages 1-10.
N. Shahzad, Fixed point results for multimaps
in CAT(0) spaces, Topology and its Applications, Vol.
textbf{156}, no.5, 997-1001, 2009.
K.K. Tan and H.K. Xu, Approximating fixed points of
nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. textbf{178} (1993), 301-308.
K.K. Tan and H.K. Xu, Fixed point iteration processes for
asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122(1994), 733-739.
Refbacks
- There are currently no refbacks.