Approximating xed points of generalized non-expansive non-self mappings in CAT(0) spaces

Document Type: Research Articles

Authors

1 Department of Mathematics, Izeh Branch, Islamic Azad University, Izeh, Iran.

2 Department of Mathematics, Takestan Branch, Islamic Azad University, Takestan, Iran.

Abstract

Suppose K is a nonempty closed convex subset of a complete CAT(0) space
X with the nearest point projection P from X onto K. Let T : K → X be a
nonself mapping, satisfying condition (C) with F(T) :={ x ε K : Tx = x}≠Φ.
Suppose fxng is generated iteratively by x1ε K, xn+1 = P((1- αn)xn+αnTP[(1- αn)xn+β nTxn]),n≥1, where {αn }and {βn } are real sequences in[ε,1-ε] for some  ε in (0,1). Then {xn} is Δ-􀀀convergence to some point x* in
F(T). This work extends a result of Laowang and Panyanak [5] to the case of
generalized nonexpansive nonself mappings.

[1] M. Bridson and A. Hae iger, Metric Spaces of Non-Positive Curvature, vol. 319
of Fundamental Principles of Mathematical Sciences, Berlin, Germany, 1999.
[2] S. Dhompongsa and W. Kirk and B. Sims, xed point of uniformly lipschitzian
mappings, Nonlinear Anal. 65 (2006), 762{772.
[3] S. Dhompongsa and B. Panyanak, On -convergence theorems in CAT(0)
spaces. Computers and Mathematics with Applications. 56 (2008), 2572{2579.
[4] W. Kirk and B. Panyanak, A concept of convergence in geodesic spaces. Nonlinear
Anal. 68 (2008), 3689{3696.
[5] W. Laowang and B. Panyanak, Approximating xed points of nonexpansive non-
self mappings in CAT(0) spaces. Fixed Point Theory Appl. Article ID 367274
(2010), 11 pages.
[6] T. Suzuki, Fixed point theorems and convergence theorems for some generalized
nonexpansive mapping, Math. Anal. Appl. 340 (2008), 1088{1095.
[7] A. Razani and H. Salahifard, Invariant approximation for CAT(0) spaces, Non-
linear Anal. 72 (2010), 2421{2425.
[8] S. Dhompongsa, W. A. Kirk and B. Panyanak, Nonexpansive set-valued mappings
in metric and Banach spaces, J. Nonlinear and Convex Anal. 8 (2007), 35-45.