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Research articles

ScienceAsia 37 (2011): 145-151 |doi: 10.2306/scienceasia1513-1874.2011.37.145

Halpern iteration of Cesàro means for asymptotically nonexpansive mappings

Qingnian Zhanga, Yisheng Songb,*

ABSTRACT:     Using a new proof technique which is independent of the approximation fixed point of T (limn→∞|xnTxn|=0) and the convergence of the Browder type iteration path (zt=tu+(1−t)Tzt), the strong convergence of the Halpern iteration {xn} of Cesàro means for asymptotically nonexpansive self-mappings T, defined by xn+1nu+(1−αn)(n+1)−1j=0nTjxn for n≥0, is proved in a uniformly convex Banach space E with a uniformly Gâteaux differentiable norm whenever {αn} is a real sequence in (0,1) satisfying the conditions limn→∞bnn=0 and limn→∞αn=0 and ∑n=0αn=∞.

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a College of Mathematics and Information Science, North China University of Water Conservancy and Electric Power, ZhengZhou 450011, China
b College of Mathematics and Information Science, Henan Normal University, XinXiang, HeNan 453007, China

* Corresponding author, E-mail: songyisheng123@yahoo.com.cn

Received 18 Jun 2010, Accepted 1 May 2011