Approximating fixed points of nonexpansive mapping by the Ishikawa process in Banach space
Yong Liu · Journal of Natural Science of Heilongjiang University · 2004
Let be a uniformly convex Banach space whose dual has the KK property. Let C be a bounded closed convex subset of X. Let T:C~1→C be a nonexpansive mapping. It is shown that for any initial guess x_0∈C, the Ishikawa iteration {x_n}~_(n=0)~(+∞) defined by x_(n+1)=t_nT(S_nTx_n+(1-s_n)x_n)+(1+t_n)x_n, with the restriction that limsup_(n→+∞)s_n is less than 1 and sum from k=o to +∞(t_(n_k)(1-t_(n_k))) diverges for any subsequence {n_k}_(k=0)~(+∞) of {n}_(n=0)~(+∞), converges weakly to a fixed point of T. This generalizes the corresponding results in [3, 5-6]. It should be emphasized that the condition X~* has the KK property is strictly weaker than the condition X has a Frechet differentiable norm which is required in paper [6].