Iterative Methods For Approximating Fixed Points For Nonexpansive Mappings

In this paper, firstly for a uniformly smooth Banach space X, supposed C be a nonempty closed convex subset of X, we proved that the iterative sequence {x_n} strongly converges to a common fixed point of nonexpansive mappings T and S. This result extends and improves the corresponding ones of Tae-Hwa Kim and Hong-Kun Xu[2, Nonlinear Anal(TMA)61(2005), 51-60].Subsequently, in a uniformly convex Banach space X which has a uniformly Gateaux norm, we proposed a new viscosity iterative sequence {x_n} for nonexpansive self-mapping, We proved that {x_n} converges strongly to a fixed point of nonexpansive mapping T as n → ∞. Our result extends and improves the corresponding results by A.Moudafi[22, J.Math.Anal.Appl. 241(2000)No.1, 46-55], Xu Hong-kun[8, J.Math. Anal.Anal.298(2004)279-291] and etc. Because the non-expansibility of m — accretive operators, we extended the viscosity iteration of nonexpansive nonself-mapping and considered the viscosity iteration of accretive operators. Then we proved its strong convergence in reflexive Banach space, the results extend and improve the corresponding ones of Hong-kun Xu[29, J.Math.Anal.Appl. 314(2006)631-643].Finally for nonexpansive nonself-mapping, we firstly give two new viscosity iterations in a uniformly smooth Banach space. We proved {x_t}和{x_n} strongly converge to a fixed point of T by sunny nonexpansive retraction. Then, in a Hilbert space, we proved the iteration {x_n} strongly converges to a fixed point of nonexpansive nonself-mapping in the weaker condition Tx_n+1 — Tx_n → 0, (n → ∞) by Banach Limit method. The results presented extend and improve the corresponding ones of Hong-kun Xu[8, J.Math.Anal.Anal. 298(2004)], Yi-sheng Song, Ru-dong Chen[18, J.Math.Anal.Anal. (2006)] and Witmann [14, Archs.Math. 58(1992)486-491] and so on.