Some Results For Convergence For Generalized Asymptotically Quasi-nonexpansive Mappings

Quasi-nonexpansive type mapping is always the main themes in the world nowadays.Many domestic and international of scholars have studied it. And in 2004 Tian Youxian proved necessary and sufficient condition for the Ishikawa iterative sequence to converge to a fixed point of quasi-nonexpansive type mapping in convex metric space and he spreaded the theme.In 2005 Xiang Changhe in the cultural heritage[8] put forward the definit of generalized quasi-nonexpansive type mapping in the Banach space:Let E is a real Banach space with C is its nonempty subset.Recall that a mapping T:C→C. F(T) = {x∈C,Tx = x} is all the fixed point of T. T is generalized quasi-nonexpansive type mapping,if F(T) is nonempty and exists {k_n} (?) [0,+∞), (?)k_n = 0 ,such that (?).Because in 2005 Xiang Changhe just discussed a fixed point of quasi-nonexpansive type mapping in the Banach space. And the author be inspire by it and lead arious new of convex structure and quasi-nonexpansive type mapping in the convex metric space . Further do, {u_n}, {v_n }are boundary, F (T) is boundary and nonempty, get Ishikawa itertion process {x_n }_n=1^∞converge to a fixed point of quasi-nonexpansive type mapping is main result of part l,and Ishikawa itertion process with errors {x_n }_n=1^∞converge to a fixed point of quasi-nonexpansive type mapping is main result of part 2 that suppose {x_n }_n=1^∞is Ishikawa itertion process with errors,if and only if (?)infd(x_n,F(T)) = 0, {x_n }_n=1^∞converge to a fixed point of T.But in the convex metric space, Ishikawa itertion process isγ_n =γ'_n =0 and u_n = v_n =0 of special situation of it.Same, in recent years, the problem of Mann and Ishikawa itertion process with errors converges is researched by many scientists. S.H.Khan^[15] discussed necessary and sufficient condition for the Mann and Ishikawa itertion process with errors converge to a fixed point of quasi-nonexpansive type mapping and strong converge to a fixed point. The author in this paper get the definit of Mann and Ishikawa itertion process with errors for generalized quasi-nonexpansive type mapping.And in real Banach space use the technique of[16], get the necessary and sufficient condition for the Mann and Ishikawa itertion process with errors converge to a fixed point of two generalized quasi-nonexpansive type mappings and in uniformly convex Banach space it strong converge to a fixed point of two generalized quasi-nonexpansive type mappings...