Convergence Theorems of Finite-step Iteration with Errors for Non-self Asymptotically Nonexpansive in the Intermediate Sense Mappings

Let K be a nonempty closed convex nonexpansive retract of a uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be non-self asymptotically nonexpansive in the intermediate sense mapping with F (T ) 6= φ. Let {α n }, {β n } and {γ n } are sequences in [0, 1] with α (i) n + β (i) n + γ (i) n = 1 for all i = 1, 2, . . . , N . From arbitrary x1 ∈ K, define the sequence {xn} iteratively by (8), where {u n } for all i = 1, 2, . . . , N are bounded sequences in K with ∑∞ n=1 u (i) n < ∞. (i) If the dual E∗ of E has the Kadec-Klee property, then {xn} converges weakly to a fixed point of T ; (ii) if T satisfies condition (A), then {xn} converges strongly to a fixed point of T . The results presented in this paper extend and improve the corresponding results of Rhoades [1], Chidume et al. [4, 6], Schu [11, 12], Osilike and Aniagbosor [18], Tan and Xu [17], Plubtieng and Wangkeeree [21], Su and Qin [31] and many others.