CONVERGENCE THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE NONSELF MAPPINGS IN THE INTERMEDIATE SENSE IN UNIFORMLY CONVEX BANACH SPACES

Abstract

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 a nonself asymp-
totically nonexpansive in the intermediate sense mapping with F(T) = \emptyset .
Let {\alpha_n^i}, {\beta_n^i} and {\gamma_n^i} are sequences in [0,1] with
\alpha_n^i + \beta_n^i + \gamma_n^i =0 for all i=1,2,3. From arbitrary x_1 in K,
define the sequence {x_n} iteratively by (1.8), where {u_n^i} for all i = 1,2,3 are
bounded sequences in K with \sum_{n=1}^{\infty}u_n^i < \infty. (i) If the dual E^* of
E has the Kadec-Klee property, then {x_n} converges weakly to a xed point of T;
(ii) if T satises condition (A), then {x_n}, {y_n} and {z_n} converges strongly to a fixed
point of T. The results presented in this paper extend and improve the results in
[1, 4, 6, 11, 12, 17, 18, 22, 32] and many others