Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be a non-self mapping which is asymptotically nonexpansive in the intermediate sense with F(T) := {x ∈ K : Tx = x} ≠ Ø. A demiclosed principle for T is proved. Moreover, if T is completely continuous, an iterative sequence {x_n} is constructed which converges strongly to some x* ∈ F(T). If T is not assumed to be completely continuous but the dual E* of E is assumed to have the Kadec-Klee property, then {x_n} converges weakly to some x* ∈ F(T). The operator P which plays a central role in our proofs is, in this case, the Banach space analogue of the proximity map in Hilbert spaces.