Convergence Theorems for Mappings Which Are Asymptotically Nonexpansive in the Intermediate Sense

Abstract

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} ≠ ∅. 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.