Iterative methods for fixed points of asymptotically weakly contractive maps

Suppose K is a closed convex nonexpansive retract of a real uniformly smooth Banach space E with P as the nonexpansive retraction. Suppose T : K → E is an asymptotically d-weakly contractive map with sequence {k _n }, k _n ≥ 1, lim k _n = 1 and with F(T) _n int (K) ≠ øF(T):= {x ∈ K: Tx = x}. Suppose {x _n } is iteratively defined by x _n+1 = P((l − k _n α _n )x _n +k _n α _n T(PT) ^n−l x _n ), n = 1,2,.., x ₁ ∈ K, where α _n ∈ (0, l) satisfies lim α _n = 0 and Σα _n = ∞. It is proved that {x _n } converges strongly to some x ^* ∈ F(T)∩ int K. Furthermore, if K is a closed convex subset of an arbitrary real Banach space and T is, in addition uniformly continuous, with F(T) ≠ ø, it is proved that {x _n } converges strongly to some x ^* ∈ F(T). ^† The author undertook this work when he was visiting the Abdus Salam International Center for Theoretical Physics, Trieste, Italy, as a postdoctoral fellow. ^‡ Department of Mathematics, University of Nigeria, Nsukka.