Iterative Methods for Fixed Points of Asymptotically Weakly Contractive Maps
Robert Pertsch Gilbert, C.E. Chidume, Habtu Zegeye, S. J. Aneke · Applicable Analysis · 2003
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 {kn }, kn ≥ 1, lim kn = 1 and with F(T) n int (K) ≠ ø F(T):= {x ∈ K: Tx = x}. Suppose {x n } is iteratively defined by x n+1 = P((l − knαn )x n +k n α n T(PT) n−l xn ), n = 1,2,...,x 1 ∈ K, where αn∈ (0,l) satisfies lim αn = 0 and Σαn = ∞. 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 {xn } converges strongly to some x * ∈ F(T).