Strong convergence of approximants to fixed points of Lipschitzian pseudocontractive maps

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, L_p or l_p spaces, 1 < p < ∞) and K be a nonempty closed convex and bounded subset of E with φ ≠ int (K). Let T: K → K be a Lipschitzian pseudocontractive mapping such that for z ∈ int (K), ∥z - Tz∥ < ∥x - Tx∥, for all x ∈ ∂(K). Then for z₀ ∈ K arbitrary, the iteration process (z_n) defined by z_n+1: = (1 - μ_n+1)z+μ_n+1y_n; y_n: = (1 - α_n)z_n + α_nTz_n converges strongly to a fixed point of T, provided that (μ_n) and (α_n) satisfy certain conditions. Moreover, if T is strictly pseudocontractive with a nonempty fixed-point set, then it is proved that the Mann type iteration scheme converges strongly to a fixed point of T.