Strong convergence of approximants to fixed points of Lipschitzian pseudocontractive maps

Abstract

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 ϵ int ( K), ∥ z − Tz∥ < ∥ x − Tx∥,for all x ϵ ∂( K). Then for z 0 ϵ K arbitrary, the iteration process { z n } defined by z n + 1 ≔ (1 − μ n + 1) z + μ n + 1 y n ; y n ≔ ( 1 −- α n) z n + α n Tz 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.