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, Lp or lp 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 z0 ε K arbitrary, the iteration process {zn} defined by zn + 1 (1 − μn + 1)z + μn + 1yn; yn ( 1 −- αn)zn + αn Tzn converges strongly to a fixed point of T, provided that {μn} and {αn}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.