Strong convergence theorems for uniformly continuous pseudocontractive maps

Let K be a nonempty closed convex subset of a real Banach space E and let T :K →K be a uniformly continuous pseudocontraction. Fix any u ∈ K. Let {xn} be defined by the iterative process: x0 ∈ K, xn+1 := μn(αnT xn + (1 − αn)xn) + (1 − μn)u. Let δ( ) denote the modulus of continuity of T with pseudoinverse φ. If {φ(t)/t: 0 < t <1} and {xn} are bounded then, under some mild conditions on the sequences {αn}n and {μn}n, the strong convergence of {xn} to a fixed point of T is proved. In the special case where T is Lipschitz, it is shown that the boundedness assumptions on {φ(t)/t: 0 < t <1} and {xn} can be dispensed with. © 2005 Elsevier Inc. All rights reserved