Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, T :K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n 0 ⊂ [1,+∞), limn→∞kn = 1 such that F(T ) = ∅. Let {αn}n 0 ⊂ [0, 1] be such that n 0 αn =∞, n 0 α2 n <∞ and n 0 αn(kn −1) <∞. Suppose {xn}n 0 is iteratively defined by xn+1 = (1 − αn)xn + αnT nxn, n 0, and suppose there exists a strictly increasing continuous function φ : [0,+∞)→[0,+∞), φ(0) = 0 such that T nx − x∗, j (x − x∗) kn x − x∗ 2 − φ( x − x∗ ), ∀x ∈ K. It is proved that {xn}n 0 converges strongly to x∗ ∈ F(T ). It is also proved that the sequence of iteration {xn} defined by xn+1 = anxn+bnT nxn+cnun, n 0 (where {un}n 0 is a bounded sequence in K and {an}n 0, {bn}n 0, {cn}n 0 are sequences in [0, 1] satisfying appropriate conditions) converges strongly to a fixed point of T . © 2005 Elsevier Inc. All rights reserved.