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 {k(n)}(n >= 0) subset of [1, +infinity), lim(n ->infinity) k(n) = 1 such that F(T) not equal 0. Let {alpha(n)}(n >= 0) subset of [0, 1] be such that Sigma(n >= 0) alpha(n) = infinity, Sigma(n >= 0)alpha(2)(n) < infinity and Sigma(n >= 0)alpha(n)(k(n-1)) < infinity. Suppose {x(n)}(n >= 0) is iteratively defined by phi:[0, +infinity)->[0, +infinity), phi(0) = 0 n >= 0, and suppose there exists a strictly increasing continuous function phi: [0, +infinity) -> [0, +infinity), phi(0) = 0 such that < T(n)x-x*, j(x-x*)> <= k(n)parallel to x-x*parallel to(2)-phi(parallel to x-x*parallel to), for all x is an element of K. It is proved that {x(n)}(n >= 0) converges strongly to x* is an element of F(T). It is also proved that the sequence of iteration {x(n)} defined by x(n+1) = a(n)x(n) + b(n) T-n x(n)+c(n)u(n), n >= 0 (where {u(n)}(n >= 0) is abounded sequence in K and {a(n)}(n >= 0), {b(n)}(n >= 0), {c(n)}(n >= 0) are sequences in [0, 1] satisfying appropriate conditions) converges strongly to a fixed point of T. (c) 2005 Elsevier Inc. All rights reserved.