Some convergence theorems for asymptotically pseudocontractive mappings

Let K be a nonempty closed convex subset of a real Banach space E,T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n ≥ 0 $\subset$ [1, ∞), limn → ∞ kn = 1 such that p $\in$ F(T) = {x $\in$ K : Tx = x}. Let {αn}n ≥ 0 $\subset$ [0,1] be such that ∑n ≥ 0 αn = ∞ and limn → ∞ αn = 0. For arbitrary x0 $\in$ K and {vn}n ≥ 0 in K let {xn}n ≥ 0 be iteratively defined by xn + 1 = (1 - αn)xn + αn Tnvn, n ≥ 0, satisfying limn → ∞ ||vn - xn|| = 0. Suppose there exists a strictly increasing function φ : [0, ∞) → [0, ∞), φ (0) = 0 such that ≤ kn ||x - p||2 - φ (||x - p||), ∀x $\in$ K. Then {xn}n ≥ 0 converges strongly to p $\in$ F (T). The remark at the end is important.