Some convergence theorems for asymptotically pseudocontractive mappings

Abstract

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 {k_n}_{n ≥ 0} $\subset$ [1, ∞), lim_{n → ∞} k_n = 1 such that p $\in$ F(T) = {x $\in$ K : Tx = x}. Let {α_n}_{n ≥ 0} $\subset$ [0,1] be such that ∑_{n ≥ 0} α_n = ∞ and lim_{n → ∞} α_n = 0. For arbitrary x₀ $\in$ K and {v_n}_{n ≥ 0} in K let {x_n}_{n ≥ 0} be iteratively defined by

x_{n + 1} = (1 - α_n)x_n + α_n Tⁿv_n, n ≥ 0,

satisfying lim_{n → ∞} ||v_n - x_n|| = 0. Suppose there exists a strictly increasing function φ : [0, ∞) → [0, ∞), φ (0) = 0 such that

<Tⁿx - p, j (x - p)> ≤ k_n ||x - p||² - φ (||x - p||), ∀x $\in$ K.

Then {x_n}_{n ≥ 0} converges strongly to p $\in$ F (T).

The remark at the end is important.