An iterative algorithm on approximating fixed points of pseudocontractive mappings

Let e be a real reflexive banach space with a uniformly gteaux differentiable norm. let k be a nonempty bounded closed convex subset of e,and every nonempty closed convex bounded subset of k has the fixed point property for non-expansive self-mappings. let f: k → k a contractive mapping and t: k → k be a uniformly continuous pseudocontractive mapping with f (t) ≠ φ. let {λ n} ⊂ (0,1/2) be a sequence satisfying the following conditions: (i) lim n→∞ λ n = 0; (ii) σ n=0 ∞ λ n = ∞. define the sequence {x n} in k by x 0 ∈ k,x n+1 = λ n f(x n) + (1 - 2λ n)x n + λ ntx n,for all n ≥ 0. under some appropriate assumptions,we prove that the sequence {x n} converges strongly to a fixed point p ∈ f (t) which is the unique solution of the following variational inequality: f (p) - p,j (z - p) ≤ 0,for all z ∈ f (t). © copyright 2012 youli yu.