An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

Let E be a real reflexive Banach space with a uniformly Gâteaux 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)≠∅. Let {λ_n}⊂(0, 1/2) be a sequence satisfying the following conditions: (i) lim_n→∞λ_n = 0; (ii) $$. Define the sequence {x_n} in K by x₀ ∈ K, x_n+1 = λ_nf(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).