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 {λn} ⊂ 0, 1/2 be a sequence satisfying the following conditions: i limn→∞λn 0; ii ∑∞ n 0 λn ∞. Define the sequence {xn} in K by x0 ∈ K, xn 1 λnf xn 1 − 2λn xn λnTxn, for all n ≥ 0. Under some appropriate assumptions, we prove that the sequence {xn} 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 .