An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

Abstract EN

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).