Abstract
Let E be a reflexive Banach space with a uniformly Gâteaux
differentiable norm, let K be a nonempty closed convex subset of
E, and let T:K→K be a uniformly continuous
pseudocontraction. If f:K→K is any contraction map
on K and if every nonempty closed convex and bounded subset of
K has the fixed point property for nonexpansive self-mappings,
then it is shown, under appropriate conditions on the sequences of
real numbers {αn}, {μn}, that the iteration process z1∈K, zn+1=μn(αnTzn+(1−αn)zn)+(1−μn)f(zn), n∈ℕ, strongly converges to the fixed point of T, which is the unique solution of some variational inequality,
provided that K is bounded.