Abstract
Let 
be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of 
has the fixed point property for nonexpansive mappings. Let 
be a nonempty closed convex subset of 
, 
a contractive mapping (or a weakly contractive mapping), and 
nonexpansive mapping with the fixed point set 
. Let 
be generated by a new composite iterative scheme: 
, 
, 
. It is proved that 
converges strongly to a point in 
, which is a solution of certain variational inequality provided that the sequence 
satisfies 
and 
, 
for some 
and the sequence 
is asymptotically regular.