The article deals with the equilibrium problem
where is a nonempty closed convex subset of a real Banach space with the dual , . The special case of this problem is the known problem of solving a variational inequality; in this problem , . It is assumed that is a strictly convex reflexive Banach space with the Kadec–Klee property and Fréchet differentiable norm, a family of closed hemi-relatively nonexpansive mappings of into itself with a common fixed point, a sequence in converging to zero. The following iterative scheme is studied:
here is the duality mapping, , is the generalized projection from onto . It is proved that converges strongly to , where is the generalized projection from onto , is the set of common fixed points of . The following modified iterative scheme
is considered too; in this case it is assumed that satisfies the following properties: (A1) ; (2) ; (3) ; (4) the functions , , are convex and lower semicontinuous.