Summary: Let \(E\) be a Banach space, \(C\) a nonempty closed convex subset of \(E,f: C\to C\) a contraction, and \(T_u: C\to C\) a nonexpansive mapping with nonempty \(F:=\bigcap_{i=1}^N \text{Fix}(T_i)\), where \(N\geq 1\) is an integer and \(\text{Fix}(T_i)\) is the set of fixed points of \(T_i\). Let \(\{x_t^n\}\) be the sequence defined by \(x_t^n= tf(x_t^n)+(1-t)T_{n+N} T_{n+N-1}\cdots T_{n+1}x_t^n\) \((0<t<1)\). First, it is shown that as \(t\to 0\), the sequence \(\{x_t^n\}\) converges strongly to a solution in \(F\) of certain variational inequality provided that \(E\) is reflexive and has a weakly sequentially continuous duality mapping. Then it is proved that the iterative algorithm \(x_{n+1}= \lambda_{n+1}f(x_n)+ (1-\lambda_{n+1})T_{n+1}x_n\) \((n>0)\) converges strongly to a solution in \(F\) of a certain variational inequality in the same Banach space, provided that the sequence \(\{\lambda_n\}\) satisfies certain conditions and the sequence \(\{x_n\}\) is weakly asymptotically regular. Applications to the convex feasibility problem are included.