Summary: We introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of [G. Marino
and H. K. Xu
, J. Math. Anal. Appl. 318, No. 1, 43–52 (2006; Zbl 1095.47038
); S. Takahashi
and W. Takahashi
, ibid. 331, No. 1, 506–515 (2007; Zbl 1122.47056
)], and many others.