Summary: We propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction mapping in the setting of real Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our proposed scheme. Our results combine the ideas of G. Marino
and H. K. Xu
[J. Math. Anal. Appl. 329, No. 1, 336–346 (2007; Zbl 1116.47053
)] and S. Takahashi
and W. Takahashi
[J. Math. Anal. Appl. 331, No. 1, 506–515 (2007; Zbl 1122.47056
)]. In particular, necessary and sufficient conditions for the strong convergence of our iterative scheme are obtained.