In this study, the authors proposed a new hybrid iteration sequence to find the common point of the set of solutions of the equilibrium problem and the set of fixed points of the mapping satisfying the conditions, establishing the convergence of this loop sequence in Banach space is smooth and convex. From this theorem, the authors obtained a consequence of the convergence of the iteration sequence for equilibrium and mapping problems satisfying conditions in real Hilbert space. At the same time, an example is given to illustrate the sequence convergence of the equilibrium problem and the fixed point problem of condition-satisfying mapping. The results of this study are extensive and improved by some of the results in the reference.