In this paper, we introduce two iterative shemes (one implicit and one explicit) by a modified Krasnoselskii-Mann algorithm for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of nonexpansive mappings in Hilbert spaces. We prove that both approaches converge strongly to a common element of the set of the equilibrium points and the set of fixed points of nonexpansive mappings. Such common element is the unique solution of a variational inequality, which is the minimum-norm element of the above two sets. Applications to split feasibility problem and optimization problem are given. Finally, numerical example is given to demonstrate the implementability of our algorithm.