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We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If T : C → H is a non-self and non-expansive mapping, we can define a map h : C → R by h(x) := inf{λ ≥ 0 : λx + (1 -λ)Tx ∈ C}. Then, for a fixed x 0 ∈ C and for α 0 := max{1/2, h(x 0 )}, we define the Krasnoselskii-Mann algorithm x n+1 = α n x n + (1 -α n )Tx n , where α n+1 = max{α n , h(x n+1 )}. We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.

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