Let \(H\) be a Hilbert space and \(C\) a closed convex subset of \(H\). An equilibrium function is a mapping \(G:H\times H\to \mathbb R\) such that (A\(_1\)) \(G(x,x)=0\) for all \(x\in H\). A strongly positive operator is a bounded linear operator \(A:H\to H\) such that for all \(x\in H\), \(\langle Ax,x\rangle\geq \bar{\gamma}\|x\|^2\) for some \(\bar{\gamma}>0\). Supposing that the equilibrium function \(G\) satisfies further the conditions (A\(_2\)) for all \(x,y\in C\), \(G(x,y)+G(y,x)\leq 0\) (i.e., \(G\) is monotone); (A\(_3\)) for all \(x,y,z \in C\) \(\limsup_{t\to 0}G(tz+(1-t)x,y)\leq G(x,y)\), and (A\(_4\)) for all \(x\in C\), the mapping \(G(x,\cdot)\) is convex and lsc. S.Plubtieng and R.Punpaeng [J. Math.Anal.Appl.336, No.1, 455–469 (2007; Zbl 1127.47053)] proposed an iteration procedure to find the unique solution \(z\in\) \(\text{Fix}(T)\cap\text{SEP}(G)\) of the variational inequality: (1) \(\langle(A-\gamma f)z,z-x\rangle\leq 0, \) for all \(x\in\text{Fix}(T)\cap\text{SEP}(G)\). Here, \(T\) is a nonexpansive mapping on \(H\), \(A\) is a strongly positive operator on \(H\), \(f\) an \(\alpha\)-contraction on \(H\) and \(\gamma > 0\) an appropriate constant. In this paper, the authors extend the above result by considering a family \(G_i,\, i=1,\dots,K,\) of equilibrium functions satisfying (A\(_2\))–(A\(_4\)) and a family \((T_n)_{n\in \mathbb N}\) of nonexpansive mappings. Supposing that \(D:=\cap_{i=1}^K\)SEP\((G_i)\cap\cap_{n\in \mathbb N}\)Fix\((T_n)\neq \varnothing\). They propose an implicit iteration procedure for finding the unique solution of the variational inequality (1) on the set \(D\) and prove the strong convergence of this procedure.