Let $H$ be a real Hilbert space with inner product $\langle \cdot, \cdot \rangle$. Let $\xi, \eta$ be elements of $H$, $\rho$ and $\beta$ be positive constants, and $K$ a nonempty closed convex subset of $H$. Let $f, g, N: H \to H$ and $M: H \times H \to H$ be nonlinear mappings with $K \subset g(H)$. The paper is concerned with the following problem: find $x, y \in H$ such that $f(x), g(y) \in K$ and $\langle \rho(M(x, g(y))-\xi)+f(x)-g(y), v-f(x) \rangle \geq 0$, $\langle \beta(N(x)-\eta)+g(y)-f(x), v-g(y) \rangle \geq 0$, $\forall v \in K$. The authors construct an iterative algorithm for the problem and establish strong convergence of the algorithm under appropriate monotonicity and continuity conditions on the operators.