<p>In this work, we propose some new conditions to obtain strong convergence results for solving variational inequalities in real Hilbert spaces. We consider two algorithms that require only a single projection onto the feasible set per iteration, and without requiring the monotonicity condition of the variational inequality mapping to approximate a solution under our new conditions. We prove that if the solution to the dual variational inequality is nonempty, then the sequence generated by our methods converges globally to a solution. Finally, we provide some numerical results. Additionally, we apply the proposed algorithms to a network equilibrium flow problem.</p>