Weighted variational inequalities in non‐pivot Hilbert spaces with applications

We introduce variational inequalities defined in non‐pivot Hilbert spaces and we show some existence results. Then, we prove regularity results for weighted variational inequalities in non‐pivot Hilbert space. These results have been applied to the weighted traffic equilibrium problem. The continuity of the traffic equilibrium solution allows us to present a numerical method to solve the weighted variational inequality that expresses the problem. In particular, we extend the Solodov‐Svaiter algorithm to the variational inequalities defined in finite‐dimensional non‐pivot Hilbert spaces. Then, by means of a interpolation, we construct the solution of the weighted variational inequality defined in a infinite‐dimensional space. Moreover, we present a convergence analysis of the method.