  This paper is concerned with dynamic user equilibrium with elastic travel
demand (E-DUE) when the trip demand matrix is determined endogenously. We
present an infinite-dimensional variational inequality (VI) formulation that is
equivalent to the conditions defining a continuous-time E-DUE problem. An
existence result for this VI is established by applying a fixed-point existence
theorem (Browder, 1968) in an extended Hilbert space. We present three
algorithms based on the aforementioned VI and its re-expression as a
differential variational inequality (DVI): a projection method, a self-adaptive
projection method, and a proximal point method. Rigorous convergence results
are provided for these methods, which rely on increasingly relaxed notions of
generalized monotonicity, namely mixed strongly-weakly monotonicity for the
projection method; pseudomonotonicity for the self-adaptive projection method,
and quasimonotonicity for the proximal point method. These three algorithms are
tested and their solution quality, convergence, and computational efficiency
compared. Our convergence results, which transcend the transportation
applications studied here, apply to a broad family of infinite-dimensional VIs
and DVIs, and are the weakest reported to date.
