Abstract: 

Consider a city with several facilities competing for multi-class users that are distributed continuously over space. Within the city region, the road network is relatively dense and is considered as a continuum. The demand distribution function is formulated as a logit-type function for modeling the probabilistic choice behavior of facilities for different classes of users. A minimization problem that describes this multi-class and multi-facility choice is set up for finding the user equilibrium flow pattern. This paper aims to study the existence and uniqueness of a solution of this multi-class and multi-facility user equilibrium problem with demand distribution. The assumptions and conditions for the existence and uniqueness of solution are given.