  In the framework inspired by R. L. Hughes model (Transp. Res. B, 2002) for
pedestrian evacuation in a corridor, we establish existence of a solution by a
topological fixed point argument. This argument applies to a class of models
where the dynamics of the pedestrian density $\rho$ (governed by a
discontinuous-flux Lighthill,Whitham and Richards model $\rho$t + (sign(x --
$\xi$(t))$\rho$v($\rho$)) x = 0) is coupled via an abstract operator to the
computation of a Lipschitz continuous "turning curve" $\xi$. We illustrate this
construction by several examples, including the standard Hughes' model with
affine cost, and either with open-end conditions or with conditions
corresponding to panic behaviour with capacity drop at exits. Other examples
put forward versions of the Hughes model with inertial dynamics of the turning
curve and general costs.
