We consider the nonlinear propagation of electron-acoustic waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The properties of small but finite amplitude electron-acoustic solitary waves (EASWs) are studied by employing the reductive perturbation technique, whereas those of arbitrary amplitude EASWs are investigated by means of the pseudopotential approach. It is found that the present plasma model supports EASWs having a positive potential, which corresponds to a hole (hump) in the cold (hot) electron number density. It is also shown that as the trapped electron temperature increases, the amplitude of the EASWs increases but their width decreases. The application of our results in interpreting the salient features of the broadband electrostatic noise in the dayside auroral zone is discussed.