The purpose of this paper is to report about a self-consistent study of acceleration of multiply reflected pickup protons at a quasi-perpendicular solar wind termination shock and the influence of these protons on the shock's structure. A time-dependent formulation is developed by treating the solar wind and low-energy pickup protons as one fluid under the influence of the accelerated pickup protons as a second fluid. The results suggest that the termination shock would experience modest mediation by such accelerated pickup protons if the width of the termination shock's ramp/subramp would be equal to the thermal electron inertial length. Although such a narrow ramp/subramp width has not been detected yet, there are examples in observations of the Earth's bow shock with a width close to this value. The shock compression ratio is reduced at most to ~2.9 compared with the initial value of ~3.1 when the distribution of multiply reflected ions (MRI) is assumed to be fMRI∝p-3 and the fraction of reflected pickup protons is ~14%. Qualitatively, stronger mediation is predicted for a harder MRI spectrum or a larger fraction of reflected pickup protons. The accelerated pickup protons reach a maximum energy of at least ~220--240 keV, which is consistent with recent hybrid calculations performed in the test particle limit. Such extended MRI spectra for pickup protons, if realistic, might enable a direct injection of these particles into the process of diffusive shock acceleration at the termination shock. Therefore the model also provides a theoretical basis for a study of the formation of anomalous cosmic rays. ¿ 2000 American Geophysical Union