  Constraints on cosmological models from strong gravitational lensing
statistics are investigated. We pay particular attention to the role of the
velocity function in the calculation of the lensing probability. The velocity
function derived from the observed galaxy luminosity function, which is used in
most previous work, is unable to predict the large separation lensing events.
In this paper, we also use the Press-Schechter theory to construct a velocity
function theoretically. Model predictions are compared with the observed
velocity function and the HST snapshot survey. Comparison with the latter
observation shows that the predictions based on the theoretical velocity
function are consistent with the observed large separation events in COBE
normalized low-density models, especially with a non-vanishing cosmological
constant. Adopting the COBE normalization, however, we could find no model
which simultaneously satisfies both the observed velocity function and the HST
snapshot survey. We systematically investigate various uncertainties in the
gravitational lensing statistics including finite core radius, the distance
formula, magnification bias, and dust obscuration. The results are very
sensitive to these effects as well as theoretical models for the velocity
function, implying that current limits on the cosmological parameters should be
interpreted with caution. Predictions for future surveys are also presented.
