Decaying $\mathrm{\Lambda }$ cosmologies and statistical properties of gravitational lenses

In this paper we investigate the statistical properties of gravitational lenses for models in which a cosmological term decreases with time as $\Lambda \propto a^{-m}$,where$a$isthescalefactorand$m$isaparameter($0 \leq m < 3$). We show that for given low values of the present matter density parameter ${\mathrm{\Omega }}_{m0}$, there is a wide range of values for $m$ for which the lensing rate is significantly smaller than that in cosmological constant ($\mathrm{\Lambda }$) models. We also show that models with low ${\mathrm{\Omega }}_{m0}$ and $m2$ have high likelihood to reproduce the observed lens statistics in the HST snapshot survey.