Decaying $\Lambda$ 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 $a$ is the scale factor and $m$ is a parameter ($0 \leq m < 3$). We show that for given low values of the present matter density parameter $\Omega_{m0}$, there is a wide range of values for $m$ for which the lensing rate is significantly smaller than that in cosmological constant ($\Lambda$) models. We also show that models with low $\Omega_{m0}$ and $m\stackrel{>}{\sim}2$ have high likelihood to reproduce the observed lens statistics in the HST snapshot survey.