In this paper we constrain four alternative models to the late cosmic acceleration in the Universe: Chevallier-Polarski-Linder (CPL), interacting dark energy (IDE), Ricci holographic dark energy (HDE), and modified polytropic Cardassian (MPC). Strong lensing (SL) images of background galaxies produced by the galaxy cluster Abell $1689$ are used to test these models. To perform this analysis we modify the LENSTOOL lens modeling code. The value added by this probe is compared with other complementary probes: Type Ia supernovae (SNIa), baryon acoustic oscillations (BAO), and cosmic microwave background (CMB). We found that the CPL constraints obtained of the SL data are consistent with those estimated using the other probes. The IDE constraints are consistent with the complementary bounds only if large errors in the SL measurements are considered. The Ricci HDE and MPC constraints are weak but they are similar to the BAO, SNIa and CMB estimations. We also compute the figure-of-merit as a tool to quantify the goodness of fit of the data. Our results suggest that the SL method provides statistically significant constraints on the CPL parameters but weak for those of the other models. Finally, we show that the use of the SL measurements in galaxy clusters is a promising and powerful technique to constrain cosmological models. The advantage of this method is that cosmological parameters are estimated by modelling the SL features for each underlying cosmology. These estimations could be further improved by SL constraints coming from other galaxy clusters.
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