  We investigate cosmological solutions of the chameleon model with a
non-minimal coupling between the matter and the scalar field through a
conformal factor with gravitational strength. By considering the spatially flat
FLRW metric and the matter density as a non-relativistic perfect fluid, we
focus on the matter-dominated phase and the late-time accelerated phase of the
universe. In this regard, we manipulate and scrutinize the related field
equations for the density parameters of the matter and the scalar fields with
respect to the e-folding. Since the scalar field fluctuations depend on the
background and the field equations become highly non-linear, we probe and
derive the governing equations in the context of various cases of the relation
between the kinetic and potential energies of the chameleon scalar field, or
indeed, for some specific cases of the scalar field equation of state
parameter. Thereupon, we schematically plot those density parameters for two
different values of the chameleon non-minimal coupling parameter, and discuss
the results. In both considered phases, we specify that, when the kinetic
energy of the chameleon scalar field is much less than its potential energy
(i.e., when the scalar field equation of state parameter is $\simeq - 1 $), the
behavior of the chameleon model is similar to the $\Lambda CDM$ model. Such
compatibility suggests that the chameleon model is phenomenologically viable
and can be tested with the observational data.
