Cosmic Evolution in Brans-Dicke Chameleon Cosmology

We have investigated the Brans-Dicke Chameleon theory of gravity and obtained exact solutions of the scale factor $a(t)$, scalar field $\varphi (t)$, an arbitrary function $f(\varphi )$ which interact with the matter Lagrangian in the action of the Brans-Dicke Chameleon theory and potential $V(\varphi )$ for different epochs of the cosmic evolution. We plot the functions $a(t)$, $\varphi (t)$, $f(t)$ and $V(\varphi )$ for different values of the Brans-Dicke parameter. In our models, there is no accelerating solution, only decelerating one with $q>0$. The physical cosmological distances have been investigated carefully. Further the statefinder parameters pair and deceleration parameter are discussed.