  Motivated by thermodynamic analogy of black holes and Van der Waals
liquid/gas system, in this paper, we study $P-V$ criticality of both dilatonic
Born-Infeld black holes and their conformal solutions, Brans-Dicke-Born-Infeld
solutions. Due to the conformal constraint, we have to neglect the old
Lagrangian of dilatonic Born-Infeld theory and its black hole solutions, and
introduce a new one. We obtain spherically symmetric nonlinearly charged black
hole solutions in both Einstein and Jordan frames and then, we calculate the
related conserved and thermodynamic quantities. After that, we extend the phase
space by considering the proportionality of the cosmological constant and
thermodynamical pressure. We obtain critical values of thermodynamic
coordinates through numerical methods and plot relevant $P-V$ and $G-T$
diagrams. Investigation of the mentioned diagrams helps us to study
thermodynamical phase transition. We also analyze the effects of varying
different parameters on the phase transition of black holes.
