  In this paper, by establishing a Brans-Dicke (BD) cosmology by means of a
deformed phase space, in the absence of any scalar potential, cosmological
constant and ordinary matter, we show that it is feasible to overcome obstacles
reported in the corresponding commutative (non-deformed) frameworks. More
concretely, by applying the Hamiltonian formalism and introducing a dynamical
deformation, between the momenta associated to the FLRW scale factor and the BD
scalar field, we obtain the modified equations of motion. In particular, these
equations reduce to their standard counterparts when the noncommutative (NC)
parameter is switched off. By focusing on a specific branch of solutions, in
contrast to standard frameworks (even with a varying BD coupling parameter), we
show that we can obtain an adequate appropriate inflationary epoch possessing a
suitable graceful exit. In other words, in the Jordan frame (JF), such branch
of solutions properly satisfy the sufficient condition required for
satisfactory inflation, which is equivalent to get an inflationary phase in the
conformal Einstein frame (EF) without branch change. Concerning the
cosmological dynamics, we further show that our NC framework bears close
resemblance to the $R^2$ (Starobinsky) inflationary model.
