Modified Brans-Dicke cosmology with Minimum Length Uncertainty

We consider a modification of the Brans-Dicke gravitational Action Integral inspired by the existence of a minimum length uncertainty for the scalar field. In particular, the kinetic part of the Brans-Dicke scalar field is modified such that the equation of motion for the scalar field be modified according to the quadratic Generalized Uncertainty Principle (GUP). For the background geometry, we assume the homogeneous and isotropic Friedmann--Lema\^{\i}tre--Robertson--Walker metric. We investigate the dynamics and the cosmological evolution for the dynamical variables of the theory and we compare the results with the unmodified Brans-Dicke theory. It follows that in consideration because of the additional degrees of freedom in the energy-momentum tensor the dynamical variables describe various aspects of the cosmological history. This is one of the first studies on the effects of GUP in a Machian gravitational theory.