  In Brans-Dicke theory a non-linear self interaction of a scalar field allows
a possibility of realizing the late-time cosmic acceleration, while recovering
the General Relativistic behavior at early cosmological epochs. We extend this
to more general modified gravitational theories in which a de Sitter solution
for dark energy exists without using a field potential. We derive a condition
for the stability of the de Sitter point and study the background cosmological
dynamics of such theories. We also restrict the allowed region of model
parameters from the demand for the avoidance of ghosts and instabilities. A
peculiar evolution of the field propagation speed allows us to distinguish
those theories from the LCDM model.
