  We consider a multidimensional cosmological model with nonlinear quadratic
$R^2$ and quartic $R^4$ actions. As a matter source, we include a monopole form
field, D-dimensional bare cosmological constant and tensions of branes located
in fixed points. In the spirit of the Universal Extra Dimensions models, the
Standard Model fields are not localized on branes but can move in the bulk. We
define conditions which ensure the stable compactification of the internal
space in zero minimum of the effective potentials. Such effective potentials
may have rather complicated form with a number of local minima, maxima and
saddle points. Then, we investigate inflation in these models. It is shown that
$R^2$ and $R^4$ models can have up to 10 and 22 e-foldings, respectively. These
values are not sufficient to solve the homogeneity and isotropy problem but big
enough to explain the recent CMB data. Additionally, $R^4$ model can provide
conditions for eternal topological inflation. However, the main drawback of the
given inflationary models consists in a value of spectral index $n_s$ which is
less than observable now $n_s\approx 1$. For example, in the case of $R^4$
model we find $n_s \approx 0.61$.
