  This paper is devoted to a simple nonlocal de Sitter gravity model and its
exact vacuum cosmological solutions. In the Einstein-Hilbert action with
$\Lambda$ term, we introduce nonlocality by the following way: $R - 2 \Lambda =
\sqrt{R-2\Lambda}\ \sqrt{R-2\Lambda} \to \sqrt{R-2\Lambda}\ F(\Box)\
\sqrt{R-2\Lambda} ,$ where ${F} (\Box) = 1 + \sum_{n= 1}^{+\infty} \big( f_n
\Box^n + f_{-n} \Box^{-n} \big) $ is an analytic function of the
d'Alembert-Beltrami operator $\Box$ and its inverse $\Box^{-1}$. By this way,
$R$ and $\Lambda$ enter with the same form into nonlocal version as they are in
the local one, and nonlocal operator $F(\Box)$ is dimensionless. The
corresponding equations of motion for gravitational field $g_{\mu\nu}$ are
presented. The first step in finding some exact cosmological solutions is
solving the equation $\Box \sqrt{R-2\Lambda} = q \sqrt{R-2\Lambda} , $ where $
q =\zeta \Lambda \quad (\zeta \in \mathbb{R})$ is an eigenvalue and
$\sqrt{R-2\Lambda}$ is an eigenfunction of the operator $\Box .$ We presented
and discussed several exact cosmological solutions for homogeneous and
isotropic universe. One of these solutions mimics effects that are usually
assigned to dark matter and dark energy. Some other solutions are examples of
the nonsingular bounce ones in flat, closed and open universe. There are also
singular and cyclic solutions. All these cosmological solutions are a result of
nonlocality and do not exist in the local de Sitter case.
