  In this paper, we show that for finite $CW$-complexes $X$ and two-stage space
$Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the
rational homotopy type of $\map(X, Y)$ is determined by the cohomology algebra
$H^*(X; \Q)$ and the rational homotopy type of $Y$. From this, we deduce the
existence of H-structures on a component of the mapping space $\map(X, Y)$,
assuming the cohomology algebras of $X$ and $Y$ are isomorphism. Finally, we
will show that $\map(X, Y; f)\simeq\map(X, Y; f')$ if the corresponding
\emph{Maurer-Cartan elements} are connected by an algebra automorphism of
$H^\ast(X, \Q)$.
