  We introduce the notion of pseudo-Poisson Nijenhuis manifolds. These
manifolds are generalizations of Poisson Nijenhuis manifolds by Magri and
Morosi \cite{MM}. We show that any pseudo-Poisson Nijenhuis manifold has an
associated quasi-Lie bialgebroid as in the case of Poisson quasi-Nijenhuis
manifolds by Sti$\acute{\mathrm{e}}$non and Xu \cite{SX}. Hence, since a
quasi-Lie bialgebroid has an associated Courant algebroid, we have new
materials to construct Courant algebroids. In the "nondegenerate" case, we show
that the conditions of pseudo-Poisson Nijenhuis structures can be reduced.
Therefore we can provide lots of non-trivial examples of pseudo-Poisson
Nijenhuis manifolds.
