  Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper
ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the
finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated
$R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatest
non-negative integer with the property that $\operatorname{H}^i_{\mathfrak
a}(N)$, the $i$-th local cohomology module of $N$ with respect to $\mathfrak
a$, is non-zero. It is known that $\operatorname{H}^i_{\mathfrak a}(M, N)$, the
$i$-th generalized local cohomology module of $M$ and $N$ with respect to
$\mathfrak a$, is zero for all $i>p+c$. In this paper, we obtain the
coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$.
Using this, in the case when $R$ is a local ring and $c$ is equal to the
dimension of $N$, we give a necessary and sufficient condition for the
vanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends the
Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology
modules.
