  Let $(R,\fm)$ be a relative Cohen-Macaulay local ring with respect to an
ideal $\fa$ of $R$ and set $c:=\h\fa$. In this paper, we investigate some
properties of the Matlis dual $\H_{\fa}^c(R)^{\vee}$ of the $R$-module
$\H_{\fa}^c(R)$ and we show that such modules treat like canonical modules over
Cohen-Macaulay local rings. Also, we provide some duality and equivalence
results with respect to the module $\H_{\fa}^c(R)^{\vee}$ and so these results
lead to achieve generalizations of some known results, such as the Local
Duality Theorem, which have been provided over a Cohen-Macaulay local ring
which admits a canonical module.
