  Let $(R,\fm)$ be a local ring and $(-)^{\vee}$ denote the Matlis duality
functor. We investigate the relationship between Foxby equivalence and local
duality through generalized local cohomology modules. Assume that $R$ possesses
a normalized dualizing complex $D$ and $X$ and $Y$ are two homologically
bounded complexes of $R$-modules with finitely generated homology modules. We
present several duality results for $\fm$-section complex ${\bf
R}\Gamma_{\fm}({\bf R}\Hom_R(X,Y))$. In particular, if G-dimension of $X$ and
injective dimension of $Y$ are finite, then we show that $${\bf
R}\Gamma_{\fm}({\bf R}\Hom_R(X,Y))\simeq ({\bf R}\Hom_R(Y,D\otimes_ R^{{\bf
L}}X))^{\vee}.$$ We deduce several applications of these duality results. In
particular, we establish Grothendieck's non-vanishing Theorem in the context of
generalized local cohomology modules.
