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Let $(R,\mathfrak{m},k)$ denote a local ring. For $I$ and $J$ ideals of $R$,for all integer $i$, let $H^i_{I,J}(-)$ denote the $i$-th local cohomologyfunctor with respect to $(I,J)$. Here we give a generalized version of LocalDuality Theorem for local cohomology defined by a pair of ideals. Also, for $M$be a finitely generated $R$-module, we study the behavior of the endomorphismrings $H^t_{I,J}(M)$ and $D(H^t_{I,J}(M))$ where $t$ is the smallest integersuch that the local cohomology with respect to a pair of ideals is non-zero and$D(-):= {\rm Hom}_R(-,E_R(k))$ is the Matlis dual functor. We show too that if$R$ be a $d$-dimensional complete Cohen-Macaulay and $H^i_{I,J}(R)=0$ for all$i\neq t$, the natural homomorphism $R\rightarrow {\rm Hom}_R(H^t_{I,J}(K_R),H^t_{I,J}(K_R))$ is an isomorphism and for all $i\neq t$, where $K_R$ denotethe canonical module of $R$.

Some Results about Endomorphism Rings for Local Cohomology Defined by a Pair of Ideals