Characterizations of additive Jordan (α,β) derivable mappings on triangular algebras

Let Γ be a triangular algebra over a commutative ring R and α,β be the automorphisms of Γ. In this paper, under some conditions, we proved that every additive mapping ?: Γ→Γ Jordan (α,β) derivable at G is a generalized (α,β) derivation where G is an arbitrary element in Γ. In addition, we get some corollaries.