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Let T = T (n 1 ,n 2 ,••• ,n k) ⊆ M n (C) be a block upper triangular matrix algebra and let M be a 2-torsion free unital T-bimodule, where C is a commutative ring. Let Δ : T → M be a C-linear map. We show that if Δ(X)Y + XΔ(Y)+Δ(Y)X +Y Δ(X) = 0 whenever X,Y ∈ T are such that XY = Y X = 0 , then Δ(X) = D(X) + α(X) + XΔ(I) , where D : T → M is a derivation, α : T → M is an antiderivation, I is the identity matrix and Δ(I)X = XΔ(I) for all X ∈ T. We also prove that under some sufficient conditions on T , we have α = 0. As a corollary, we show that under given sufficient conditions, each Jordan derivation Δ : T → M is a derivation and this is an answer to the question raised in [9]. Some previous results are also generalized by our conclusions.

Suppose that T = Tri A , ℳ , ℬ is a 2-torsion free triangular ring, and S = A , B | A B = 0 , A , B ∈ T ∪ A , X | A ∈ T ,   X ∈ P , Q , where P is the standard idempotent of T and Q = I − P . Let δ : T ⟶ T be a mapping (not necessarily additive) satisfying, A , B ∈ S ⇒ δ A ∘ B = A ∘ δ B + δ A ∘ B , where A ∘ B = A B + B A is the Jordan product of T . We obtain various equivalent conditions for δ , specifically, we show that δ is an additive derivation. Our result generalizes various results in these directions for triangular rings. As an application, δ on nest algebras are determined.