Characterizations of derivations and Jordan derivations on Banach algebras

Let A be an algebra and M be an A-bimodule. Let X be in A and δ:A→M be a linear map which satisfies δ(AB)=δ(A)B+Aδ(B) for all A,B∈A with AB=X. It is shown that δ is a Jordan derivation if δ is continuous and X is left (or right) invertible. Also, it is shown that δ is a derivation if X is idempotent such that for M∈M the condition XA(I-X)M=0 implies (I-X)M=0 and the condition MXA(I-X)=0 implies MX=0.