Left derivable or Jordan left derivable mappings on Banach algebras

Let A be a unital Banach algebra‎, ‎M be a left A-module‎, ‎and W in Z(A) be a left separating point of M‎. ‎We show that if M is a unital left A-module and ? is a linear mapping from A into M‎, ‎then the following four conditions are equivalent‎: ‎(i) ? is a Jordan left derivation; (ii)? is left derivable at W; (iii) ? is Jordan left derivable at W; (iv)A?(B)+B?(A)=?(W) for each A,B in A with AB=BA=W‎. ‎Let A have property (B) (see Definition ???)‎, ‎M be a Banach left A-module‎, ‎and ? be a continuous linear operator from A into M‎. ‎Then ? is a generalized Jordan left derivation if and only if ? is Jordan left derivable at zero‎. ‎In addition‎, ‎if there exists an element C?Z(A) which is a left separating point of M‎, ‎and RannM(A)={0}‎, ‎then ? is a generalized left derivation if and only if ? is left derivable at zero.