Jordan left θ-derivations mapping into tile Jacobson radical

Let ? be a continuous automorphism of the Banach algebra A. The purpose of this paper is to obtain Jordan left θ-derivations that map into the Jacobson radical : (ⅰ) Let d be a spectrally bounded Jordan left θ-derivation on a Banach algebra A. If [d(x), θ(x)]?rad(A) for all x?A, then d(A) ⊆ rad(A). (ⅱ) Let d be a Jordan left θ-derivation on a unital Banach algebra A with the condition sup{r(z?¹d(z))|z?A invertible}＜∞. If [d(x), θ(x)]?rad(A) for all x?A, then d(A)⊆rad(A).