left derivable or jordan left derivable mappings on banach algebras

‎let $mathcal{a}$ be a unital banach algebra‎, ‎$mathcal{m}$ be a left $mathcal{a}$-module‎, ‎and $w$ in $mathcal{z}(mathcal{a})$ be a left separating point of $mathcal{m}$‎. ‎we show that if $mathcal{m}$ is a unital left $mathcal{a}$-module and $delta$ is a linear mapping from $mathcal{a}$ into $mathcal{m}$‎, ‎then the following four conditions are equivalent‎: ‎(i) $delta$ is a jordan left derivation; (ii)$delta$ is left derivable at $w$; (iii) $delta$ is jordan left derivable at $w$; (iv)$adelta(b)+bdelta(a)=delta(w)$ for each $a,b$ in $mathcal{a}$ with $ab=ba=w$‎. ‎let $mathcal{a}$ have property ($mathbb{b}$) (see definition ref{prop_b})‎, ‎$mathcal{m}$ be a banach left $mathcal{a}$-module‎, ‎and $delta$ be a continuous linear operator from $mathcal{a}$ into $mathcal{m}$‎. ‎then $delta$ is a generalized jordan left derivation if and only if $delta$ is jordan left derivable at zero‎. ‎in addition‎, ‎if there exists an element $cinmathcal{z}(mathcal{a})$ which is a left separating point of $mathcal{m}$‎, ‎and $rann_{mathcal{m}}(mathcal{a})={0}$‎, ‎then $delta$ is a generalized left derivation if and only if $delta$ is left derivable at zero.