  Let $\mathcal{A}$ be a unital algebra over the complex field $\mathbb{C}$. A
linear mapping $\delta$ from $\mathcal{A}$ into itself is called a weak
(\textit{m,n,l})-Jordan centralizer if
$(m+n+l)\delta(A^2)-m\delta(A)A-nA\delta(A)-lA\delta(I)A\in \mathbb{C}I$ for
every $A\in \mathcal{A}$, where $m\geq0, n\geq0, l\geq0$ are fixed integers
with $m+n+l\neq 0$. In this paper, we study weak (\textit{m,n,l})-Jordan
centralizer on generalized matrix algebras and some reflexive algebras
alg$\mathcal{L}$, where $\mathcal{L}$ is CSL or satisfies $\vee\{L: L\in
\mathcal{J}(\mathcal{L})\}=X$ or $\wedge\{L_-: L\in
\mathcal{J}(\mathcal{L})\}=(0)$, and prove that each weak
(\textit{m,n,l})-Jordan centralizer of these algebras is a centralizer when
$m+l\geq1$ and $n+l\geq1$.
