On $\mathrm{\Delta }$-weak $\varphi$-amenability of Banach algebras

Let $A$ be a Banach algebra and $\varphi \in \mathrm{\Delta }(A)\cup \{0\}$. We say that $A$ is $\mathrm{\Delta }$-weak $\varphi$-amenable if there exists an $m\in A^{\ast \ast }$ such that $m(\varphi )=0$ and $m(\psi \mathrm{.}a)=\psi (a)$ for each $\psi \in \mathrm{\Delta }(A)$ and $a\in \ker(\phi)$\mathrm{.}Itisshownthat$A$is$\Delta$-weak$\phi$-amenable if and only if $\ker (\varphi )$ has a bounded $\mathrm{\Delta }$-weak approximate identity. We examine this notion for some algebras over amenable locally compact groups. Also we prove that every $\mathrm{\Delta }$-weak $\varphi$-amenable Banach algebra has a bounded $\mathrm{\Delta }$-weak approximate identity.