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It is shown that $A$ is $\\Delta$-weak $\\phi$-amenable if and only if $\\ker(\\phi)$ has a bounded $\\Delta$-weak approximate identity. We examine this notion for some algebras over amenable locally compact groups. 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We say that $A$ is $\\Delta$-weak $\\phi$-amenable if there exists an $m\\in A^{**}$ such that $m(\\phi)=0$ and $m(\\psi.a)=\\psi(a)$ for each $\\psi\\in \\Delta(A)$ and $a\\in \\ker(\\phi)$. It is shown that $A$ is $\\Delta$-weak $\\phi$-amenable if and only if $\\ker(\\phi)$ has a bounded $\\Delta$-weak approximate identity. We examine this notion for some algebras over amenable locally compact groups. 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