Closed ideals with bounded Delta-weak approximate identities in some certain Banach algebras

It is shown that a locally compact group $G$ is amenable if and only if some certain closed ideals of the Fig\`{a}-Talamanca-Herz algebra $A_{p}(G)$ admit bounded $\Delta$-weak approximate identities. Also, similar results are obtained for the function algebras $\mathfrak{L}A(G)$ and $C_{0}^{w}(G)$.