  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 $C_{0}^{w}(G)$.
