$\varphi$-approximate biprojective and $\varphi$-approximate amenable Banach algebras

Let $A$ be a Banach algebra and $\varphi$ be a continuous homomorphism on $A$. We define the notions of a $\varphi$-approximate biprojective and $\varphi$-approximate amenable Banach algebra $A$, and consider relations between them and some their properties.