We Introduce new type of maps between normed spaces, namely,
 $p$-quasi-$\lambda$-nuclear map. We prove that the composition of
 a $q$-quasi-$\lambda$-nuclear map ($0<q\leq 1$) with a
 $p$-quasi-$\lambda$-nuclear map ($0<p\leq 1$) is a pseudo-$\lambda$-nuclear map.
 Also we prove that for a nuclear $G_\infty$-space a linear map
 $T$ between normed spaces is $p$-quasi-$\lambda$-nuclear iff it
 is $q$-quasi-$\lambda$-nuclear.