  A theoretical investigation of the modulational instability (MI) of
dust-acoustic waves (DAWs) by deriving a nonlinear Schr\"{o}dinger equation in
an electron depleted opposite polarity dusty plasma system containing
non-extensive positive ions has been presented. The conditions for MI of DAWs
and formation of envelope solitons have been investigated. The sub-extensivity
and super-extensivity of positive ions are seen to change the stable and
unstable parametric regimes of DAWs. The addition of dust grains causes to
change the width of both bright and dark envelope solitons. The findings of
this study may be helpful to understand the nonlinear features of DAWs in
Martian atmosphere, cometary tail, solar system, and in laboratory experiments,
etc.
