  Our interest in this paper is a generalization of the additive
Deligne-Simpson problem which is originally defined for Fuchsian differential
equations on the Riemann sphere. We shall extend this problem to differential
equations having an arbitrary number of unramified irregular singular points
and determine the existence of solutions of the generalized additive
Deligne-Simpson problems. Moreover we apply this result to the geometry of the
moduli spaces of stable meromorphic connections of trivial bundles on the
Riemann sphere. Namely, open embedding of the moduli spaces into quiver
varieties is given and the non-emptiness condition of the moduli spaces is
determined. Furthermore the connectedness of the moduli spaces is shown.
