  In this paper we will first show some Kollar-Enoki type injectivity theorems
on compact Kahler manifolds, by using the Hodge theory, the Bochner-
Kodaira-Nakano identity and the analytic method provided by O. Fujino and S.
Matsumura in [15, 25, 36, 39]. We have some straightforward corollaries. In
particular, we will show that our main injectivity theorem implies several
Nadel type vanishing theorems on smooth projective manifolds. Second, by
applying the transcendental method, especially the Demailly-Peternell-Schneider
equisingular approximation theorem and the Hormander L2 estimates, we will
prove some Nakano-Demailly type and Nadel type vanishing theorems for
holomorphic vector bundles on compact Kahler manifolds, twisted by
pseudo-effective line bundles and multiplier ideal sheaves. As applications, we
will show that our first main vanishing theorem generalizes the classical
Nakano-Demailly vanishing theorem while the second one contains the famous
Nadel vanishing theorem as a special case.
