Degenerate complex Monge-Amp\` ere type equations on compact Hermitian manifolds and applications

We show the existence and uniqueness of bounded solutions to the degenerate complex Monge-Ampère type equations on compact Hermitian manifolds. We also study the asymptotics of these solutions. As applications, we give partial answers to the Tosatti-Weinkove conjecture and Demailly-P\u aun conjecture.