  In this paper, we introduce new methods for solving the vacuum Einstein
constraints equations: the first one is based on Schaefer's fixed point theorem
(known methods use Schauder's fixed point theorem) while the second one uses
the concept of half-continuity coupled with the introduction of local
supersolutions. These methods allow to: unify some recent existence results,
simplify many proofs (for instance, the main theorem in arXiv:1012.2188) and
weaken the assumptions of many recent results.
