Existence and Comparison Results for Differential Equations in a Banach Space
The study of the Cauchy problem for differential equations in a Banach space has taken two different directions. One direction is to find compactness type conditions that guarantee only existence of solutions and the corresponding results are extensions of the classical Peano's theorem. The other approach is to employ monotonicity (accretive or dissipative) type conditions that assure existence as well as uniqueness of solutions. In fact, this latter method shows that uniqueness conditions imply existence of solutions also  and therefore may be regarded as extensions of the classical Picard's theorem.