This chapter is concerned with the general theory of nonlinear quasi-
-accretive operators in Banach spaces with applications to the existence theory of nonlinear elliptic boundary value problems in
-spaces and first-order quasilinear equations. While the monotone operators are defined in a duality pair (
) and, therefore, in a variational framework, the accretive operators are intrinsically related to geometric properties of the space
and are more suitable for nonvariational and nonHilbertian existence theory of nonlinear problems. The presentation is confined, however, to the essential results of this theory necessary to the construction of accretive dynamics in the next chapter.