The author proves some generalizations of the classical Poincaré- Birkhoff theorem on area-preserving homeomorphisms of the annulus which satisfy a boundary twist condition. Briefly speaking, in his generalization, the existence of positively returning disks and negatively returning disks replaces the boundary twist condition. Furthermore, the area-preserving hypothesis is replaced by the weaker condition that every point be non-wandering. It is particularly worth noticing that the author considers in Section 4 of this paper smooth maps of the annulus
into the larger annulus
which do not leave A invariant but are exact symplectic. In this setting he also obtains the existence of two fixed points. A similar but more general result was obtained by Weiyue Ding
[Proc. Am. Math. Soc. 88, 341-346 (1983; Zbl 0522.55005
)]. This kind of generalized Poincaré-Birkhoff theorems is very useful in applications for proving the existence of periodic solutions of some ordinary differential equations, for example, the Duffing equations.