Summary: Since it is very difficult to obtain the compactness of an almost periodic function set, many classical methods controlled by compact conditions such as Schauder’s fixed point theorem and the coincidence degree cannot be applied to solve almost periodic cases. Therefore, it becomes more complicated to investigate the existence, nonexistence and uniqueness of positive almost periodic solution for a certain model by the traditional methods. In this paper, the authors establish a new fixed point theorem without the compact conditions. As its application, some sufficient conditions for the existence, nonexistence and uniqueness of positive almost periodic solutions for a model of hematopoiesis [L. Glass
and M. C. Mackey
, Science 197, 287–289 (1977; d
oi:10.1126/science.267326); cf. also: Ann. New York Acad. Sci., Vol. 316, 214–235 (1979; Zbl 0427.92004
)] are obtained. Also, the technique used here is different from the usual methods employed to solve almost periodic cases such as the contraction mapping principle and the Lyapunov functional.