Almost Periodicity of the Solutions of an Integrodifferential Equation

摘要

This paper discusses the almost periodicity of bounded solutions of the integrodifferential equation x' + micron * x = f. Here x and f map R into C sub n, the prime denotes differentiation, micron is an n by n matrix valued finite measure on R, and f is either an almost periodic distribution, or an almost periodic function in the sense of Bohr, Stephanoff, Weyl or Besicovitch. In the first three cases the author gives a simple sufficient condition (countability of the set where the characteristic function of the kernel is not invertible) for bounded solutions to be almost periodic. This condition is not longer sufficient in the last two cases, as is shown with a simple counterexample. (Author)