Almost periodic solutions for an asymmetric oscillation

In this paper we study the dynamical behavior of the differential equationx″+ax+−bx−=f(t), where x+=max⁡{x,0}, x−=max⁡{−x,0}, a and b are two different positive constants, f(t) is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see [11]). Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation.