Bounded solutions of linear almost periodic differential equations

The paper deals with bounded (on R+ or R) solutions of the equation (x)over dot = A(t)x with recurrent (almost periodic) coefficients. We show that the zero solution of this equation is uniformly stable (bistable) if and only if all its solutions and the solutions of its limit equations are bounded on R+ (R). These results are generalizations of the well-known theorem of Cameron-Johnson.