Bounded and almost automorphic solutions of some nonlinear differential equations in Banach spaces

We study some properties of bounded and almost automorphic solutions of the following nonlinear differential equation: x′=f(t,x),x′=f(t,x), where XX is a Banach space and f:R×X⟶Xf:R×X⟶X is almost automorphic with a parameter. We give sufficient conditions for the existence of almost automorphic solutions when the differential equation admits at least a solution with a relatively compact range defined on the half-line of positive numbers.