Range-exclusive solutions of almost automorphic differential equations

We present a unified approach to establishing the existence of almost automorphic solutions to differential equations in Banach spaces. We introduce the concept of solutions having exclusive ranges for differential equations of the form $u^{\prime }=f(t,u)$ in a Banach space X, where f is almost automorphic in time t. The proposed methodology unifies classical techniques, covering cases where nonlinearities are globally Lipschitz with exponentially stable linear parts, as well as differential equations without global Lipschitz nonlinearities. The main result shows that, under certain conditions on f, every solution with an exclusive range is compactly almost automorphic. To illustrate the versatility of the developed approach, we provide several examples and applications, including a Bohr–Neugebauer type theorem and the analysis of differential equations possessing multiple and unique almost automorphic solutions.