A Unified Approach to Eberlein Almost Periodic Differential Equations

We develop a unified approach for proving the existence of Eberlein weakly almost periodic (E.w.a.p.) solutions of differential equations in Banach spaces. The approach avoids the classical requirement of uniqueness of bounded solutions on and relaxes the assumptions usually imposed on the time‐dependent coefficients. It applies both to equations with globally Lipschitz nonlinearities and exponentially stable linear parts, and to classes of equations with non‐globally Lipschitz nonlinearities. We then illustrate the scope of the main theorem through several applications, including models admitting either unique or multiple E.w.a.p. solutions. In particular, for the logistic equation we prove the existence of bounded solutions on under merely bounded and continuous coefficients. To the best of our knowledge, this extends earlier results, which typically require almost periodic or almost automorphic coefficients. Numerical simulations are included to illustrate the theory.