Existence theorems for some abstract nonlinear non-autonomous systems with delays

•We obtained the existence results for a general class of abstract models.•The sufficient conditions were originated from biological applications.•We expand on well-known existence results and shorten existing proofs.

For some abstract classes of nonlinear non-autonomous systems with variable and state-dependent delays existence, non-existence and multiplicity of periodic solutions are discussed. To illustrate the efficiency of the method, we obtain some well-known results for applied systems as corollaries of our existence theorems.