Summary: This article concerns the existence of solutions to initial-value problems for nonlinear first-order differential systems with nonlocal conditions of functional type. The fixed point principles by Perov, Schauder and Leray-Schauder are applied to a nonlinear integral operator split into two operators, one of Fredholm-type and the other of Volterra-type. The novelty in this article is combining this approach with the technique that uses convergent to zero matrices and vector norms.