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The aim of the is to study the existence of solutions of initial value problems for nonlinear first order differential systems with nonlocal conditions. The proof will rely on the Perov, Schauder and Leray-Schauder fixed point principles which are applied to a nonlinear integral operator splitted into two parts, one of Fredholm type for the subinterval containing the points involved by the nonlocal condition, and an another one of Volterra type for the rest of the interval. The novelty in this paper is that this approach is combined with the technique that uses convergent to zero matrices and vector norms.

Mathematics Subject Classification (2010): 34A34, 34A12, 45G10.

Keywords: Nonlinear differential system, nonlocal initial condition, fixed point theorem, vector norm, matrix convergent to zero.