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On the nonlocal initial value problem for first order differential systems. (English) Zbl 1274.34041
Summary: The aim of this paper 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 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.

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations