Implicit first order differential systems with nonlocal conditions

The present paper is devoted to the existence of solutions for implicit first order differential systems with nonlocal conditions expressed by continuous linear functionals. The lack of complete continuity of the associated integral operators, due to the implicit form of the equations, is overcome by using Krasnoselskii’s fixed point theorem for the sum of two operators. Moreover, a vectorial version of Krasnoselskii’s theorem and the technique based on vector-valued norms and matrices having the spectral radius less than one are likely to allow the system nonlinearities to behave independently as much as possible. In addition, the connection between the support of the nonlocal conditions and the constants from the growth conditions is highlighted.