Existence results for systems with coupled nonlocal initial conditions

In this paper an existence theory is developed for first-order differential systems with coupled nonlocal conditions given by Stieltjes integrals. The approach is based on the fixed point theorems of Perov, Schauder and Schaefer and on a vector method for treating systems which uses matrices having spectral radius less than one. When the nonlocal conditions depend on functionals restricted to a proper subinterval, the nonlinear integral operator associated to the system splits into two parts, one of Fredholm type and another one of Volterra type. Correspondingly, the sufficient conditions for the existence results will differ in the two parts of the interval. Some examples are presented to illustrate the theory.