Infinite first order differential systems with nonlocal initial conditions

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear" growth then the system has at least one solution. The approach relies on the application, in a suitable Fr\'echet space, of the classical Schauder-Tychonoff fixed point theorem. We show that, as a special case, our approach covers the case of a system of a finite number of differential equations. An illustrative example of application is also provided.