  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.
