The existence of nonzero periodic travelling wave solutions for a general
discrete nonlinear Schr\"odinger equation (DNLS) on finite one-dimensional
lattices is proved. The DNLS features a general nonlinear term and variable
range of interactions going beyond the usual nearest-neighbour interaction. The
problem of the existence of travelling wave solutions is converted into a fixed
point problem for an operator on some appropriate function space which is
solved by means of Schauder's Fixed Point Theorem
