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

Periodic travelling wave solutions of discrete nonlinear Schr\"odinger equations