In the paper Operators on infinite dimensional cartesian product, (Analele Univ. Vest Timişoara, Mat. Inform., 48 (2010),
253–263), by I. A. Rus and M. A. Şerban, the authors give a generalization of the Fibre contraction theorem on infinite dimensional cartesian product. In this paper we give an application of this abstract result to an infinite system of differential equations.