In view of Schaefer’s theorem, in the case of a normed linear space \((B,\parallel \cdot \parallel)\) and of a continuous mapping \(H\) of \(B\) into \(B\) which is compact on each bounded subset of \(B\), we have either (i) the equation \(x=\lambda Hx\) has a solution for \(\lambda =1\) or (ii) the set of all such solutions, for \(0<\lambda <1\), is unbounded. Using this fixed-point theorem and a Lyapunov functional, the author studies the existence of a \(T\)-periodic solution of the difference equation \[ \Delta x(n)=Dx(n)+\sum_{j=-\infty }^nC(n-j)x(j)+g(n) \] where \(D\) and \(C\) are given \(k\times k\) real matrices with \(\sum_{u=0}^\infty |C(u)|<\infty \), and \(x(n)\), \(g(n)\) are \(k\times 1\) vectors with \(g(n+T)=g(n)\). An example which gives a necessary condition for the existence of periodic solution of this equation, and a sufficient condition concerning the simpler case of \(\Delta x(n)=Dx(n)+g(n)\) are included.