Necessary and sufficient condition for existence of positive solutions of the difference equation \((E)\quad (-1)^{m-1}\Delta^ mA_ n+\sum^{\infty}_{k=0}p_ kA_{n-\ell_ k}=0\) is established, where m is a positive integer, \((p_ k)_{k\geq 0}\) is a sequence of positive real numbers, \((\ell_ k)_{k\geq 0}\) is a sequence of integers with \(0\leq \ell_ 0<\ell_ 1<\ell_ 2<...;\) \(\Delta A_ n=A_{n+1}-A_ n,\) \(\Delta^ 0A_ n=A_ n\) and \(\Delta^ iA_ n=\Delta (\Delta^{i-1}A_ n)\) \((i=1,2,...)\) for every \(n\in Z=\{...,- 1,0,1,...\}.\) The main result of the paper is the following Theorem: (i) For m even, (E) has a positive solution \((A_ n)_{n\in Z}\) with \(\limsup_{n\to \infty}A_ n<\infty\) if and only if the characteristic equation of (E), i.e., \((a)\quad -(1-\lambda)^ m+\sum^{\infty}_{k=0}p_ k\lambda^{-\ell_ k}=0\) has a root in (0,1). (ii) For m odd, (E) has a positive solution \((A_ n)_{n\in Z}\) if and only if (a) has a root in (0,1). A sufficient condition for equation (a) has no roots in (0,1) is also indicated. As noted by the authors, equation (E) can be considered as a generalization of the discrete version of a certain delay differential equation.