On existence of positive solutions and bounded oscillations for neutral difference equations (Q1191789)

The authors consider the linear homogeneous difference equation \(\Delta^ \alpha(x_ n-cx_{n-m})+p_ n x_{n-k}=0\), where \(c,p_ n\geq 0\). Here \(\Delta\) is the forward difference operator, \(\Delta x_ n=x_{n+1}-x_ n\), and \(\alpha=1,2\). It is shown that positive solutions tending to zero or \(\infty\) or remaining bounded, or oscillatory solutions, respectively, exist under appropriate conditions on \(c\) and \(p_ n\). Reviewer's remark: The right-hand sides of equations (2.17) and (2.20) should read \((n+3)/(n^ 2(n+1))\) and \((n+3)/(n(n+1))\), respectively, for these equations to have the indicated solutions.