Consider the difference equation \[ \Delta(x_n-cx_{n-m})\pm (p_nx_{n-k_1}- q_nx_{n-k_2})=0, \qquad n=0,1,2,\dots\tag \(*\) \] where \(c>0\) is a constant, \(m\), \(k_1\), \(k_2\), \(p_n\) and \(q_n\) are nonnegative. The authors obtain some sufficient and necessary conditions for the existence of bounded and unbounded positive solutions, as well as some sufficient conditions for the existence of bounded and unbounded oscillatory solutions of \((*)\).
For related results see the papers of G. Ladas [Rocky Mt. J. Math. 20, No. 4, 1051-1061 (1990; Zbl 0727.39002)] and M. P. Chen and B. G. Zhang [Bull. Inst. Math., Acad. Sin. 22, No. 4, 295-306 (1994; Zbl 0817.39003)].