On the asymptotic behaviour of nonhomogeneous linear difference equations (Q1355400)

The following linear difference equations are discussed: \[ c^r_n y_{n+r}+\cdots+c^1_ny_{n+1}+c^0_ny_n=d_n,\quad n\in\mathbb{N},\tag{E\(_1\)} \] and \[ \Delta x_n=\sum^r_{i=0} a^i_n x_{n+i}+ b_n,\quad n\in\mathbb{N},\tag{E\(_2\)} \] where \(x\), \(y\), \(c^i\), \(a^i\) \((i=0,1,\dots,r)\), \(b\) and \(d:\mathbb{N}\to\mathbb{R}\), here \(\mathbb{N}=\{1,2,\dots\}\); \(\Delta x_n:=x_{n+1}-x_n\). Firstly, a sufficient conditions for the existence of asymptotically constant solutions is proved for \((\text{E}_2)\). Theorem 1: Let \(a^0_n\neq-1\) for \(n\in\mathbb{N}\), \(\sup_{n\geq m}[\max_i|a^i_n|]>0\) for all \(m\in\mathbb{N}\), \(\sum^\infty_{j=1}|a^i_j|<\infty\) \((i=0,1,\dots,r)\) and the series \(\sum^\infty_{j=1}b_j\) converges. Then for any \(C\in\mathbb{R}\), \(C\neq 0\), there exists a solution of \((\text{E}_2)\) such that \(x_n\to C\) as \(n\to\infty\). It is proved that, if \(r\geq 2\), \(c^0_m\neq 0\), \(c^1_m\neq 0\), and \(\sup_{n\geq m}[\max_{2\leq i\leq r}|c^i_n|]>0\) for all \(m\in\mathbb{N}\), then \((\text{E}_1)\) can be transformed to an \((\text{E}_2)\)-type difference equation. Hence a corollary for \((\text{E}_1)\) then follows from Theorem 1. Finally, using this corollary, it is proved in Theorem 2 that, if there exist a positive constant \(\gamma\) and an integer \(\mu\) such that \(c^0_n=\gamma c^1_n\) for all \(n\geq\mu\), then \((\text{E}_1)\) possesses a family of eventually strictly oscillatory solutions. Here a sequence \(\{z_n\}\) is said to be eventually strictly oscillatory if for some \(m\in\mathbb{N}\) we have \(z_nz_{n+1}<0\) for all \(n\geq m\).