Oscillation and nonoscillation theorems for second order nonlinear difference equations. (Q1856978)

The authors consider the following general difference equation \[ \Delta_a (p_n \Delta_a x_n)+ q_n \Delta_a x_n= F(n,x_n, \Delta_b x_n), \quad n\in \mathbb{N}, \tag{*} \] where \(a (\neq 0)\), \(b\) are real numbers, \(F:\mathbb{N}\times \mathbb{R}^2\to \mathbb{R}\), \(x:\mathbb{N}\to \mathbb{R}\), \(\Delta _k x_n:= x_{n+1}-kx_n\) for any real number \(k\), \(\Delta _k^2 x_n:= \Delta _k(\Delta _k x_n)\), \(x_n=x(n)\), \(n\in \mathbb{N}\), and \(\{ p_n\} \) and \(\{ q_n\} \) are real sequences with \(p_n\neq 0 \) for all \(n\in \mathbb{N}. \) The authors obtain oscillation and nonoscillation theorems for (*) which generalize and improve all results proved by \textit{J. Popenda} [J. Math. Anal. Appl. 123, 34--38 (1987; Zbl 0612.39002)]. One of the theorems is the following: Theorem. If \(a>0 \) in (*), and the following conditions: (i) \(F(n,u,v)=0\), if \(v+(b-a)u=0,\) (ii) \(\frac{ v+(b-a)u}{p_{n+1}}\{ F(n,u,v)+(ap_n-q_n)[{ v+(b-a)u]}\} \geq 0\), if \(v+(b-a)u\neq 0\) are satisfied, where \( n\in \mathbb{N}\); \(u,v\in \mathbb{R},\) then all nontrivial solutions of (*) are nonoscillatory.