Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients
The authors study the higher order neutral delay difference equation of the form \[ \Delta^m(x_n+cx_{n-k})+p_nx_{n-r}-q_nx_{n-l}=0, \] where \(c\in{\mathbb R}, m\geq 1, k\geq 1, r,l\geq 0, n\geq n_0\geq 0\) are integers, \(\{p_n\}_{n=n_0}^\infty\) and \(\{q_n\}_{n=n_0}^\infty\) are sequences of nonnegative real numbers, and \(\Delta x_n=x_{n+1}-x_n\). A solution \(\{x_n\}\) of the above equation is said to be oscillatory if the numbers \(x_n\) are neither eventually all positive nor eventually all negative. Otherwise, the solution is called nonoscillatory. The authors prove the following Theorem. Suppose \(m\geq 1\) is an odd (resp. \(m\geq 2\) is an even) integer, \[ \sum_{i=n_0}^\infty i^{m-1}p_i<\infty, \qquad \sum_{i=n_0}^\infty i^{m-1}q_i<\infty, \] and one of the following two conditions is fulfilled: (i) \(c\geq 0\) and there exist a constant \(\alpha>\max\{1,c\}/(1-c)\) and a number \(N\geq n_0\) such that \(p_n\geq\alpha q_n\) (resp. \(q_n\geq\alpha p_n\)) for all \(n\geq N\); (ii) \(c<0\) and there exist a constant \(\alpha>1\) and a number \(N\geq n_0\) such that \(p_n\geq\alpha q_n\) (resp. \(q_n\geq\alpha p_n\)) for all \(n\geq N\). Then the above equation has a bounded nonoscillatory solution.
scientific article; zbMATH DE number 2082209
scientific article