We consider discrete Sturm–Liouville-type equations of the form

We present a theory of asymptotic properties of solutions which allows us to control the degree of approximation. Namely, we establish conditions under which for a given sequence y which solves the equation \(\varDelta (r_n\varDelta y_n)=b_n\), the above equation possesses a solution x with the property \(x_n=y_n+\mathrm {o}(u_n)\), where u is a given positive, nonincreasing sequence. The obtained results are applied to the study of asymptotically periodic solutions. Moreover, these results also allow us to obtain some nonoscillation criteria for the classical Sturm–Liouville equation.

我们考虑以下形式的离散Sturm–Liouville型方程

我们提出了一种解决方案的渐近性质的理论，它使我们能够控制近似度。即，我们建立在何种条件下对于给定的序列ÿ解决了方程\（\ varDelta（r_n \ varDelta y_n）= B_N \） ，上面的方程具有的溶液X与属性\（x_n = y_n + \ mathrm {Ó }（u_n）\），其中u是给定的正数（不增加）。所得结果用于渐近周期解的研究。此外，这些结果还使我们能够获得一些经典Sturm-Liouville方程的非振动准则。