Properties of Solutions of Generalized Sturm–Liouville Discrete Equations

Abstract We consider discrete Sturm–Liouville-type equations of the form $$\begin{aligned} \varDelta (r_n\varDelta x_n)=a_nf(x_{\sigma (n)})+b_n. \end{aligned}$$ ? ( r n x ) = a f ? + b . present a theory asymptotic properties solutions which allows us to control degree approximation. Namely, we establish conditions under for given sequence y solves equation $$\varDelta y_n)=b_n$$ y , above possesses solution x with property $$x_n=y_n+\mathrm {o}(u_n)$$ o u where u is positive, nonincreasing sequence. The obtained results are applied study asymptotically periodic solutions. Moreover, these also allow obtain some nonoscillation criteria classical Sturm–Liouville equation.