Properties of Solutions of Generalized Sturm–Liouville Discrete Equations

Abstract

AbstractWe consider discrete Sturm–Liouville-type equations of the form $$\begin{aligned} \varDelta (r_n\varDelta x_n)=a_nf(x_{\sigma (n)})+b_n. \end{aligned}$$ Δ ( r n Δ x n ) = a n f ( x σ ( n ) ) + b n . 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 $$\varDelta (r_n\varDelta y_n)=b_n$$ Δ ( r n Δ y n ) = b n , the above equation possesses a solution x with the property $$x_n=y_n+\mathrm {o}(u_n)$$ x n = y n + 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.