Asymptotic Properties of Solutions to Discrete Sturm-Liouville Monotone Type Equations

We investigate the discrete equations of the form Δ(rnΔxn)=anf(xσ(n))+bn. \Delta \left( {{r_n}\Delta {x_n}} \right) = {a_n}f\left( {{x_{\sigma \left( n \right)}}} \right) + {b_n}. Using the Knaster-Tarski fixed point theorem, we study solutions with prescribed asymptotic behaviour. Our technique allows us to control the degree of approximation. In particular, we present the results concerning harmonic and geometric approximations of solutions.