Asymptotic properties of solutions of third order difference equations

Abstract

We consider the difference equation of the form ∆(rn∆(pn∆xn)) = anf (x σ(n)) + bn. We present sufficient conditions under which, for a given solution y of the equation ∆(rn∆(pn∆yn)) = 0, there exists a solution x of the nonlinear equation with the asymptotic behavior xn = yn + zn, where z is a sequence convergent to zero. Our approach allows us to control the degree of approximation, i.e., the rate of convergence of the sequence z. We examine two types of approximation: harmonic approximation when zn = o(n s), s ≤ 0, and geometric approximation when zn = o(µ n), µ ∈ (0, 1).