Approximation of Solutions to Nonautonomous Difference Equations

We study the asymptotic properties of solutions to nonautonomous difference equations of the form

Δmxn=anf(n,xσ(n))+bn, f:𝕅×𝕉→𝕉, σ:𝕅→𝕅$${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$

Using the iterated remainder operator and asymptotic difference pairs we establish some results concerning approximative solutions and approximations of solutions. Our approach allows us to control the degree of approximation.