Approximation of Solutions to Nonautonomous Difference Equations

We study the asymptotic properties of solutions to nonautonomous difference equations of the form Δ m x n = a n f ( n , x σ ( n ) ) + b n ,     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.