Approximation of Solutions to Nonautonomous Difference Equations

Abstract

We study the asymptotic properties of solutions to nonautonomous difference equations of the form $${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$ 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.

Asymptotic Properties of Solutions to Fourth-Order Difference Equations on Time Scales

“…Many authors concentrate on the oscillatory and asymptotic behaviour of the second-order dynamic equation on time scales (see, for example, [2-4, 8-11, 14]). So far, they are less interested in studying higher order dynamical equations [18,[21][22][23][24]27]. Oscillation results for fourth-order nonlinear dynamic equations were considered in [1,12,13,20,25,26,28,29].…”