On asymptotic approximations of first integrals for second order difference equations

In this paper, the concept of invariance factors for second order difference equations to obtain first integrals or invariants will be presented. It will be shown that all invariance factors have to satisfy a functional equation. Van Horssen (J. Indones. Math. Soc. 13:1–15, 2007) developed a perturbation method for a single first order difference equation based on invariance factors. This perturbation method will be reviewed shortly, and will be extended to second order difference equations. Also, in this paper, we will construct approximations of first integrals for second order linear, and weakly nonlinear difference equations.