Asymptotic behavior of the solutions of second-order difference equations (Q1917918)

Consider the second order difference equation \(\Delta^2 x(n) = f(n, x(n))\). Under some assumptions to complicated to be presented here, as \(n \to \infty\), every solution with initial conditions small enough has the form \(x(n) = (\delta_1 + o(1)) n + \delta_2 + o(1)\), where \(\delta_i (i = 1,2)\) are constants. Some extensions of this result are studied also. Two illustrative exemples are included.