Asymptotic behavior of a forced difference equation
The aim of this paper is to study the asymptotic behavior of the forced difference equation (1) \(x_{n+1} -x_n+ px_{n-k} =r_n\), \(n=0,1, \dots\), where \(p\) is a real number, \(k\) is a positive integer, and \(\{r_n\}\) is a real sequence. The corresponding homogeneous equation of (1) is (2) \(y_{n+1} -y_n+ py_{n-k} =0\), \(n=0,1, \dots\). It is well known that if \(\lambda_0\) is a simple real root of the characteristic equation for (2) with the largest modulus among all the roots, then the asymptotic behavior of solutions of (2) is dominated by the solution \(\{\lambda^n\}\). In the present paper it is shown that under certain hypotheses, the solutions of (1) have the same property. Moreover, for any solution \(\{x_n\}\) of (1), \(\lim_{n\to \infty} \{\lambda_0^{-n} x_n\}\) is found explicitly in terms of the initial conditions of \(\{x_n\}\). In the case \(p>0\) a necessary and sufficient condition is established for every solution of (1) to tend to zero.
scientific article; zbMATH DE number 944361