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The asymptotic behavior of the solutions of a Volterra difference equation. (English) Zbl 0880.39011

Let \(Lx_n= \Delta(a_n\Delta x_n)+b_n\Delta x_n+c_nx_n\). Under certain summability conditions on the functions \(r\), \(g\) and \(h\) it is proved that \(Lx_n= r_n\sum_{l=0}^{n-1} g_lx_l+ h(n,x_n,\Delta x_n)\) has a solution with \(x_n=(\delta_1+ o(1))z_n^1+ (\delta_2+o(1))z_n^2\) for \(n\to\infty\), where \(\{z^1,z^2\}\) is a fundamental system of solutions of \(Lz_n=0\), and \(\delta_1,\delta_2\) are constants.

MSC:

39A11 Stability of difference equations (MSC2000)
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