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On the asymptotics of some systems of difference equations. (English) Zbl 1233.39001

The article deals with the following (at least highly cumbersome formulated) statement: if the functions \[ f_i:\;I_i \times \ldots \times I_1 \times I_k \times \ldots \times I_{i+1} \to I_i, \qquad i = 1,\ldots,k, \] (\(I_i\) are some intervals in \({\mathbb R}\)) are continuous and non-decreasing in all arguments and the sequences \(y_n^{(i)}\) and \(z_n^{(i)}\), \(i = 1,\ldots,k\), satisfy the inequalities \(y_n^{(i)} < z_n^{(i)}\) \[ \left.\begin{matrix} y_n^{(1)} &\leq& f_1(y_{n+1}^{(1)},y_n^{(k)},\ldots,y_n^{(3)},y_n^{(2)}) \\ y_n^{(2)} &\leq& f_2(y_{n+1}^{(2)},y_{n+1}^{(1)},y_n^{(k)},\ldots,y_n^{(3)}) \\ \hdotsfor 3 \\ y_n^{(k-1)} &\leq& f_{k-1}(y_{n+1}^{(k-1)},\ldots,y_{n+1}^{(1)},y_n^{(k)})\\ y_n^{(k)} &\leq& f_k(y_{n+1}^{(k)},y_{n+1}^{(k-1)},\ldots,y_{n+1}^{(1)}) \end{matrix}\right\} \]
\[ \left.\begin{matrix} z_n^{(1)} &\geq& f_1(z_{n+1}^{(1)},z_n^{(k)},\ldots,z_n^{(3)},z_n^{(2)}) \\ z_n^{(2)} &\geq& f_2(z_{n+1}^{(2)},z_{n+1}^{(1)},z_n^{(k)},\ldots,z_n^{(3)}) \\ \hdotsfor 3 \\ z_n^{(k-1)} &\geq& f_{k-1}(z_{n+1}^{(k-1)},\ldots,z_{n+1}^{(1)},z_n^{(k)})\\ z_n^{(k)} &\geq& f_k(z_{n+1}^{(k)},z_{n+1}^{(k-1)},\ldots,z_{n+1}^{(1)}) \end{matrix}\right\} \] then the system \[ \left.\begin{matrix} u_n^{(1)} &=& f_1(u_{n+1}^{(1)},u_n^{(k)},\ldots,u_n^{(3)},u_n^{(2)}) \\ u_n^{(2)} &=& f_2(u_{n+1}^{(2)},u_{n+1}^{(1)},u_n^{(k)},\ldots,u_n^{(3)}) \\ \hdotsfor 3 \\ u_n^{(k-1)} &=& f_{k-1}(u_{n+1}^{(k-1)},\ldots,u_{n+1}^{(1)},u_n^{(k)})\\ u_n^{(k)} &=& f_k(u_{n+1}^{(k)},u_{n+1}^{(k-1)},\ldots,u_{n+1}^{(1)}) \end{matrix}\right\} \] has a solution \(u_n^{(i)}\) such that \(y_n^{(i)} \leq u_n^{(i)} \leq z_n^{(i)}\). This result is applied in getting the asymptotics of some particular solutions to the difference equation \[ x_n = \frac{x_{n-k}}{1 + x_{n-1} \cdots x_{n-k+1}}. \] The cases \(k = 2,3\) are considered.

MSC:

39A10 Additive difference equations
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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References:

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