Global attractivity in a nonlinear difference equation
Consider the nonlinear difference equation \[ \text{(E)} \qquad x_ n = a + \sum^ m_{k=1} {b_ k \over x_ n - k}\qquad(n = 0,1,2, \dots) \] where \(a, b_ 1,\dots,b_ m\) are nonnegative numbers with \(b = \sum^ m_{k=1} b_ k > 0\). The equation has a unique positive equilibrium point \(L = {a \over 2} + \sqrt {({a \over 2})^ 2 + B}\). Theorem: (i) Assume that \(a > 0\). Then \(L\) is a global attractor of all positive solutions of (E). (ii) Assume that \(a = 0\). Let \(\nu\), \(1 \leq \nu \leq m\), be an integer such that \(b_ \nu > 0\), and suppose that there exists a positive integer \(\mu\) with \(2 \mu \nu \leq m\) such that \(b_{2 \mu \nu} > 0\). Then \(L = \sqrt B\) is a global attractor of all positive solutions of (E).
scientific article; zbMATH DE number 637513
scientific article