Bounded and monotone properties of solutions of second-order quasilinear forced difference equations
The authors consider the quasilinear difference equation (*) \(\Delta(p_{n-1}(\Delta y_{n-1})^\alpha)=q_n y_n^\beta+r_n\), where \(p_0>0\), \(q_0\geq 0\) for \(n\geq N_0\), and \(q_n\not\equiv 0\) for \(n\geq N_1\) for any \(N_1\geq N_0\). Although the results are valid for \(\alpha>0\) and \(\beta>0\), for simplicity of notation the discussion is carried out only in the case \(\alpha\) and \(\beta\) are quotients of odd positive integers. A solution \(y_n\) is of class \(A\) or \(B\) as \(y_n\Delta y_n>0\) or \(y_n\Delta y_n<0\) for some integer \(n\geq N\). If \(r_n\geq 0\), it is shown that every solution of (*) is nonoscillatory and that it belongs either to class \(A\) or class \(B\). A necessary and sufficient condition is given for solutions of class \(A\) to be bounded. A necessary condition is given for the existence of solutions of class \(B\) by means of fixed-point techniques. Lastly, fixed-point and limit-cycle problems are studied. Examples illustrate the results.
second-order quasilinear forced difference equations
scientific article; zbMATH DE number 1391617