Non-oscillating solutions of a generalized system of ODEs with derivative terms (Q2063276)

A system of ordinary differential equations \[ x_1^{(n_1)}(t)+h_1(t,x_1,x_1',\dots,x_1^{(N_1)},x_2,x_2',\dots,x_2^{(N_2)})=0, \] \[ x_2^{(n_2)}(t)+h_2(t,x_1,x_1',\dots,x_1^{(N_1)},x_2,x_2',\dots,x_2^{(N_2)})=0, \] where \(h_1\), \(h_2\) are real valued continuous functions on \([t_0,\infty)\times \mathbb{R}^{N_1+N_2}\), \(t\geq t_0>0\), and \(n_i>1\) are integers with \(0 \leq N_i\leq n_i-1\), \(i=1,2\), is considered. The existence of a solution which does not oscillate is proved. The fixed point technique is applied to show that under certain conditions there exists at least one solution to the system which is not only non-oscillating, but also asymptotically constant.