Nonoscillatory solutions of nonlinear differential systems
Here, the system of \(n\) ordinary differential equations \[ \begin{aligned} x'_i&=a_i(t)f_i(x_{i+1}), \qquad\text{for }i=1,\dots,n-1, \\ x'_n&=-a_n(t)f_n(x_1) \end{aligned} \] is studied. The functions \(a_i(t)\) are supposed to be positive and continuous on \([t_0,\infty)\) for \(i=1,\dots,n\), and the functions \(f_i(u)\) are supposed to be nondecreasing continuous functions which satisfy the sign condition \(uf_i(u)>0\) for \(u\neq 0\), \(i=1,\dots,n\). The author divides the nonoscillatory solutions of the system belonging to the first open quadrant \(\{x=(x_1,\dots,x_n)\in {\mathbb{R}}^n,\;x_k>0,\;k=1..n\}\) into three classes and presents a necessary and sufficient condition for every class which ensures the existence of a nonoscillatory solution belonging to this class. The existence of a nonoscillatory solution is proved via Schauder's and Knaster's fixed-point theorem. The paper extends results from \textit{Wan-Tong Li} and \textit{Sui Sun Cheng} [Proc. Edinb. Math. Soc., II. Ser. 43, No.3, 457-473 (2000; Zbl 0979.34029)].
scientific article; zbMATH DE number 2082610