Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach

Abstract

In this paper an attempt is made to depict a clear picture of the overall structure of nonoscillatory solutions of the first order half-linear differential system \begin{equation*} x'-p(t)\varphi_{1/\alpha}(y)=0,\qquad y'+q(t)\varphi_{\alpha}(x)=0, \tag{A} \end{equation*} where $\alpha>0$ is a constant, $p(t)$ and $q(t)$ are positive continuous functions on $[0,\infty)$, and $\varphi_{\gamma}(u)=|u|^{\gamma}\textrm{sgn}\;u, \;u\in {\mathbb R},\;\gamma>0$. A systematic analysis of the existence and asymptotic behavior of solutions of (A) is proposed for this purpose. A special mention should be made of the fact that all possible types of nonoscillatory solutions of (A) can be constructed by solving the Riccati type differential equations associated with (A). Worthy of attention is that all the results for (A) can be applied to the second order half-linear differential equation \begin{equation*} (p(t)\varphi_{\alpha}(x'))'+q(t)\varphi_{\alpha}(x)=0, \tag{E} \end{equation*} to build automatically a nonoscillation theory for (E).

Keywords

• half-linear differential systems
• nonoscillatory solutions
• riccati equation