Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach
Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach
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DOI：
10.14232/ejqtde.2018.1.92
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发表时间：
2018
影响因子：
1.1
通讯作者：
J. Jaros;T. Kusano;T. Tanigawa
中科院分区：
文献类型：
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作者：
J. Jaros;T. Kusano;T. Tanigawa
本文试图描绘一阶半线性微分系统非振荡解的整体结构 x′ − p(t)φ1/α(y) = 0, y′ + q(t)φα(x) = 0, (A) 其中 α > 0 是常数，p(t) 和 q(t) 是 [0, ∞) 上的正连续函数，并且 φγ(u) = |u|γsgn u, u ∈ R, γ > 0。为此目的，提出了对 (A) 解的存在性和渐近行为的系统分析。应该特别提及的是，(A) 的所有可能类型的非振荡解都可以通过求解与 (A) 相关的 Riccati 型微分方程来构造。值得注意的是，（A）的所有结果都可以应用于二阶半线性微分方程（p（t）φα（x））+ q（t）φα（x）= 0，（E），自动构建（E）的非振荡理论。
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 x′ − p(t)φ1/α(y) = 0, y′ + q(t)φα(x) = 0, (A) where α > 0 is a constant, p(t) and q(t) are positive continuous functions on [0, ∞), and φγ(u) = |u|γsgn u, u ∈ R, γ > 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 (p(t)φα(x)) + q(t)φα(x) = 0, (E) to build automatically a nonoscillation theory for (E).