带阻尼项的二阶奇异耦合系统的正周期解

Positive Periodic Solutions of Second-Order Singular Coupled Systems with Damping Terms

中文关键词:  正周期解  奇异耦合系统  Schauder不动点定理  弱奇异性

英文关键词:positive periodic solutions  singular coupled systems  Schauder's fixed point theorem  weak singularities

中文摘要:

      运用Schauder不动点定理和反极大值原理, 为二阶奇异耦合系统$$\left\{ \aligned x''+p_1(t)x'+q_1(t)x=f_1(t,y(t))+c_1(t),\\ y''+p_2(t)y'+q_2(t)y=f_2(t,x(t))+c_2(t)\\ \endaligned\right.$$ 建立了正周期解的存在性定理, 其中$p_i,\ q_i,\ c_i\in C(\mathbb{R}/T\mathbb{Z};\mathbb{R})$, $i=1,2$; $f_1,\ f_2\in C(\mathbb{R}/T\mathbb{Z}\times(0,\infty),\mathbb{R})$且在$0$处有奇性. 本文的主要结果推广和发展了已有文献的相应结论.

英文摘要:

      We establish the existence of positive periodic solutions of the second-order singular coupled systems $$\left\{ \aligned x''+p_1(t)x'+q_1(t)x=f_1(t,y(t))+c_1(t),\\ y''+p_2(t)y'+q_2(t)y=f_2(t,x(t))+c_2(t),\\ \endaligned\right.$$ where $p_i,\ q_i,\ c_i\in C(\mathbb{R}/T\mathbb{Z};\mathbb{R}),\ i=1,2$;\ \ $f_1,\ f_2\in C(\mathbb{R}/T\mathbb{Z}\times(0,\infty),\mathbb{R})$ and may be singular near the zero. The proof relies on Schauder's fixed point theorem and anti-maximum principle. Our main results generalize and improve those available in the literature.