依赖于一阶导数的二阶脉冲微分方程边值问题的正解(英)

Positive Solutions for Second Order Impulsive Differential Equations with Dependence on First Order Derivative

中文关键词:  脉冲微分方程  不动点定理  Green函数.

英文关键词:impulsive differential equation  fixed point theorem  Green function.

中文摘要:

      本文研究一类二阶脉冲微分方程:$$\left\{\begin{array}{ll}x''(t)+f(t, x(t),x'(t))=0,&t\neq t_i\\ \triangle x(t_i)=I_i(x(t_i),x'(t_i)),&i=1, 2, \cdots, k\\ \triangle x'(t_i)=J_i(x(t_i),x'(t))\\ x(0)=0=x(1)-\alpha x(\eta)\end{array}\right.$$ 的正解存在性.其中, $0<\eta<1, 0<\alpha<1$, $f:[0,1]\times [0,\infty)\times R \rightarrow [0,\infty)$, $I_i:[0,\infty)\times R\rightarrow R, J_i:[0,\infty)\times R\rightarrow R, (i=1, 2, \cdots, k)$均为连续函数. 本文所用方法是文献[5]推广的 Krasnoselskii 不动点定理,此定理为解决依赖于一阶导数的边值问题提供了理论依据.基于此定理,获得了问题正解存在性定理. 特别地,我们获得此类问题的Green函数,使问题的解决更直观和简单.

英文摘要:

      We study positive solutions for second order three-point boundary value problem: $$\left\{\begin{array}{ll} x''(t)+f(t, x(t),x'(t))=0,&t\neq t_i \\ \triangle x(t_i)=I_i(x(t_i),x'(t_i)),&i=1, 2, \cdots, k \\ \triangle x'(t_i)=J_i(x(t_i),x'(t)), \\ x(0)=0=x(1)-\alpha x(\eta),\end{array}\right. $$ where $0<\eta<1, 0<\alpha<1$, and $f:[0,1]\times [0,\infty)\times R \rightarrow [0,\infty)$, $I_i:[0,\infty)\times R\rightarrow R, J_i:[0,\infty)\times R\rightarrow R, (i=1, 2, \cdots, k)$ are continuous. Based on a new extension of Krasnoselskii fixed-point theorem (which was established by Guo Yan-ping and GE Wei-gao$^{[5]}$), the existence of positive solutions for the boundary value problems is obtained. In particular, we obtain the Green function of the problem, which makes the problem simpler.