依赖于一阶导数的二阶脉冲微分方程边值问题的正解(英)

本文研究一类二阶脉冲微分方程：■的正解存在性．其中,0＜η＜1,0＜α＜1,f：0,1]×0,∞)×R→0,∞),I_i：0,∞)×R→R,J_i：0,∞)×R→R,(i=1,2,…,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．

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

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 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 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.