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 $0<\eta<1, 0<\alpha<1$, 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.