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Positive solutions for second order impulsive differential equations with dependence on first order derivative. (English) Zbl 1128.34015
Summary: We study positive solutions for the second order three-point boundary value problem: $$\cases x''(t)+f(t,x(t),x'(t))=0,\quad & t\ne t_i\\ \Delta x(t_i)=I_i(x(t_i), x'(t_i)),& i = 1,2,\dots,k\\ \Delta x'(t_i)=J_i(x(t_i),\ x'(t)),\\ x(0)=0=x(1) - \alpha x(\eta),\endcases$$ where $0 < \eta < 1$, $0 <\alpha< 1$, and $f: [0,1]\times [0,\infty)\times \Bbb R\to [0,\infty)$, $I_i: [0,\infty)\times \Bbb R\to \Bbb R$, $J_i:[0,\infty)\times \Bbb R\to \Bbb R$, $(i = 1, 2,\dots,k)$ are continuous. Based on a new extension of Krasnoselskii fixed-point theorem (which was established by {\it Y. Guo} and {\it W. Ge} [J. Math. Anal. Appl. 290, No. 1, 291--301 (2004; Zbl 1054.34023)], 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.
34B37Boundary value problems for ODE with impulses
34B18Positive solutions of nonlinear boundary value problems for ODE