Positive solutions for three-point boundary value problems with dependence on the first order derivative. (Q1426997)

The existence of positive solutions for the second-order three-point boundary value problem \(x''+f(t,x,x')=0\), \(0<t<1\), \(x(0)=0\), \(x(1)=\alpha x(\eta)\), is considered. It is supposed that \(\alpha>0\), \(0<\eta<1\), \(1-\alpha\eta>0\), and \(f:[0,1]\times [0,\infty)\times \mathbb R\to[0,\infty]\) is continuous. Under some growth conditions, the authors prove that the problem has at least one positive solution. As a tool for the proof a new fixed-point theorem in a cone is used.