## Positive solutions for three-point boundary value problems with dependence on the first order derivative.(English)Zbl 1054.34025

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.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47H11 Degree theory for nonlinear operators 47N20 Applications of operator theory to differential and integral equations
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