## Positive solutions of a three-point boundary-value problem on a time scale.(English)Zbl 1047.34015

Let T be a time scale such that $$0,T\in {\mathbf T}$$. The author utilizes a theoretic fixed-point theorem in a cone to show the existence of positive solutions of the second-order boundary value problem $u^{\nabla\nabla}(t)+a(t)f(u(t))=0, \quad t\in (0,T)\cap {\mathbf T},$
$u(0)=0, \quad \alpha u(\eta)=y(T),$ where $$\eta\in (0,\rho(T))\cap {\mathbf T}$$, and $$0<\alpha<T/\eta.$$

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 39A12 Discrete version of topics in analysis

### Keywords:

Time scale; boundary value problem; positive solutions.
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