Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 114, pp. 1-10.

Title: Existence of positive solutions for semipositone
       dynamic system on time scales
  
Authors: Na-Na Shao    (Lanzhou Univ., Gansu, China)
         You-Wei Zhang (Hexi Univ., Gansu, China)
 
Abstract: 
 In this paper, we study the following  semipositone
 dynamic system on time scales
 $$\displaylines{
 -x^{\Delta\Delta}(t)=f(t,y)+p(t),  \quad  t\in(0,T)_{\mathbb{T}},\cr
 -y^{\Delta\Delta}(t)=g(t,x),  \quad  t\in(0,T)_{\mathbb{T}},\cr
 x(0)=x(\sigma^{2}(T))=0, \cr
 \alpha{y(0)}-\beta{y^{\Delta}{(0)}}=
 \gamma{y(\sigma(T))}+\delta{y^{\Delta}(\sigma(T))}=0.
 }$$
 Using fixed point index theory, we show the existence of at
 least one positive solution. The interesting point is the
 that nonlinear term is allowed to change sign and may tend to
 negative infinity.

Submitted April 8, 2008. Published August 20, 2008.
Math Subject Classifications: 34B15, 39A10
Key Words: Positive solution; semipositone dynamic system;
           cone; fixed point index; time scales.