Existence of positive solutions for semipositone dynamic system on time scales

Abstract

In this paper, we study the following semipositone dynamic system on time scales $$displaylines{ -x^{DeltaDelta}(t)=f(t,y)+p(t), quad tin(0,T)_{mathbb{T}},cr -y^{DeltaDelta}(t)=g(t,x), quad tin(0,T)_{mathbb{T}},cr x(0)=x(sigma^{2}(T))=0, cr alpha{y(0)}-eta{y^{Delta}{(0)}}= gamma{y(sigma(T))}+delta{y^{Delta}(sigma(T))}=0. }$$ 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.

Keywords

• Positive solution
• semipositone dynamic system
• cone
• fixed point index
• time scales