On the number of positive solutions of systems of nonlinear dynamic equations on time scales

In this paper we consider the following n-dimensional second-order nonlinear system on time scalesuΔΔ(t)+λa(t)f(uσ(t))=0,t∈[a,b]Twith the Sturm-Liouville boundary conditionsαu(a)-βuΔ(a)=0,γu(σ(b))+δuΔ(σ(b))=0,where u=(u1,…,un), α=diag[α1,…,αn], β=diag[β1,…,βn], γ=diag[γ1,…,γn], δ=diag[δ1,…,δn]. Let f0=∑i=1nlim∥u∥→0fi(u)/∥u∥ and f∞=∑i=1nlim∥u∥→∞fi(u)/∥u∥. Define i0= number of zeros in the set {f0,f∞} and i∞= number of infinities in the set {f0,f∞}. By using fixed point index theory, we show that: