Existence of positive solutions for a dynamic equation on measure chains

In this paper we consider the following dynamic equation on a measure chain TLx(t)=−xΔΔ(t)+p(t)xΔ(t)=f(t,x(σ(t))),t∈[a,b], with the boundary value conditions x(a)=0=x(σ2(b)). Unlike many other dynamic equations on measure chains, it involves the term xΔ(t), so it is hard to get Green's function. We obtain Green's function of this equation. And the Leray-Schauder fixed point theorem will be used to prove the existence of at least one solution.