Existence of positive solutions for a dynamic equation on measure chains

In this paper we consider the following dynamic equation on a measure chain T L x ( t ) = − x Δ Δ ( t ) + p ( t ) x Δ ( t ) = f ( t , x ( σ ( t ) ) ) , t ∈ [ a , b ] , with the boundary value conditions x ( a ) = 0 = x ( σ 2 ( b ) ) . Unlike many other dynamic equations on measure chains, it involves the term x Δ ( 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.