Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain

We are concerned with proving existence of one or more than one positive solution of a general two point boundary value problem for the nonlinear equation Lx(t)≔−[p(t)xΔ(t)]Δ+q(t)xσ(t)=λa(t)f(t,xσ(t)). We shall also obtain criteria which leads to nonexistence of positive solutions. Here the independent variable t is in a “measure chain”. We will use fixed point theorems for operators on a Banach space.