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

摘要

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)x~Δ(t)]~Δ + q(t)x~σ(t) = λa(t)f(t,x~σ(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.