Criteria for Existence and Nonexistence of Positive Solutions to a Discrete Periodic Boundary Value Problem

摘要

We are concerned with proving the existence of positive solutions of a periodic boundary value problem for a discrete nonlinear equation -Δ[[(n-1)Δu(n-1)]+q(n)u(n)=λf(n,u)(n)). We shall also obtain criteria which leads to nonexistence of positive solutions. Index theory for the positive smooth function f and lower and upper solutions when the function f is allowed to have nonconstant sign and to be singular at 0 are employed.