The existence and nonexistence of positive solutions to a discrete fractional boundary value problem with a parameter

In this paper, we study the existence and nonexistence of positive solutions for the boundary value problem with a parameter { − Δ ν y ( t ) = λ f ( t + ν − 1 , y ( t + ν − 1 ) ) , y ( ν − 2 ) = y ( ν + b + 1 ) = 0 , where t ∈ [ 0 , b + 1 ] N , 1 < ν ≤ 2 is a real number, f : [ ν − 1 , ν + b ] N ν − 1 × R → ( 0 , + ∞ ) is a continuous function, b ≥ 2 is an integer, λ is a parameter. The eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered by the properties of the Green function and Guo–Krasnosel’skii fixed point theorem on cones, some sufficient conditions of the nonexistence of positive solutions for the boundary value problem are established. As applications, we give some examples to illustrate the main results.