Discrete third-order three-point boundary value problem

We are concerned with the discrete boundary value problem Lx ( t ) ≔ Δ 3 x ( t ) = f ( t , x ( t + 1 ) ) , t 1 ⩽ t ⩽ t 3 - 1 , x ( t 1 ) = 0 , α x ( t 2 ) - β Δ x ( t 2 ) = 0 , γ x ( t 3 ) + δ Δ 2 x ( t 3 ) = 0 , and the eigenvalue problem Δ 3 x ( t ) = λ f ( t , x ( t + 1 ) ) with the same boundary conditions where t 1 < t 2 < t 3 are distinct integers. Under various assumptions on f and λ , we prove the existence of positive solutions of both problems by applying a fixed point theorem.