Existence of solutions to higher-order discrete three-point problems

Abstract

We are concerned with the higher-order discrete three-point boundary-value problem (∆n x)(t) = ƒ(t, x(t + θ)), t1 ≤ t ≤ t3 - 1, -τ ≤ θ ≤ 1 (∆ix)(t1) = 0, 0 ≤ i ≤ n - 4, n ≥ 4 α(∆n-3x)(t) - β(∆n-2x)(t) = η(t), t1 - τ - 1 ≤ t ≤ t1 (∆n-2x)(t2) = (∆n-1x)(t3) = 0. By placing certain restrictions on the nonlinearity and the distance between boundary points, we prove the existence of at least one solution of the boundary value problem by applying the Krasnoselskii fixed point theorem.