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

Abstract

We are concerned with the higher-order discrete three-point boundary-value problem $$displaylines{ (Delta^n x)(t)=f(t,x(t+heta)), quad t_1le tle t_3-1, quad -aule hetale 1cr (Delta^i x)(t_1)=0, quad 0le ile n-4, quad nge 4 cr alpha (Delta^{n-3}x)(t)-Beta (Delta^{n-2}x)(t)=Beta(t), quad t_1-au-1le tle t_1 cr (Delta^{n-2}x)(t_2)=(Delta^{n-1}x)(t_3)=0. }$$ 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.

Keywords

• Difference equations
• boundary-value problem
• Green's function
• fixed points
• cone.