On Two-Point Boundary Value Problems for Second-Order Difference Equation

Abstract

In this paper, we aim to investigate the difference equation $$∆^2 y(t − 1) + |y(t)| = 0, t ∈ [1, T]_{\mathbb{Z}}$$ with different boundary conditions $y(0) = 0$ or $∆y(0) = 0$ and $y(T + 1) = B$ or $∆y(T) = B,$ where $T ≥ 1$ is an integer and $B ∈\mathbb{R}.$ We will show that how the values of $T$ and $B$ influence the existence and uniqueness of the solutions to the about problem. In details, for the different problems, the $TB$-plane explicitly divided into different parts according to the number of solutions to the above problems. These parts of $TB$-plane for the value of $T$ and $B$ guarantee the uniqueness, the existence and the nonexistence of solutions respectively.