On Two-point Boundary Value Problems for Second-order Difference Equation

In this paper, we aim to investigate the difference equationbegin{align*}Delta^{2}y(t-1)+|y(t)|=0, tin[1,T]_{mathbb{Z}}end{align*}with different boundary conditions $y(0)=0$ or $Delta y(0)=0$ and $y(T+1)=B$ or $Delta y(T)=B$, where $Tgeq 1$ is an integer and $Binmathbb{R}$. 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 the 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.