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On Two-point Boundary Value Problems for Second-order Difference Equation
  
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KeyWord:Second-order difference equation; Different boundary conditions; Boundary value problems
Author NameAffiliation
Huijuan Li Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China 
Gaofeng Du Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China 
Cunyan Yue Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China 
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Abstract:
      In this paper, we aim to investigate the difference equation \begin{align*} \Delta^{2}y(t-1)+|y(t)|=0, \ \ \ \ \ t\in[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\ $T\geq 1$ is an integer and $B\in\mathbb{R}$. 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 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.