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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 Name | Affiliation | | 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. |
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