# Solvability of second-order $m$-point difference equation boundary value problems on infinite intervals

Volume 10, Issue 11, pp 5734--5743 Publication Date: November 15, 2017       Article History
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### Authors

Changlong Yu - College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P. R. China. Jufang Wang - College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P. R. China. Yanping Guo - College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P. R. China. Surong Miao - College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P. R. China.

### Abstract

In this paper, we study second-order $m$-point difference boundary value problems on infinite intervals $\left\{\begin{array}{l} \Delta^{2}x(k-1)+f(k,x(k),\Delta x(k-1))=0,~k\in N,\\ x(0)=\sum\limits_{i=1}^{m-2}\alpha_{i}x(\eta_{i}),~\lim\limits_{k \rightarrow\infty }\Delta x(k)=0, \end{array} \right.$ where $N=\{1,2,\cdots\},\ f:N\times R^{2}\rightarrow R$ is continuous, $\alpha_{i}\in R,~\sum\limits_{i=1}^{m-2}\alpha_{i}\neq1,~\eta_{i}\in N,~0<\eta_{1}<\eta_{2}<\cdots<\infty$ and $\Delta x(k)=x(k+1)-x(k),$ the nonlinear term is dependent in a difference of lower order on infinite intervals. By using Leray-Schauder continuation theorem, the existence of solutions are investigated. Finally, we give one example to demonstrate the use of the main result.

### Keywords

• Difference equation
• boundary value problem
• Leray-Schauder continuation theorem
• infinite interval

•  34B15
•  34B16
•  34B18
•  34G20

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