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

Yu, Changlong ¹ ; Wang, Jufang ¹ ; Guo, Yanping ¹ ; Miao, Surong ¹

¹ College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P. R. China

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 11, p. 5734-5743.

Résumé

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.

Détail

DOI : 10.22436/jnsa.010.11.11

Classification : 34B15, 34B16, 34B18, 34G20
Keywords: Difference equation, boundary value problem, Leray-Schauder continuation theorem, infinite interval

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