Positive Solutions of Second Order Discrete Problem on Infinite Intervals

Abstract:

      In this paper, by using the discrete Arzel\'{a}-Ascoli Lemma and the fixed-point theorem in cones,~we discuss the existence of positive solutions of the following second order discrete Sturm-Liouville boundary value problem on infinite intervals $$\left\{\begin{array}{l}-\Delta^{2}u(x-1)=f(x, u(x), \Delta u(x-1)),~~x\in\mathbb{N},\\ u(0)-a\Delta u(0)=B,~~\Delta u(\infty)=C,\end{array}\right.$$ where $\Delta u(x)=u(x+1)-u(x)$ is the forward difference operator, $\mathbb{N}=\{1, 2, \ldots, \infty\}$, $f:\mathbb{N}\times\mathbb{R_{+}}\times\mathbb{R_{+}}\rightarrow\mathbb{R_{+}}$ is continuous, $a>0$, $B$ and $C$ are nonnegative constants, $\mathbb{R_{+}}=[0, +\infty),\ \Delta u(\infty)=\lim_{x\rightarrow\infty}\Delta u(x)$.