带有平均曲率算子的离散问题在无穷区间上解的存在性

Existence of Solutions for a Discrete Problems with the Mean Curvature Operator on Infinite Interval

摘要: 运用Avery-Peterson不动点定理讨论无穷区间上带有平均曲率算子的离散问$\{\begin{array}{l}\Delta \left(\phi \left(\Delta u\left(x-1\right)\right)\right)+q\left(x\right)f\left(x,u\left(x\right),\Delta u\left(x-1\right)\right)=0,x\in N,\\ \Delta u\left(0\right)=\lambda u\left(0\right),\Delta u\left(\infty \right)=0\end{array}$ 解的存在性。其中$\Delta u\left(x\right)=u\left(x+1\right)-u\left(x\right)$ 为前项差分算子，$\mathbb{N}=\left\{1,2,\cdots ,\infty \right\},f:\mathbb{N}\times {ℝ}_{+}\times \left(-1,1\right)\to {ℝ}_{+}$ ，$q:\mathbb{N}\to {ℝ}_{+}$ 连续，$\lambda >0$ 为常数，${ℝ}_{+}=\left[0,+\infty \right)$ ，$\Delta u\left(\infty \right)=\underset{x\to \infty }{lim}\Delta u\left(x\right)$ ，$\phi \left(s\right)=\frac{s}{\sqrt{1-s^2}},s\in \left(-1,1\right)$ 。

Abstract: In this paper, by using the Avery-Peterson theory, we are concerned with the existence of the following problems: $\{\begin{array}{l}\Delta \left(\phi \left(\Delta u\left(x-1\right)\right)\right)+q\left(x\right)f\left(x,u\left(x\right),\Delta u\left(x-1\right)\right)=0,x\in N,\\ \Delta u\left(0\right)=\lambda u\left(0\right),\Delta u\left(\infty \right)=0\end{array}$ where $\Delta u\left(x\right)=u\left(x+1\right)-u\left(x\right)$ is the forward difference operator, $\mathbb{N}=\left\{1,2,\cdots ,\infty \right\},f:\mathbb{N}\times {ℝ}_{+}\times \left(-1,1\right)\to {ℝ}_{+}$ , $q:\mathbb{N}\to {ℝ}_{+}$ are continuous, $\lambda >0$ is a constant, ${ℝ}_{+}=\left[0,+\infty \right)$ , $\Delta u\left(\infty \right)=\underset{x\to \infty }{lim}\Delta u\left(x\right)$ , $\phi \left(s\right)=\frac{s}{\sqrt{1-s^2}},s\in \left(-1,1\right)$ .