非线性边界条件下二阶奇异差分方程的正解

Positive Solutions for Second-Order Singular Difference Equation with Nonlinear Boundary Conditions

中文关键词:  差分方程  非线性边界条件  正解

英文关键词:difference equation  nonlinear boundary conditions  positive solutions

中文摘要:

      本文我们考虑如下二阶奇异差分边值问题\begin{equation*}\begin{cases}-\Delta^{2} u(t-1)=\lambda g(t)f(u) ,\ t\in [1,T]_\mathbb{Z},\\u(0)=0,\\ \Delta u(T)+c(u(T+1))u(T+1)=0,\end{cases}\end{equation*}正解的存在性. 其中, $\lambda>0$, $f:(0,\infty)\rightarrow \mathbb{R}$ 是连续的,且允许在~$0$ 处奇异.通过引入一个新的全连续算子, 我们建立正解的存在性.

英文摘要:

      In this paper, we discuss the existence of positive solutions for the second-order singular difference equation boundary value problem $$\left\{\begin{array}{ll}-\Delta^{2} u(t-1)=\lambda g(t)f(u), &t\in [1,T]_\mathbb{Z},\\u(0)=0,\\ \Delta u(T)+c(u(T+1))u(T+1)=0,\end{array}\right.$$ where $\lambda>0$ is a positive parameter, $f:(0,\infty)\rightarrow \mathbb{R}$ is continuous, and is allowed to be singular at $0$. The existence of positive solutions is established via introducing a new complete continuous operator.