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

Abstract:

      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.