We discuss the discrete $p$-Laplacian eigenvalue problem, $$\left\{\begin{array}{c}\Delta \left({\phi }_p\left(\Delta u(k-1)\right)\right)+\lambda a\left(k\right)g\left(u\left(k\right)\right)=0,k\in \{1,2,...,T\},\hfill \\ u\left(0\right)=u(T+1)=0,\hfill \end{array}\right.$$ where $T>1$ is a given positive integer and ${\phi }_p\left(x\right):={\left|x\right|}^{p-2}x$, $p>1$. First, the existence of an unbounded continuum $𝒞$ of positive solutions emanating from $(\lambda ,u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $𝒞$ is a monotone continuous curve globally defined for all $\lambda >0$.

By means of Riccati transformation techniques, we establish some new oscillation criteria for second-order nonlinear difference equation which are sharp.

This paper is concerned with the nonlinear advanced difference equation with constant coefficients $$x_{n+1}-x_n+\sum _{i=1}^m{p}_i{f}_i\left(x_{n-k_i}\right)=0,n=0,1,\cdots$$ where $p_i\in (-\infty ,0)$ and $k_i\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$. We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients $$x_{n+1}-x_n+\sum _{i=1}^m{p}_{in}{f}_i\left(x_{n-k_i}\right)=0,n=0,1,\cdots$$ where $p_{in}\le 0$ and $k_i\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$.