Sufficient conditions for the absence of absolutely continuous spectrum for unbounded Jacobi operators are given. A class of unbounded Jacobi operators with purely singular continuous spectrum is constructed as well.

We investigate the boundedness nature of positive solutions of the difference equation $$x_{n+1}=max\left\{\frac{A_n}{X_n},\frac{B_n}{X_{n-2}}\right\},n=0,1,...,$$ where A nn=0∞ and B nn=0∞ are periodic sequences of positive real numbers.

We study k th order systems of two rational difference equations $$x_n=\frac{\alpha +{\sum }_{i=1}^k{\beta }_i{x}_{n-i}+{\sum }_{i=1}^k{\gamma }_i{y}_{n-i}}{A+{\sum }_{j=1}^k{B}_j{x}_{n-j}+{\sum }_{j=1}^k{C}_j{y}_{n-j}},n\in \mathbb{N},$$ In particular we assume non-negative parameters and non-negative initial conditions. We develop several approaches which allow us to prove that unbounded solutions exist for certain initial conditions in a range of the parameters.

In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let $f\left(z\right)$ and $g\left(z\right)$ be two transcendental entire functions of finite order, and $\alpha \left(z\right)$ a small function with respect to both $f\left(z\right)$ and $g\left(z\right)$. Suppose that $c$ is a non-zero complex constant and $n\ge 7$ (or $n\ge 10$) is an integer. If $f^n\left(z\right)(f\left(z\right)-1)f(z+c)$ and $g^n\left(z\right)(g\left(z\right)-1)g(z+c)$ share “$\left(\alpha \right(z),2)$” (or ${(\alpha \left(z\right),2)}^{*}$), then $f\left(z\right)\equiv g\left(z\right)$. Our results extend and generalize some well known previous results....