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....

This paper is devoted to value distribution and uniqueness problems for difference polynomials of entire functions such as fⁿ(f-1)f(z+c). We also consider sharing value problems for f(z) and its shifts f(z+c), and improve some recent results of Heittokangas et al. [J. Math. Anal. Appl. 355 (2009), 352-363]. Finally, we obtain some results on the existence of entire solutions of a difference equation of the form $fⁿ+P\left(z\right){\left({\Delta }_{c}f\right)}^m=Q\left(z\right).$

In this paper, we investigate the value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations, and obtain the results on the relations between the order of the solutions and the convergence exponents of the zeros, poles, a-points and small function value points of the solutions, which show the relations in the case of non-homogeneous equations are sharper than the ones in the case of homogeneous equations.

A new approach to differentiation on a time scale is presented. We give a suitable generalization of the Vitali Lemma and apply it to prove that every increasing function f: → ℝ has a right derivative f₊’(x) for ${\mu }_{\Delta }$-almost all x ∈ . Moreover, ${\int }_{[a,b)}f{₊}^{\text{'}}\left(x\right)d{\mu }_{\Delta }\le f\left(b\right)-f\left(a\right)$.