A two-sided sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m}+c_{n-m}=2cₙcₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ is bounded if $su{p}_{n\in \mathbb{Z}}\left|\right|cₙ\left|\right|<\infty$. A (bounded) group decomposition for a cosine sequence $c={\left(cₙ\right)}_{n\in \mathbb{Z}}$ is a representation of c as $cₙ=(bⁿ+b^{-n})/2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $su{p}_{n\in \mathbb{Z}}\left|\right|bⁿ\left|\right|<\infty$, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, the so-called...

A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation $$\Delta \left(a_n{\left|\Delta x_n\right|}^{\alpha }sgn\Delta x_n\right)+b_n{\left|x_{n+1}\right|}^{\beta }sgn{x}_{n+1}=0$$ is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.

We have established sufficient conditions for oscillation of a class of first order neutral impulsive difference equations with deviating arguments and fixed moments of impulsive effect.

In this work, oscillatory behaviour of solutions of a class of fourth-order neutral functional difference equations of the form $${\Delta }^2\left(r\left(n\right){\Delta }^2(y\left(n\right)+p\left(n\right)y(n-m))\right)+q\left(n\right)G\left(y(n-k)\right)=0$$ is studied under the assumption $$\sum _{n=0}^{\infty }\frac{n}{r\left(n\right)}<\infty .$$ New oscillation criteria have been established which generalize some of the existing results in the literature.

We collect and generalize various known definitions of principal iteration semigroups in the case of multiplier zero and establish connections among them. The common characteristic property of each definition is conjugating of an iteration semigroup to different normal forms. The conjugating functions are expressed by suitable formulas and satisfy either Böttcher’s or Schröder’s functional equation.

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, ${\phi |}_{(-\infty ,0]}=0$, ${\phi |}_{[1,\infty )}=1$, = φ: ℝ → ℝ|φ is continuous, ${\phi |}_{(-\infty ,0]}=0$, ${\phi |}_{[1,\infty )}=1$. We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection...