This paper deals with basic stability properties of a two-term linear autonomous fractional difference system involving the Riemann-Liouville difference. In particular, we focus on the case when eigenvalues of the system matrix lie on a boundary curve separating asymptotic stability and unstability regions. This issue was posed as an open problem in the paper J. Čermák, T. Kisela, and L. Nechvátal (2013). Thus, the paper completes the stability analysis of the corresponding fractional difference...

We give a review of results proved and published mostly in recent years, concerning real-valued convex functions as well as almost convex functions defined on a (not necessarily convex) subset of a group. Analogues of such classical results as the theorems of Jensen, Bernstein-Doetsch, Blumberg-Sierpiński, Ostrowski, and Mehdi are presented. A version of the Hahn-Banach theorem with a convex control function is proved, too. We also study some questions specific for the group setting, for instance...

We investigate the criticality of the one term $2n$-order difference operators $l{\left(y\right)}_k={\Delta }^n\left(r_k{\Delta }^n{y}_k\right)$. We explicitly determine the recessive and the dominant system of solutions of the equation $l{\left(y\right)}_k=0$. Using their structure we prove a criticality criterion.

In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix...

In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q{y}_n=0,n\ge 1,$$ where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q_{n}g\left(y_n\right)=f_{n-1},n\ge 1,$$ where, unlike earlier works, $f_n\ge 0$ or $\le 0$ (but $\neg \equiv 0)$ for large $n$. Further, these results are used to obtain...

The paper discusses basics of calculus of backward fractional differences and sums. We state their definitions, basic properties and consider a special two-term linear fractional difference equation. We construct a family of functions to obtain its solution.

Consider the first order linear difference equation with general advanced argument and variable coefficients of the form $$\nabla x\left(n\right)-p\left(n\right)x\left(\tau \right(n\left)\right)=0,n⩾1,$$ where p(n) is a sequence of nonnegative real numbers, τ(n) is a sequence of positive integers such that $$\tau \left(n\right)⩾n+1,n⩾1,$$ and ▿ denotes the backward difference operator ▿x(n) = x(n) − x(n − 1). Sufficient conditions which guarantee that all solutions oscillate are established. Examples illustrating the results are given.

In this paper, sufficient conditions are obtained for oscillation of all solutions of third order difference equations of the form $$y_{n+3}+r_n{y}_{n+2}+q_n{y}_{n+1}+p_n{y}_n=0,n\ge 0.$$ These results are generalization of the results concerning difference equations with constant coefficients $$y_{n+3}+r{y}_{n+2}+q{y}_{n+1}+p{y}_n=0,n\ge 0.$$ Oscillation, nonoscillation and disconjugacy of a certain class of linear third order difference equations are discussed with help of a class of linear second order difference equations.

We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem–Mahler–Lech theorem to rational function coefficients. The second problem is whether or not for a given linear difference equation there exists a Picard–Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell–Lang conjecture. This allows us to deduce...

Oscillatory properties of solutions to the system of first-order linear difference equations $$\begin{array}{cc}\hfill \Delta u_k& =q_k{v}_k\hfill \\ \hfill \Delta v_k& =-p_k{u}_{k+1},\hfill \end{array}$$ are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).

In this paper we use the fixed point method to prove asymptotic stability results of the zero solution of a generalized linear neutral difference equation with variable delays. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Y. N. Raffoul (2006), E. Yankson (2009), M. Islam and E. Yankson (2005).